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doubleSYcomputational(3) LAPACK doubleSYcomputational(3)

NAME

doubleSYcomputational - double

SYNOPSIS

Functions


subroutine dla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. double precision function dla_syrcond (UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
DLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix. subroutine dla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. double precision function dla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. subroutine dlasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method. subroutine dlasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)
DLASYF_AA subroutine dlasyf_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. subroutine dlasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. subroutine dsycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSYCON subroutine dsycon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSYCON_3 subroutine dsycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
DSYCON_ROOK subroutine dsyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO)
DSYCONV subroutine dsyconvf (UPLO, WAY, N, A, LDA, E, IPIV, INFO)
DSYCONVF subroutine dsyconvf_rook (UPLO, WAY, N, A, LDA, E, IPIV, INFO)
DSYCONVF_ROOK subroutine dsyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
DSYEQUB subroutine dsygs2 (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm). subroutine dsygst (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
DSYGST subroutine dsyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSYRFS subroutine dsyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
DSYRFSX subroutine dsytd2 (UPLO, N, A, LDA, D, E, TAU, INFO)
DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). subroutine dsytf2 (UPLO, N, A, LDA, IPIV, INFO)
DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). subroutine dsytf2_rk (UPLO, N, A, LDA, E, IPIV, INFO)
DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). subroutine dsytf2_rook (UPLO, N, A, LDA, IPIV, INFO)
DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). subroutine dsytrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD subroutine dsytrd_2stage (VECT, UPLO, N, A, LDA, D, E, TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)
DSYTRD_2STAGE subroutine dsytrd_sy2sb (UPLO, N, KD, A, LDA, AB, LDAB, TAU, WORK, LWORK, INFO)
DSYTRD_SY2SB subroutine dsytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
DSYTRF subroutine dsytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
DSYTRF_AA subroutine dsytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO)
DSYTRF_AA_2STAGE subroutine dsytrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). subroutine dsytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
DSYTRF_ROOK subroutine dsytri (UPLO, N, A, LDA, IPIV, WORK, INFO)
DSYTRI subroutine dsytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
DSYTRI2 subroutine dsytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO)
DSYTRI2X subroutine dsytri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO)
DSYTRI_3 subroutine dsytri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO)
DSYTRI_3X subroutine dsytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO)
DSYTRI_ROOK subroutine dsytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DSYTRS subroutine dsytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO)
DSYTRS2 subroutine dsytrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO)
DSYTRS_3 subroutine dsytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)
DSYTRS_AA subroutine dsytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO)
DSYTRS_AA_2STAGE subroutine dsytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DSYTRS_ROOK subroutine dtgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
DTGSYL subroutine dtrsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
DTRSYL

Detailed Description

This is the group of double computational functions for SY matrices

Function Documentation

subroutine dla_syamv (integer UPLO, integer N, double precision ALPHA, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)

DLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.

Purpose:


DLA_SYAMV performs the matrix-vector operation
y := alpha*abs(A)*abs(x) + beta*abs(y),
where alpha and beta are scalars, x and y are vectors and A is an
n by n symmetric matrix.
This function is primarily used in calculating error bounds.
To protect against underflow during evaluation, components in
the resulting vector are perturbed away from zero by (N+1)
times the underflow threshold. To prevent unnecessarily large
errors for block-structure embedded in general matrices,
"symbolically" zero components are not perturbed. A zero
entry is considered "symbolic" if all multiplications involved
in computing that entry have at least one zero multiplicand.

Parameters

UPLO


UPLO is INTEGER
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
follows:
UPLO = BLAS_UPPER Only the upper triangular part of A
is to be referenced.
UPLO = BLAS_LOWER Only the lower triangular part of A
is to be referenced.
Unchanged on exit.

N


N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.

ALPHA


ALPHA is DOUBLE PRECISION .
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.

A


A is DOUBLE PRECISION array, dimension ( LDA, n ).
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients.
Unchanged on exit.

LDA


LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, n ).
Unchanged on exit.

X


X is DOUBLE PRECISION array, dimension
( 1 + ( n - 1 )*abs( INCX ) )
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.

INCX


INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.

BETA


BETA is DOUBLE PRECISION .
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.

Y


Y is DOUBLE PRECISION array, dimension
( 1 + ( n - 1 )*abs( INCY ) )
Before entry with BETA non-zero, the incremented array Y
must contain the vector y. On exit, Y is overwritten by the
updated vector y.

INCY


INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


Level 2 Blas routine.
-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
-- Modified for the absolute-value product, April 2006
Jason Riedy, UC Berkeley

double precision function dla_syrcond (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, double precision, dimension( * ) C, integer INFO, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK)

DLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.

Purpose:


DLA_SYRCOND estimates the Skeel condition number of op(A) * op2(C)
where op2 is determined by CMODE as follows
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)
The Skeel condition number cond(A) = norminf( |inv(A)||A| )
is computed by computing scaling factors R such that
diag(R)*A*op2(C) is row equilibrated and computing the standard
infinity-norm condition number.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF


AF is DOUBLE PRECISION array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDAF


LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

CMODE


CMODE is INTEGER
Determines op2(C) in the formula op(A) * op2(C) as follows:
CMODE = 1 op2(C) = C
CMODE = 0 op2(C) = I
CMODE = -1 op2(C) = inv(C)

C


C is DOUBLE PRECISION array, dimension (N)
The vector C in the formula op(A) * op2(C).

INFO


INFO is INTEGER
= 0: Successful exit.
i > 0: The ith argument is invalid.

WORK


WORK is DOUBLE PRECISION array, dimension (3*N).
Workspace.

IWORK


IWORK is INTEGER array, dimension (N).
Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, double precision, dimension( * ) RES, double precision, dimension( * ) AYB, double precision, dimension( * ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO)

DLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:


DLA_SYRFSX_EXTENDED improves the computed solution to a system of
linear equations by performing extra-precise iterative refinement
and provides error bounds and backward error estimates for the solution.
This subroutine is called by DSYRFSX to perform iterative refinement.
In addition to normwise error bound, the code provides maximum
componentwise error bound if possible. See comments for ERR_BNDS_NORM
and ERR_BNDS_COMP for details of the error bounds. Note that this
subroutine is only resonsible for setting the second fields of
ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters

PREC_TYPE


PREC_TYPE is INTEGER
Specifies the intermediate precision to be used in refinement.
The value is defined by ILAPREC(P) where P is a CHARACTER and P
= 'S': Single
= 'D': Double
= 'I': Indigenous
= 'X' or 'E': Extra

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right-hand-sides, i.e., the number of columns of the
matrix B.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF


AF is DOUBLE PRECISION array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDAF


LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

COLEQU


COLEQU is LOGICAL
If .TRUE. then column equilibration was done to A before calling
this routine. This is needed to compute the solution and error
bounds correctly.

C


C is DOUBLE PRECISION array, dimension (N)
The column scale factors for A. If COLEQU = .FALSE., C
is not accessed. If C is input, each element of C should be a power
of the radix to ensure a reliable solution and error estimates.
Scaling by powers of the radix does not cause rounding errors unless
the result underflows or overflows. Rounding errors during scaling
lead to refining with a matrix that is not equivalent to the
input matrix, producing error estimates that may not be
reliable.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right-hand-side matrix B.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

Y


Y is DOUBLE PRECISION array, dimension (LDY,NRHS)
On entry, the solution matrix X, as computed by DSYTRS.
On exit, the improved solution matrix Y.

LDY


LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

BERR_OUT


BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
On exit, BERR_OUT(j) contains the componentwise relative backward
error for right-hand-side j from the formula
max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. This is computed by DLA_LIN_BERR.

N_NORMS


N_NORMS is INTEGER
Determines which error bounds to return (see ERR_BNDS_NORM
and ERR_BNDS_COMP).
If N_NORMS >= 1 return normwise error bounds.
If N_NORMS >= 2 return componentwise error bounds.

ERR_BNDS_NORM


ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP


ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * slamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * slamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * slamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
This subroutine is only responsible for setting the second field
above.
See Lapack Working Note 165 for further details and extra
cautions.

RES


RES is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate residual.

AYB


AYB is DOUBLE PRECISION array, dimension (N)
Workspace. This can be the same workspace passed for Y_TAIL.

DY


DY is DOUBLE PRECISION array, dimension (N)
Workspace to hold the intermediate solution.

Y_TAIL


Y_TAIL is DOUBLE PRECISION array, dimension (N)
Workspace to hold the trailing bits of the intermediate solution.

RCOND


RCOND is DOUBLE PRECISION
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.

ITHRESH


ITHRESH is INTEGER
The maximum number of residual computations allowed for
refinement. The default is 10. For 'aggressive' set to 100 to
permit convergence using approximate factorizations or
factorizations other than LU. If the factorization uses a
technique other than Gaussian elimination, the guarantees in
ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

RTHRESH


RTHRESH is DOUBLE PRECISION
Determines when to stop refinement if the error estimate stops
decreasing. Refinement will stop when the next solution no longer
satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
default value is 0.5. For 'aggressive' set to 0.9 to permit
convergence on extremely ill-conditioned matrices. See LAWN 165
for more details.

DZ_UB


DZ_UB is DOUBLE PRECISION
Determines when to start considering componentwise convergence.
Componentwise convergence is only considered after each component
of the solution Y is stable, which we define as the relative
change in each component being less than DZ_UB. The default value
is 0.25, requiring the first bit to be stable. See LAWN 165 for
more details.

IGNORE_CWISE


IGNORE_CWISE is LOGICAL
If .TRUE. then ignore componentwise convergence. Default value
is .FALSE..

INFO


INFO is INTEGER
= 0: Successful exit.
< 0: if INFO = -i, the ith argument to DLA_SYRFSX_EXTENDED had an illegal
value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

double precision function dla_syrpvgrw (character*1 UPLO, integer N, integer INFO, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK)

DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.

Purpose:


DLA_SYRPVGRW computes the reciprocal pivot growth factor
norm(A)/norm(U). The "max absolute element" norm is used. If this is
much less than 1, the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that the
solution X, estimated condition numbers, and error bounds could be
unreliable.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.

INFO


INFO is INTEGER
The value of INFO returned from DSYTRF, .i.e., the pivot in
column INFO is exactly 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF


AF is DOUBLE PRECISION array, dimension (LDAF,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDAF


LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

WORK


WORK is DOUBLE PRECISION array, dimension (2*N)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlasyf (character UPLO, integer N, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer INFO)

DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.

Purpose:


DLASYF computes a partial factorization of a real symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The partial
factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )
A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

NB


NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB


KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.
If UPLO = 'L':
Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.

W


W is DOUBLE PRECISION array, dimension (LDW,NB)

LDW


LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dlasyf_aa (character UPLO, integer J1, integer M, integer NB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WORK)

DLASYF_AA

Purpose:


DLATRF_AA factorizes a panel of a real symmetric matrix A using
the Aasen's algorithm. The panel consists of a set of NB rows of A
when UPLO is U, or a set of NB columns when UPLO is L.
In order to factorize the panel, the Aasen's algorithm requires the
last row, or column, of the previous panel. The first row, or column,
of A is set to be the first row, or column, of an identity matrix,
which is used to factorize the first panel.
The resulting J-th row of U, or J-th column of L, is stored in the
(J-1)-th row, or column, of A (without the unit diagonals), while
the diagonal and subdiagonal of A are overwritten by those of T.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

J1


J1 is INTEGER
The location of the first row, or column, of the panel
within the submatrix of A, passed to this routine, e.g.,
when called by DSYTRF_AA, for the first panel, J1 is 1,
while for the remaining panels, J1 is 2.

M


M is INTEGER
The dimension of the submatrix. M >= 0.

NB


NB is INTEGER
The dimension of the panel to be facotorized.

A


A is DOUBLE PRECISION array, dimension (LDA,M) for
the first panel, while dimension (LDA,M+1) for the
remaining panels.
On entry, A contains the last row, or column, of
the previous panel, and the trailing submatrix of A
to be factorized, except for the first panel, only
the panel is passed.
On exit, the leading panel is factorized.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

IPIV


IPIV is INTEGER array, dimension (M)
Details of the row and column interchanges,
the row and column k were interchanged with the row and
column IPIV(k).

H


H is DOUBLE PRECISION workspace, dimension (LDH,NB).

LDH


LDH is INTEGER
The leading dimension of the workspace H. LDH >= max(1,M).

WORK


WORK is DOUBLE PRECISION workspace, dimension (M).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dlasyf_rk (character UPLO, integer N, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer INFO)

DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:


DLASYF_RK computes a partial factorization of a real symmetric
matrix A using the bounded Bunch-Kaufman (rook) diagonal
pivoting method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )
A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L',
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

NB


NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB


KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,N-KB+1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,N-KB+1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,N-KB+1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the submatrix A(1:N,1:KB).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the submatrix A(1:N,1:KB).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

W


W is DOUBLE PRECISION array, dimension (LDW,NB)

LDW


LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dlasyf_rook (character UPLO, integer N, integer NB, integer KB, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldw, * ) W, integer LDW, integer INFO)

DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.

Purpose:


DLASYF_ROOK computes a partial factorization of a real symmetric
matrix A using the bounded Bunch-Kaufman ("rook") diagonal
pivoting method. The partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12**T U22**T )
A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
DLASYF_ROOK is an auxiliary routine called by DSYTRF_ROOK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

NB


NB is INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.

KB


KB is INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
Only the last KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
Only the first KB elements of IPIV are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

W


W is DOUBLE PRECISION array, dimension (LDW,NB)

LDW


LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsycon (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYCON

Purpose:


DSYCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by DSYTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

ANORM


ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.

RCOND


RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK


WORK is DOUBLE PRECISION array, dimension (2*N)

IWORK


IWORK is INTEGER array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsycon_3 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYCON_3

Purpose:


DSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
computed by DSYTRF_RK or DSYTRF_BK:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver DSYTRS_3.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_RK or DSYTRF_BK.

ANORM


ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.

RCOND


RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK


WORK is DOUBLE PRECISION array, dimension (2*N)

IWORK


IWORK is INTEGER array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsycon_rook (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYCON_ROOK

Purpose:


DSYCON_ROOK estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
A = U*D*U**T or A = L*D*L**T computed by DSYTRF_ROOK.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF_ROOK.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_ROOK.

ANORM


ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.

RCOND


RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.

WORK


WORK is DOUBLE PRECISION array, dimension (2*N)

IWORK


IWORK is INTEGER array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


April 2012, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsyconv (character UPLO, character WAY, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) E, integer INFO)

DSYCONV

Purpose:


DSYCONV convert A given by TRF into L and D and vice-versa.
Get Non-diag elements of D (returned in workspace) and
apply or reverse permutation done in TRF.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

WAY


WAY is CHARACTER*1
= 'C': Convert
= 'R': Revert

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

E


E is DOUBLE PRECISION array, dimension (N)
E stores the supdiagonal/subdiagonal of the symmetric 1-by-1
or 2-by-2 block diagonal matrix D in LDLT.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsyconvf (character UPLO, character WAY, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)

DSYCONVF

Purpose:


If parameter WAY = 'C':
DSYCONVF converts the factorization output format used in
DSYTRF provided on entry in parameter A into the factorization
output format used in DSYTRF_RK (or DSYTRF_BK) that is stored
on exit in parameters A and E. It also converts in place details of
the intechanges stored in IPIV from the format used in DSYTRF into
the format used in DSYTRF_RK (or DSYTRF_BK).
If parameter WAY = 'R':
DSYCONVF performs the conversion in reverse direction, i.e.
converts the factorization output format used in DSYTRF_RK
(or DSYTRF_BK) provided on entry in parameters A and E into
the factorization output format used in DSYTRF that is stored
on exit in parameter A. It also converts in place details of
the intechanges stored in IPIV from the format used in DSYTRF_RK
(or DSYTRF_BK) into the format used in DSYTRF.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix A.
= 'U': Upper triangular
= 'L': Lower triangular

WAY


WAY is CHARACTER*1
= 'C': Convert
= 'R': Revert

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
1) If WAY ='C':
On entry, contains factorization details in format used in
DSYTRF:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = 'U': multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = 'L': multipliers used to obtain factor L
in the superdiagonal part of A.
On exit, contains factorization details in format used in
DSYTRF_RK or DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
2) If WAY = 'R':
On entry, contains factorization details in format used in
DSYTRF_RK or DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, contains factorization details in format used in
DSYTRF:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = 'U': multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = 'L': multipliers used to obtain factor L
in the superdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
1) If WAY ='C':
On entry, just a workspace.
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
2) If WAY = 'R':
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
On exit, is not changed

IPIV


IPIV is INTEGER array, dimension (N)
1) If WAY ='C':
On entry, details of the interchanges and the block
structure of D in the format used in DSYTRF.
On exit, details of the interchanges and the block
structure of D in the format used in DSYTRF_RK
( or DSYTRF_BK).
1) If WAY ='R':
On entry, details of the interchanges and the block
structure of D in the format used in DSYTRF_RK
( or DSYTRF_BK).
On exit, details of the interchanges and the block
structure of D in the format used in DSYTRF.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


November 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dsyconvf_rook (character UPLO, character WAY, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)

DSYCONVF_ROOK

Purpose:


If parameter WAY = 'C':
DSYCONVF_ROOK converts the factorization output format used in
DSYTRF_ROOK provided on entry in parameter A into the factorization
output format used in DSYTRF_RK (or DSYTRF_BK) that is stored
on exit in parameters A and E. IPIV format for DSYTRF_ROOK and
DSYTRF_RK (or DSYTRF_BK) is the same and is not converted.
If parameter WAY = 'R':
DSYCONVF_ROOK performs the conversion in reverse direction, i.e.
converts the factorization output format used in DSYTRF_RK
(or DSYTRF_BK) provided on entry in parameters A and E into
the factorization output format used in DSYTRF_ROOK that is stored
on exit in parameter A. IPIV format for DSYTRF_ROOK and
DSYTRF_RK (or DSYTRF_BK) is the same and is not converted.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix A.
= 'U': Upper triangular
= 'L': Lower triangular

WAY


WAY is CHARACTER*1
= 'C': Convert
= 'R': Revert

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
1) If WAY ='C':
On entry, contains factorization details in format used in
DSYTRF_ROOK:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = 'U': multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = 'L': multipliers used to obtain factor L
in the superdiagonal part of A.
On exit, contains factorization details in format used in
DSYTRF_RK or DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
2) If WAY = 'R':
On entry, contains factorization details in format used in
DSYTRF_RK or DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, contains factorization details in format used in
DSYTRF_ROOK:
a) all elements of the symmetric block diagonal
matrix D on the diagonal of A and on superdiagonal
(or subdiagonal) of A, and
b) If UPLO = 'U': multipliers used to obtain factor U
in the superdiagonal part of A.
If UPLO = 'L': multipliers used to obtain factor L
in the superdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
1) If WAY ='C':
On entry, just a workspace.
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
2) If WAY = 'R':
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
On exit, is not changed

IPIV


IPIV is INTEGER array, dimension (N)
On entry, details of the interchanges and the block
structure of D as determined:
1) by DSYTRF_ROOK, if WAY ='C';
2) by DSYTRF_RK (or DSYTRF_BK), if WAY ='R'.
The IPIV format is the same for all these routines.
On exit, is not changed.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


November 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dsyequb (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, double precision, dimension( * ) WORK, integer INFO)

DSYEQUB

Purpose:


DSYEQUB computes row and column scalings intended to equilibrate a
symmetric matrix A (with respect to the Euclidean norm) and reduce
its condition number. The scale factors S are computed by the BIN
algorithm (see references) so that the scaled matrix B with elements
B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
the smallest possible condition number over all possible diagonal
scalings.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The N-by-N symmetric matrix whose scaling factors are to be
computed.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

S


S is DOUBLE PRECISION array, dimension (N)
If INFO = 0, S contains the scale factors for A.

SCOND


SCOND is DOUBLE PRECISION
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.

AMAX


AMAX is DOUBLE PRECISION
Largest absolute value of any matrix element. If AMAX is
very close to overflow or very close to underflow, the
matrix should be scaled.

WORK


WORK is DOUBLE PRECISION array, dimension (2*N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

References:

Livne, O.E. and Golub, G.H., 'Scaling by Binormalization',
Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.
DOI 10.1023/B:NUMA.0000016606.32820.69
Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

subroutine dsygs2 (integer ITYPE, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).

Purpose:


DSYGS2 reduces a real symmetric-definite generalized eigenproblem
to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.
B must have been previously factorized as U**T *U or L*L**T by DPOTRF.

Parameters

ITYPE


ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T *A*L.

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored, and how B has been factorized.
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrices A and B. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

B


B is DOUBLE PRECISION array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by DPOTRF.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsygst (integer ITYPE, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYGST

Purpose:


DSYGST reduces a real symmetric-definite generalized eigenproblem
to standard form.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
B must have been previously factorized as U**T*U or L*L**T by DPOTRF.

Parameters

ITYPE


ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T*A*L.

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored and B is factored as
U**T*U;
= 'L': Lower triangle of A is stored and B is factored as
L*L**T.

N


N is INTEGER
The order of the matrices A and B. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

B


B is DOUBLE PRECISION array, dimension (LDB,N)
The triangular factor from the Cholesky factorization of B,
as returned by DPOTRF.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsyrfs (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYRFS

Purpose:


DSYRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, and
provides error bounds and backward error estimates for the solution.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF


AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or
A = L*D*L**T as computed by DSYTRF.

LDAF


LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X


X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DSYTRS.
On exit, the improved solution matrix X.

LDX


LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

FERR


FERR is DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.

BERR


BERR is DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).

WORK


WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK


IWORK is INTEGER array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Internal Parameters:


ITMAX is the maximum number of steps of iterative refinement.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsyrfsx (character UPLO, character EQUED, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DSYRFSX

Purpose:


DSYRFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, and
provides error bounds and backward error estimates for the
solution. In addition to normwise error bound, the code provides
maximum componentwise error bound if possible. See comments for
ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.
The original system of linear equations may have been equilibrated
before calling this routine, as described by arguments EQUED and S
below. In this case, the solution and error bounds returned are
for the original unequilibrated system.


Some optional parameters are bundled in the PARAMS array. These
settings determine how refinement is performed, but often the
defaults are acceptable. If the defaults are acceptable, users
can pass NPARAMS = 0 which prevents the source code from accessing
the PARAMS argument.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

EQUED


EQUED is CHARACTER*1
Specifies the form of equilibration that was done to A
before calling this routine. This is needed to compute
the solution and error bounds correctly.
= 'N': No equilibration
= 'Y': Both row and column equilibration, i.e., A has been
replaced by diag(S) * A * diag(S).
The right hand side B has been changed accordingly.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The symmetric matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular
part of the matrix A, and the strictly lower triangular
part of A is not referenced. If UPLO = 'L', the leading
N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AF


AF is DOUBLE PRECISION array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or A =
L*D*L**T as computed by DSYTRF.

LDAF


LDAF is INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

S


S is DOUBLE PRECISION array, dimension (N)
The scale factors for A. If EQUED = 'Y', A is multiplied on
the left and right by diag(S). S is an input argument if FACT =
'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
= 'Y', each element of S must be positive. If S is output, each
element of S is a power of the radix. If S is input, each element
of S should be a power of the radix to ensure a reliable solution
and error estimates. Scaling by powers of the radix does not cause
rounding errors unless the result underflows or overflows.
Rounding errors during scaling lead to refining with a matrix that
is not equivalent to the input matrix, producing error estimates
that may not be reliable.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
The right hand side matrix B.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

X


X is DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DGETRS.
On exit, the improved solution matrix X.

LDX


LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).

RCOND


RCOND is DOUBLE PRECISION
Reciprocal scaled condition number. This is an estimate of the
reciprocal Skeel condition number of the matrix A after
equilibration (if done). If this is less than the machine
precision (in particular, if it is zero), the matrix is singular
to working precision. Note that the error may still be small even
if this number is very small and the matrix appears ill-
conditioned.

BERR


BERR is DOUBLE PRECISION array, dimension (NRHS)
Componentwise relative backward error. This is the
componentwise relative backward error of each solution vector X(j)
(i.e., the smallest relative change in any element of A or B that
makes X(j) an exact solution).

N_ERR_BNDS


N_ERR_BNDS is INTEGER
Number of error bounds to return for each right hand side
and each type (normwise or componentwise). See ERR_BNDS_NORM and
ERR_BNDS_COMP below.

ERR_BNDS_NORM


ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
normwise relative error, which is defined as follows:
Normwise relative error in the ith solution vector:
max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
max_j abs(X(j,i))
The array is indexed by the type of error information as described
below. There currently are up to three pieces of information
returned.
The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_NORM(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated normwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*A, where S scales each row by a power of the
radix so all absolute row sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.

ERR_BNDS_COMP


ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
For each right-hand side, this array contains information about
various error bounds and condition numbers corresponding to the
componentwise relative error, which is defined as follows:
Componentwise relative error in the ith solution vector:
abs(XTRUE(j,i) - X(j,i))
max_j ----------------------
abs(X(j,i))
The array is indexed by the right-hand side i (on which the
componentwise relative error depends), and the type of error
information as described below. There currently are up to three
pieces of information returned for each right-hand side. If
componentwise accuracy is not requested (PARAMS(3) = 0.0), then
ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
the first (:,N_ERR_BNDS) entries are returned.
The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
right-hand side.
The second index in ERR_BNDS_COMP(:,err) contains the following
three fields:
err = 1 "Trust/don't trust" boolean. Trust the answer if the
reciprocal condition number is less than the threshold
sqrt(n) * dlamch('Epsilon').
err = 2 "Guaranteed" error bound: The estimated forward error,
almost certainly within a factor of 10 of the true error
so long as the next entry is greater than the threshold
sqrt(n) * dlamch('Epsilon'). This error bound should only
be trusted if the previous boolean is true.
err = 3 Reciprocal condition number: Estimated componentwise
reciprocal condition number. Compared with the threshold
sqrt(n) * dlamch('Epsilon') to determine if the error
estimate is "guaranteed". These reciprocal condition
numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
appropriately scaled matrix Z.
Let Z = S*(A*diag(x)), where x is the solution for the
current right-hand side and S scales each row of
A*diag(x) by a power of the radix so all absolute row
sums of Z are approximately 1.
See Lapack Working Note 165 for further details and extra
cautions.

NPARAMS


NPARAMS is INTEGER
Specifies the number of parameters set in PARAMS. If <= 0, the
PARAMS array is never referenced and default values are used.

PARAMS


PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
Specifies algorithm parameters. If an entry is < 0.0, then
that entry will be filled with default value used for that
parameter. Only positions up to NPARAMS are accessed; defaults
are used for higher-numbered parameters.
PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
refinement or not.
Default: 1.0D+0
= 0.0: No refinement is performed, and no error bounds are
computed.
= 1.0: Use the double-precision refinement algorithm,
possibly with doubled-single computations if the
compilation environment does not support DOUBLE
PRECISION.
(other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
computations allowed for refinement.
Default: 10
Aggressive: Set to 100 to permit convergence using approximate
factorizations or factorizations other than LU. If
the factorization uses a technique other than
Gaussian elimination, the guarantees in
err_bnds_norm and err_bnds_comp may no longer be
trustworthy.
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
will attempt to find a solution with small componentwise
relative error in the double-precision algorithm. Positive
is true, 0.0 is false.
Default: 1.0 (attempt componentwise convergence)

WORK


WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK


IWORK is INTEGER array, dimension (N)

INFO


INFO is INTEGER
= 0: Successful exit. The solution to every right-hand side is
guaranteed.
< 0: If INFO = -i, the i-th argument had an illegal value
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
has been completed, but the factor U is exactly singular, so
the solution and error bounds could not be computed. RCOND = 0
is returned.
= N+J: The solution corresponding to the Jth right-hand side is
not guaranteed. The solutions corresponding to other right-
hand sides K with K > J may not be guaranteed as well, but
only the first such right-hand side is reported. If a small
componentwise error is not requested (PARAMS(3) = 0.0) then
the Jth right-hand side is the first with a normwise error
bound that is not guaranteed (the smallest J such
that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
the Jth right-hand side is the first with either a normwise or
componentwise error bound that is not guaranteed (the smallest
J such that either ERR_BNDS_NORM(J,1) = 0.0 or
ERR_BNDS_COMP(J,1) = 0.0). See the definition of
ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
about all of the right-hand sides check ERR_BNDS_NORM or
ERR_BNDS_COMP.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytd2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAU, integer INFO)

DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Purpose:


DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
form T by an orthogonal similarity transformation: Q**T * A * Q = T.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

D


D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E


E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

TAU


TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

subroutine dsytf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

Purpose:


DSYTF2 computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method:
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**T is the transpose of U, and D is symmetric and
block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k-1) < 0, then rows and columns
k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block.
If UPLO = 'L':
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) = IPIV(k+1) < 0, then rows and columns
k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
is a 2-by-2 diagonal block.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:


09-29-06 - patch from
Bobby Cheng, MathWorks
Replace l.204 and l.372
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
by
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company

subroutine dsytf2_rk (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, integer INFO)

DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose:


DSYTF2_RK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

INFO


INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


TODO: put further details

Contributors:


December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA

subroutine dsytf2_rook (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).

Purpose:


DSYTF2_ROOK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U**T is the transpose of U, and D is symmetric and
block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:


November 2013, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

subroutine dsytrd (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRD

Purpose:


DSYTRD reduces a real symmetric matrix A to real symmetric
tridiagonal form T by an orthogonal similarity transformation:
Q**T * A * Q = T.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

D


D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E


E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

TAU


TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
For optimum performance LWORK >= N*NB, where NB is the
optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(1:i-1,i+1), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

subroutine dsytrd_2stage (character VECT, character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAU, double precision, dimension( * ) HOUS2, integer LHOUS2, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRD_2STAGE

Purpose:


DSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric
tridiagonal form T by a orthogonal similarity transformation:
Q1**T Q2**T* A * Q2 * Q1 = T.

Parameters

VECT


VECT is CHARACTER*1
= 'N': No need for the Housholder representation,
in particular for the second stage (Band to
tridiagonal) and thus LHOUS2 is of size max(1, 4*N);
= 'V': the Householder representation is needed to
either generate Q1 Q2 or to apply Q1 Q2,
then LHOUS2 is to be queried and computed.
(NOT AVAILABLE IN THIS RELEASE).

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the band superdiagonal
of A are overwritten by the corresponding elements of the
internal band-diagonal matrix AB, and the elements above
the KD superdiagonal, with the array TAU, represent the orthogonal
matrix Q1 as a product of elementary reflectors; if UPLO
= 'L', the diagonal and band subdiagonal of A are over-
written by the corresponding elements of the internal band-diagonal
matrix AB, and the elements below the KD subdiagonal, with
the array TAU, represent the orthogonal matrix Q1 as a product
of elementary reflectors. See Further Details.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

D


D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T.

E


E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T.

TAU


TAU is DOUBLE PRECISION array, dimension (N-KD)
The scalar factors of the elementary reflectors of
the first stage (see Further Details).

HOUS2


HOUS2 is DOUBLE PRECISION array, dimension (LHOUS2)
Stores the Householder representation of the stage2
band to tridiagonal.

LHOUS2


LHOUS2 is INTEGER
The dimension of the array HOUS2.
If LWORK = -1, or LHOUS2 = -1,
then a query is assumed; the routine
only calculates the optimal size of the HOUS2 array, returns
this value as the first entry of the HOUS2 array, and no error
message related to LHOUS2 is issued by XERBLA.
If VECT='N', LHOUS2 = max(1, 4*n);
if VECT='V', option not yet available.

WORK


WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK


LWORK is INTEGER
The dimension of the array WORK. LWORK = MAX(1, dimension)
If LWORK = -1, or LHOUS2=-1,
then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK = MAX(1, dimension) where
dimension = max(stage1,stage2) + (KD+1)*N
= N*KD + N*max(KD+1,FACTOPTNB)
+ max(2*KD*KD, KD*NTHREADS)
+ (KD+1)*N
where KD is the blocking size of the reduction,
FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196

subroutine dsytrd_sy2sb (character UPLO, integer N, integer KD, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRD_SY2SB

Purpose:


DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric
band-diagonal form AB by a orthogonal similarity transformation:
Q**T * A * Q = AB.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

KD


KD is INTEGER
The number of superdiagonals of the reduced matrix if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
The reduced matrix is stored in the array AB.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if UPLO = 'U', the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= 'L', the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

AB


AB is DOUBLE PRECISION array, dimension (LDAB,N)
On exit, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

LDAB


LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.

TAU


TAU is DOUBLE PRECISION array, dimension (N-KD)
The scalar factors of the elementary reflectors (see Further
Details).

WORK


WORK is DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, or if LWORK=-1,
WORK(1) returns the size of LWORK.

LWORK


LWORK is INTEGER
The dimension of the array WORK which should be calculated
by a workspace query. LWORK = MAX(1, LWORK_QUERY)
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
where FACTOPTNB is the blocking used by the QR or LQ
algorithm, usually FACTOPTNB=128 is a good choice otherwise
putting LWORK=-1 will provide the size of WORK.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


Implemented by Azzam Haidar.
All details are available on technical report, SC11, SC13 papers.
Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
Parallel reduction to condensed forms for symmetric eigenvalue problems
using aggregated fine-grained and memory-aware kernels. In Proceedings
of 2011 International Conference for High Performance Computing,
Networking, Storage and Analysis (SC '11), New York, NY, USA,
Article 8 , 11 pages.
http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013.
An improved parallel singular value algorithm and its implementation
for multicore hardware, In Proceedings of 2013 International Conference
for High Performance Computing, Networking, Storage and Analysis (SC '13).
Denver, Colorado, USA, 2013.
Article 90, 12 pages.
http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
A novel hybrid CPU-GPU generalized eigensolver for electronic structure
calculations based on fine-grained memory aware tasks.
International Journal of High Performance Computing Applications.
Volume 28 Issue 2, Pages 196-209, May 2014.
http://hpc.sagepub.com/content/28/2/196


If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
A(i,i+kd+1:n), and tau in TAU(i).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(k), where k = n-kd.
Each H(i) has the form
H(i) = I - tau * v * v**T
where tau is a real scalar, and v is a real vector with
v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
A(i+kd+2:n,i), and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( ab ab/v1 v1 v1 v1 ) ( ab )
( ab ab/v2 v2 v2 ) ( ab/v1 ab )
( ab ab/v3 v3 ) ( v1 ab/v2 ab )
( ab ab/v4 ) ( v1 v2 ab/v3 ab )
( ab ) ( v1 v2 v3 ab/v4 ab )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i)..fi

subroutine dsytrf (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF

Purpose:


DSYTRF computes the factorization of a real symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method. The form of the
factorization is
A = U**T*D*U or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', then A = U**T*D*U, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

subroutine dsytrf_aa (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_AA

Purpose:


DSYTRF_AA computes the factorization of a real symmetric matrix A
using the Aasen's algorithm. The form of the factorization is
A = U**T*T*U or A = L*T*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a symmetric tridiagonal matrix.
This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the tridiagonal matrix is stored in the diagonals
and the subdiagonals of A just below (or above) the diagonals,
and L is stored below (or above) the subdiaonals, when UPLO
is 'L' (or 'U').

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The length of WORK. LWORK >= MAX(1,2*N). For optimum performance
LWORK >= N*(1+NB), where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrf_aa_2stage (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_AA_2STAGE

Purpose:


DSYTRF_AA_2STAGE computes the factorization of a real symmetric matrix A
using the Aasen's algorithm. The form of the factorization is
A = U**T*T*U or A = L*T*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and T is a symmetric band matrix with the
bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is
LU factorized with partial pivoting).
This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, L is stored below (or above) the subdiaonal blocks,
when UPLO is 'L' (or 'U').

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TB


TB is DOUBLE PRECISION array, dimension (LTB)
On exit, details of the LU factorization of the band matrix.

LTB


LTB is INTEGER
The size of the array TB. LTB >= 4*N, internally
used to select NB such that LTB >= (3*NB+1)*N.
If LTB = -1, then a workspace query is assumed; the
routine only calculates the optimal size of LTB,
returns this value as the first entry of TB, and
no error message related to LTB is issued by XERBLA.

IPIV


IPIV is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of A were interchanged with the
row and column IPIV(k).

IPIV2


IPIV2 is INTEGER array, dimension (N)
On exit, it contains the details of the interchanges, i.e.,
the row and column k of T were interchanged with the
row and column IPIV2(k).

WORK


WORK is DOUBLE PRECISION workspace of size LWORK

LWORK


LWORK is INTEGER
The size of WORK. LWORK >= N, internally used to select NB
such that LWORK >= N*NB.
If LWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the WORK array,
returns this value as the first entry of the WORK array, and
no error message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, band LU factorization failed on i-th column

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrf_rk (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose:


DSYTRF_RK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.
For more information see Further Details section.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK


WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned
by ILAENV.
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the WORK
array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued
by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


TODO: put correct description

Contributors:


December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsytrf_rook (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRF_ROOK

Purpose:


DSYTRF_ROOK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
The form of the factorization is
A = U*D*U**T or A = L*D*L**T
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U':
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k-1 and -IPIV(k-1) were inerchaged,
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
If UPLO = 'L':
If IPIV(k) > 0, then rows and columns k and IPIV(k)
were interchanged and D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
columns k and -IPIV(k) were interchanged and rows and
columns k+1 and -IPIV(k+1) were inerchaged,
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The length of WORK. LWORK >=1. For best performance
LWORK >= N*NB, where NB is the block size returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:


If UPLO = 'U', then A = U*D*U**T, where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If UPLO = 'L', then A = L*D*L**T, where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

Contributors:


April 2012, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsytri (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer INFO)

DSYTRI

Purpose:


DSYTRI computes the inverse of a real symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
DSYTRF.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by DSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

WORK


WORK is DOUBLE PRECISION array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytri2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRI2

Purpose:


DSYTRI2 computes the inverse of a DOUBLE PRECISION symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
DSYTRF. DSYTRI2 sets the LEADING DIMENSION of the workspace
before calling DSYTRI2X that actually computes the inverse.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by DSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

WORK


WORK is DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3)

LWORK


LWORK is INTEGER
The dimension of the array WORK.
WORK is size >= (N+NB+1)*(NB+3)
If LWORK = -1, then a workspace query is assumed; the routine
calculates:
- the optimal size of the WORK array, returns
this value as the first entry of the WORK array,
- and no error message related to LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytri2x (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( n+nb+1,* ) WORK, integer NB, integer INFO)

DSYTRI2X

Purpose:


DSYTRI2X computes the inverse of a real symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
DSYTRF.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the NNB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by DSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the NNB structure of D
as determined by DSYTRF.

WORK


WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3)

NB


NB is INTEGER
Block size

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytri_3 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRI_3

Purpose:


DSYTRI_3 computes the inverse of a real symmetric indefinite
matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
DSYTRI_3 sets the leading dimension of the workspace before calling
DSYTRI_3X that actually computes the inverse. This is the blocked
version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the symmetric inverse of the original
matrix.
If UPLO = 'U': the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = 'L': the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_RK or DSYTRF_BK.

WORK


WORK is DOUBLE PRECISION array, dimension (N+NB+1)*(NB+3).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The length of WORK. LWORK >= (N+NB+1)*(NB+3).
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the optimal
size of the WORK array, returns this value as the first
entry of the WORK array, and no error message related to
LWORK is issued by XERBLA.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


November 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dsytri_3x (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

DSYTRI_3X

Purpose:


DSYTRI_3X computes the inverse of a real symmetric indefinite
matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
On exit, if INFO = 0, the symmetric inverse of the original
matrix.
If UPLO = 'U': the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = 'L': the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_RK or DSYTRF_BK.

WORK


WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3).

NB


NB is INTEGER
Block size.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

subroutine dsytri_rook (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK, integer INFO)

DSYTRI_ROOK

Purpose:


DSYTRI_ROOK computes the inverse of a real symmetric
matrix A using the factorization A = U*D*U**T or A = L*D*L**T
computed by DSYTRF_ROOK.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by DSYTRF_ROOK.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_ROOK.

WORK


WORK is DOUBLE PRECISION array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


April 2012, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsytrs (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYTRS

Purpose:


DSYTRS solves a system of linear equations A*X = B with a real
symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by DSYTRF.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs2 (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer INFO)

DSYTRS2

Purpose:


DSYTRS2 solves a system of linear equations A*X = B with a real
symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by DSYTRF and converted by DSYCONV.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF.
Note that A is input / output. This might be counter-intuitive,
and one may think that A is input only. A is input / output. This
is because, at the start of the subroutine, we permute A in a
"better" form and then we permute A back to its original form at
the end.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

WORK


WORK is DOUBLE PRECISION array, dimension (N)

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs_3 (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYTRS_3

Purpose:


DSYTRS_3 solves a system of linear equations A * X = B with a real
symmetric matrix A using the factorization computed
by DSYTRF_RK or DSYTRF_BK:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This algorithm is using Level 3 BLAS.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

E


E is DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_RK or DSYTRF_BK.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


June 2017, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dsytrs_aa (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSYTRS_AA

Purpose:


DSYTRS_AA solves a system of linear equations A*X = B with a real
symmetric matrix A using the factorization A = U**T*T*U or
A = L*T*L**T computed by DSYTRF_AA.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U**T*T*U;
= 'L': Lower triangular, form is A = L*T*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
Details of factors computed by DSYTRF_AA.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by DSYTRF_AA.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))

LWORK


LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N-2).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs_aa_2stage (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYTRS_AA_2STAGE

Purpose:


DSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a real
symmetric matrix A using the factorization A = U**T*T*U or
A = L*T*L**T computed by DSYTRF_AA_2STAGE.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U**T*T*U;
= 'L': Lower triangular, form is A = L*T*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
Details of factors computed by DSYTRF_AA_2STAGE.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TB


TB is DOUBLE PRECISION array, dimension (LTB)
Details of factors computed by DSYTRF_AA_2STAGE.

LTB


LTB is INTEGER
The size of the array TB. LTB >= 4*N.

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges as computed by
DSYTRF_AA_2STAGE.

IPIV2


IPIV2 is INTEGER array, dimension (N)
Details of the interchanges as computed by
DSYTRF_AA_2STAGE.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dsytrs_rook (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)

DSYTRS_ROOK

Purpose:


DSYTRS_ROOK solves a system of linear equations A*X = B with
a real symmetric matrix A using the factorization A = U*D*U**T or
A = L*D*L**T computed by DSYTRF_ROOK.

Parameters

UPLO


UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

N


N is INTEGER
The order of the matrix A. N >= 0.

NRHS


NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by DSYTRF_ROOK.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV


IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_ROOK.

B


B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:


April 2012, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester

subroutine dtgsyl (character TRANS, integer IJOB, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldc, * ) C, integer LDC, double precision, dimension( ldd, * ) D, integer LDD, double precision, dimension( lde, * ) E, integer LDE, double precision, dimension( ldf, * ) F, integer LDF, double precision SCALE, double precision DIF, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

DTGSYL

Purpose:


DTGSYL solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
D * R - L * E = scale * F
where R and L are unknown m-by-n matrices, (A, D), (B, E) and
(C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
respectively, with real entries. (A, D) and (B, E) must be in
generalized (real) Schur canonical form, i.e. A, B are upper quasi
triangular and D, E are upper triangular.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
scaling factor chosen to avoid overflow.
In matrix notation (1) is equivalent to solve Zx = scale b, where
Z is defined as
Z = [ kron(In, A) -kron(B**T, Im) ] (2)
[ kron(In, D) -kron(E**T, Im) ].
Here Ik is the identity matrix of size k and X**T is the transpose of
X. kron(X, Y) is the Kronecker product between the matrices X and Y.
If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
which is equivalent to solve for R and L in
A**T * R + D**T * L = scale * C (3)
R * B**T + L * E**T = scale * -F
This case (TRANS = 'T') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using DLACON.
If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z. See [1-2] for more
information.
This is a level 3 BLAS algorithm.

Parameters

TRANS


TRANS is CHARACTER*1
= 'N': solve the generalized Sylvester equation (1).
= 'T': solve the 'transposed' system (3).

IJOB


IJOB is INTEGER
Specifies what kind of functionality to be performed.
= 0: solve (1) only.
= 1: The functionality of 0 and 3.
= 2: The functionality of 0 and 4.
= 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB = 1 is used).
= 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( DGECON on sub-systems is used ).
Not referenced if TRANS = 'T'.

M


M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.

N


N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.

A


A is DOUBLE PRECISION array, dimension (LDA, M)
The upper quasi triangular matrix A.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).

B


B is DOUBLE PRECISION array, dimension (LDB, N)
The upper quasi triangular matrix B.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N).

C


C is DOUBLE PRECISION array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.

LDC


LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).

D


D is DOUBLE PRECISION array, dimension (LDD, M)
The upper triangular matrix D.

LDD


LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).

E


E is DOUBLE PRECISION array, dimension (LDE, N)
The upper triangular matrix E.

LDE


LDE is INTEGER
The leading dimension of the array E. LDE >= max(1, N).

F


F is DOUBLE PRECISION array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.

LDF


LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).

DIF


DIF is DOUBLE PRECISION
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'T', DIF is not touched.

SCALE


SCALE is DOUBLE PRECISION
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, C and F hold the
solutions R and L, respectively, to the homogeneous system
with C = F = 0. Normally, SCALE = 1.

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK


LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

IWORK


IWORK is INTEGER array, dimension (M+N+6)

INFO


INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:


[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
Appl., 15(4):1045-1060, 1994
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with
Condition Estimators for Solving the Generalized Sylvester
Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
July 1989, pp 745-751.

subroutine dtrsyl (character TRANA, character TRANB, integer ISGN, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldc, * ) C, integer LDC, double precision SCALE, integer INFO)

DTRSYL

Purpose:


DTRSYL solves the real Sylvester matrix equation:
op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,
where op(A) = A or A**T, and A and B are both upper quasi-
triangular. A is M-by-M and B is N-by-N; the right hand side C and
the solution X are M-by-N; and scale is an output scale factor, set
<= 1 to avoid overflow in X.
A and B must be in Schur canonical form (as returned by DHSEQR), that
is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
each 2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.

Parameters

TRANA


TRANA is CHARACTER*1
Specifies the option op(A):
= 'N': op(A) = A (No transpose)
= 'T': op(A) = A**T (Transpose)
= 'C': op(A) = A**H (Conjugate transpose = Transpose)

TRANB


TRANB is CHARACTER*1
Specifies the option op(B):
= 'N': op(B) = B (No transpose)
= 'T': op(B) = B**T (Transpose)
= 'C': op(B) = B**H (Conjugate transpose = Transpose)

ISGN


ISGN is INTEGER
Specifies the sign in the equation:
= +1: solve op(A)*X + X*op(B) = scale*C
= -1: solve op(A)*X - X*op(B) = scale*C

M


M is INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.

N


N is INTEGER
The order of the matrix B, and the number of columns in the
matrices X and C. N >= 0.

A


A is DOUBLE PRECISION array, dimension (LDA,M)
The upper quasi-triangular matrix A, in Schur canonical form.

LDA


LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

B


B is DOUBLE PRECISION array, dimension (LDB,N)
The upper quasi-triangular matrix B, in Schur canonical form.

LDB


LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).

C


C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N right hand side matrix C.
On exit, C is overwritten by the solution matrix X.

LDC


LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M)

SCALE


SCALE is DOUBLE PRECISION
The scale factor, scale, set <= 1 to avoid overflow in X.

INFO


INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: A and B have common or very close eigenvalues; perturbed
values were used to solve the equation (but the matrices
A and B are unchanged).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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