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complex16SYauxiliary(3) LAPACK complex16SYauxiliary(3)

NAME

complex16SYauxiliary - complex16

SYNOPSIS

Functions


subroutine zlaesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1)
ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. double precision function zlansy (NORM, UPLO, N, A, LDA, WORK)
ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix. subroutine zlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. subroutine zsymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSYMV computes a matrix-vector product for a complex symmetric matrix. subroutine zsyr (UPLO, N, ALPHA, X, INCX, A, LDA)
ZSYR performs the symmetric rank-1 update of a complex symmetric matrix. subroutine zsyswapr (UPLO, N, A, LDA, I1, I2)
ZSYSWAPR subroutine ztgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

This is the group of complex16 auxiliary functions for SY matrices

Function Documentation

subroutine zlaesy (complex*16 A, complex*16 B, complex*16 C, complex*16 RT1, complex*16 RT2, complex*16 EVSCAL, complex*16 CS1, complex*16 SN1)

ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.

Purpose:


ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
( ( A, B );( B, C ) )
provided the norm of the matrix of eigenvectors is larger than
some threshold value.
RT1 is the eigenvalue of larger absolute value, and RT2 of
smaller absolute value. If the eigenvectors are computed, then
on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
[ CS1 SN1 ] . [ A B ] . [ CS1 -SN1 ] = [ RT1 0 ]
[ -SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]

Parameters

A


A is COMPLEX*16
The ( 1, 1 ) element of input matrix.

B


B is COMPLEX*16
The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element
is also given by B, since the 2-by-2 matrix is symmetric.

C


C is COMPLEX*16
The ( 2, 2 ) element of input matrix.

RT1


RT1 is COMPLEX*16
The eigenvalue of larger modulus.

RT2


RT2 is COMPLEX*16
The eigenvalue of smaller modulus.

EVSCAL


EVSCAL is COMPLEX*16
The complex value by which the eigenvector matrix was scaled
to make it orthonormal. If EVSCAL is zero, the eigenvectors
were not computed. This means one of two things: the 2-by-2
matrix could not be diagonalized, or the norm of the matrix
of eigenvectors before scaling was larger than the threshold
value THRESH (set below).

CS1


CS1 is COMPLEX*16

SN1


SN1 is COMPLEX*16
If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector
for RT1.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

double precision function zlansy (character NORM, character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)

ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.

Purpose:


ZLANSY returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
complex symmetric matrix A.

Returns

ZLANSY


ZLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

Parameters

NORM


NORM is CHARACTER*1
Specifies the value to be returned in ZLANSY as described
above.

UPLO


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

N


N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANSY is
set to zero.

A


A is COMPLEX*16 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(N,1).

WORK


WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zlaqsy (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)

ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.

Purpose:


ZLAQSY equilibrates a symmetric matrix A using the scaling factors
in the vector S.

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 COMPLEX*16 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 EQUED = 'Y', the equilibrated matrix:
diag(S) * A * diag(S).

LDA


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

S


S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.

SCOND


SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).

AMAX


AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.

EQUED


EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).

Internal Parameters:


THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zsymv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y, integer INCY)

ZSYMV computes a matrix-vector product for a complex symmetric matrix.

Purpose:


ZSYMV performs the matrix-vector operation
y := alpha*A*x + beta*y,
where alpha and beta are scalars, x and y are n element vectors and
A is an n by n symmetric matrix.

Parameters

UPLO


UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
follows:
UPLO = 'U' or 'u' Only the upper triangular part of A
is to be referenced.
UPLO = 'L' or 'l' Only the lower triangular part of A
is to be referenced.
Unchanged on exit.

N


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

ALPHA


ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.

A


A is COMPLEX*16 array, dimension ( LDA, N )
Before entry, with UPLO = 'U' or 'u', the leading n by n
upper triangular part of the array A must contain the upper
triangular part of the symmetric matrix and the strictly
lower triangular part of A is not referenced.
Before entry, with UPLO = 'L' or 'l', the leading n by n
lower triangular part of the array A must contain the lower
triangular part of the symmetric matrix and the strictly
upper triangular part of A is not referenced.
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 COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element 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 COMPLEX*16
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 COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the n
element 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.

subroutine zsyr (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX, complex*16, dimension( lda, * ) A, integer LDA)

ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.

Purpose:


ZSYR performs the symmetric rank 1 operation
A := alpha*x*x**H + A,
where alpha is a complex scalar, x is an n element vector and A is an
n by n symmetric matrix.

Parameters

UPLO


UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
follows:
UPLO = 'U' or 'u' Only the upper triangular part of A
is to be referenced.
UPLO = 'L' or 'l' Only the lower triangular part of A
is to be referenced.
Unchanged on exit.

N


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

ALPHA


ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.

X


X is COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element 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.

A


A is COMPLEX*16 array, dimension ( LDA, N )
Before entry, with UPLO = 'U' or 'u', the leading n by n
upper triangular part of the array A must contain the upper
triangular part of the symmetric matrix and the strictly
lower triangular part of A is not referenced. On exit, the
upper triangular part of the array A is overwritten by the
upper triangular part of the updated matrix.
Before entry, with UPLO = 'L' or 'l', the leading n by n
lower triangular part of the array A must contain the lower
triangular part of the symmetric matrix and the strictly
upper triangular part of A is not referenced. On exit, the
lower triangular part of the array A is overwritten by the
lower triangular part of the updated matrix.

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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zsyswapr (character UPLO, integer N, complex*16, dimension( lda, n ) A, integer LDA, integer I1, integer I2)

ZSYSWAPR

Purpose:


ZSYSWAPR applies an elementary permutation on the rows and the columns of
a symmetric matrix.

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 COMPLEX*16 array, dimension (LDA,N)
On entry, the NB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by ZSYTRF.
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).

I1


I1 is INTEGER
Index of the first row to swap

I2


I2 is INTEGER
Index of the second row to swap

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine ztgsy2 (character TRANS, integer IJOB, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF, double precision SCALE, double precision RDSUM, double precision RDSCAL, integer INFO)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:


ZTGSY2 solves the generalized Sylvester equation
A * R - L * B = scale * C (1)
D * R - L * E = scale * F
using Level 1 and 2 BLAS, 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. A, B, D and E are upper triangular
(i.e., (A,D) and (B,E) in generalized Schur form).
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
scaling factor chosen to avoid overflow.
In matrix notation solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as
Z = [ kron(In, A) -kron(B**H, Im) ] (2)
[ kron(In, D) -kron(E**H, Im) ],
Ik is the identity matrix of size k and X**H is the conjuguate transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.
If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b
is solved for, which is equivalent to solve for R and L in
A**H * R + D**H * L = scale * C (3)
R * B**H + L * E**H = scale * -F
This case is used to compute an estimate of Dif[(A, D), (B, E)] =
= sigma_min(Z) using reverse communication with ZLACON.
ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
of an upper bound on the separation between to matrix pairs. Then
the input (A, D), (B, E) are sub-pencils of two matrix pairs in
ZTGSYL.

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: A contribution from this subsystem to a Frobenius
norm-based estimate of the separation between two matrix
pairs is computed. (look ahead strategy is used).
=2: A contribution from this subsystem to a Frobenius
norm-based estimate of the separation between two matrix
pairs is computed. (DGECON on sub-systems is used.)
Not referenced if TRANS = 'T'.

M


M is INTEGER
On entry, M specifies the order of A and D, and the row
dimension of C, F, R and L.

N


N is INTEGER
On entry, N specifies the order of B and E, and the column
dimension of C, F, R and L.

A


A is COMPLEX*16 array, dimension (LDA, M)
On entry, A contains an upper triangular matrix.

LDA


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

B


B is COMPLEX*16 array, dimension (LDB, N)
On entry, B contains an upper triangular matrix.

LDB


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

C


C is COMPLEX*16 array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1).
On exit, if IJOB = 0, C has been overwritten by the solution
R.

LDC


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

D


D is COMPLEX*16 array, dimension (LDD, M)
On entry, D contains an upper triangular matrix.

LDD


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

E


E is COMPLEX*16 array, dimension (LDE, N)
On entry, E contains an upper triangular matrix.

LDE


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

F


F is COMPLEX*16 array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1).
On exit, if IJOB = 0, F has been overwritten by the solution
L.

LDF


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

SCALE


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

RDSUM


RDSUM is DOUBLE PRECISION
On entry, the sum of squares of computed contributions to
the Dif-estimate under computation by ZTGSYL, where the
scaling factor RDSCAL (see below) has been factored out.
On exit, the corresponding sum of squares updated with the
contributions from the current sub-system.
If TRANS = 'T' RDSUM is not touched.
NOTE: RDSUM only makes sense when ZTGSY2 is called by
ZTGSYL.

RDSCAL


RDSCAL is DOUBLE PRECISION
On entry, scaling factor used to prevent overflow in RDSUM.
On exit, RDSCAL is updated w.r.t. the current contributions
in RDSUM.
If TRANS = 'T', RDSCAL is not touched.
NOTE: RDSCAL only makes sense when ZTGSY2 is called by
ZTGSYL.

INFO


INFO is INTEGER
On exit, if INFO is set to
=0: Successful exit
<0: If INFO = -i, input argument number i is illegal.
>0: The matrix pairs (A, D) and (B, E) have common or very
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.

Author

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