doublePTcomputational(3) LAPACK doublePTcomputational(3)

NAME¶

doublePTcomputational

SYNOPSIS¶

Functions¶

subroutine dptcon (N, D, E, ANORM, RCOND, WORK, INFO)
DPTCON subroutine dpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
DPTEQR subroutine dptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
DPTRFS subroutine dpttrf (N, D, E, INFO)
DPTTRF subroutine dpttrs (N, NRHS, D, E, B, LDB, INFO)
DPTTRS subroutine dptts2 (N, NRHS, D, E, B, LDB)
DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Detailed Description¶

This is the group of double computational functions for PT matrices

Function Documentation¶

subroutine dptcon (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision ANORM, double precision RCOND, double precision, dimension( * ) WORK, integer INFO)¶

DPTCON

Purpose:

DPTCON computes the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite tridiagonal matrix
using the factorization A = L*D*L**T or A = U**T*D*U computed by
DPTTRF.
Norm(inv(A)) is computed by a direct method, and the reciprocal of
the condition number is computed as
RCOND = 1 / (ANORM * norm(inv(A))).

Parameters

N

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

D

D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
factorization of A, as computed by DPTTRF.

E

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the unit bidiagonal factor
U or L from the factorization of A, as computed by DPTTRF.

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 the
1-norm of inv(A) computed in this routine.

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

NAG Ltd.

Date

December 2016

Further Details:

The method used is described in Nicholas J. Higham, "Efficient
Algorithms for Computing the Condition Number of a Tridiagonal
Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

subroutine dpteqr (character COMPZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)¶

DPTEQR

Purpose:

DPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF, and then calling DBDSQR to compute the singular
values of the bidiagonal factor.
This routine computes the eigenvalues of the positive definite
tridiagonal matrix to high relative accuracy. This means that if the
eigenvalues range over many orders of magnitude in size, then the
small eigenvalues and corresponding eigenvectors will be computed
more accurately than, for example, with the standard QR method.
The eigenvectors of a full or band symmetric positive definite matrix
can also be found if DSYTRD, DSPTRD, or DSBTRD has been used to
reduce this matrix to tridiagonal form. (The reduction to tridiagonal
form, however, may preclude the possibility of obtaining high
relative accuracy in the small eigenvalues of the original matrix, if
these eigenvalues range over many orders of magnitude.)

Parameters

COMPZ

COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvectors of original symmetric
matrix also. Array Z contains the orthogonal
matrix used to reduce the original matrix to
tridiagonal form.
= 'I': Compute eigenvectors of tridiagonal matrix also.

N

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

D

D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal
matrix.
On normal exit, D contains the eigenvalues, in descending
order.

E

E is DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.

Z

Z is DOUBLE PRECISION array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix used in the
reduction to tridiagonal form.
On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
original symmetric matrix;
if COMPZ = 'I', the orthonormal eigenvectors of the
tridiagonal matrix.
If INFO > 0 on exit, Z contains the eigenvectors associated
with only the stored eigenvalues.
If COMPZ = 'N', then Z is not referenced.

LDZ

LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

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

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is:
<= N the Cholesky factorization of the matrix could
not be performed because the i-th principal minor
was not positive definite.
> N the SVD algorithm failed to converge;
if INFO = N+i, i off-diagonal elements of the
bidiagonal factor did not converge to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

subroutine dptrfs (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) DF, double precision, dimension( * ) EF, 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 INFO)¶

DPTRFS

Purpose:

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

Parameters

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.

D

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

E

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.

DF

DF is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
factorization computed by DPTTRF.

EF

EF is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal factor
L from the factorization computed by DPTTRF.

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 DPTTRS.
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 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).

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 (2*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

NAG Ltd.

Date

December 2016

subroutine dpttrf (integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, integer INFO)¶

DPTTRF

Purpose:

DPTTRF computes the L*D*L**T factorization of a real symmetric
positive definite tridiagonal matrix A. The factorization may also
be regarded as having the form A = U**T*D*U.

Parameters

N

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

D

D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix
A. On exit, the n diagonal elements of the diagonal matrix
D from the L*D*L**T factorization of A.

E

E is DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A. On exit, the (n-1) subdiagonal elements of the
unit bidiagonal factor L from the L*D*L**T factorization of A.
E can also be regarded as the superdiagonal of the unit
bidiagonal factor U from the U**T*D*U factorization of A.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
> 0: if INFO = k, the leading minor of order k is not
positive definite; if k < N, the factorization could not
be completed, while if k = N, the factorization was
completed, but D(N) <= 0.

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

subroutine dpttrs (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

DPTTRS

Purpose:

DPTTRS solves a tridiagonal system of the form
A * X = B
using the L*D*L**T factorization of A computed by DPTTRF. D is a
diagonal matrix specified in the vector D, L is a unit bidiagonal
matrix whose subdiagonal is specified in the vector E, and X and B
are N by NRHS matrices.

Parameters

N

N is INTEGER
The order of the tridiagonal 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.

D

D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
L*D*L**T factorization of A.

E

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal factor
L from the L*D*L**T factorization of A. E can also be regarded
as the superdiagonal of the unit bidiagonal factor U from the
factorization A = U**T*D*U.

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side vectors B for the system of
linear equations.
On exit, the solution vectors, 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 = -k, the k-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

subroutine dptts2 (integer N, integer NRHS, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldb, * ) B, integer LDB)¶

DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:

DPTTS2 solves a tridiagonal system of the form
A * X = B
using the L*D*L**T factorization of A computed by DPTTRF. D is a
diagonal matrix specified in the vector D, L is a unit bidiagonal
matrix whose subdiagonal is specified in the vector E, and X and B
are N by NRHS matrices.

Parameters

N

N is INTEGER
The order of the tridiagonal 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.

D

D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
L*D*L**T factorization of A.

E

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal factor
L from the L*D*L**T factorization of A. E can also be regarded
as the superdiagonal of the unit bidiagonal factor U from the
factorization A = U**T*D*U.

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the right hand side vectors B for the system of
linear equations.
On exit, the solution vectors, X.

LDB

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

Author

Univ. of Tennessee

Univ. of California Berkeley