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realPTcomputational(3) LAPACK realPTcomputational(3)

NAME

realPTcomputational - real

SYNOPSIS

Functions


subroutine sptcon (N, D, E, ANORM, RCOND, WORK, INFO)
SPTCON subroutine spteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
SPTEQR subroutine sptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
SPTRFS subroutine spttrs (N, NRHS, D, E, B, LDB, INFO)
SPTTRS subroutine sptts2 (N, NRHS, D, E, B, LDB)
SPTTS2 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 real computational functions for PT matrices

Function Documentation

subroutine sptcon (integer N, real, dimension( * ) D, real, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) WORK, integer INFO)

SPTCON

Purpose:


SPTCON 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
SPTTRF.
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 REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
factorization of A, as computed by SPTTRF.

E


E is REAL 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 SPTTRF.

ANORM


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

RCOND


RCOND is REAL
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 REAL 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.

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 spteqr (character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO)

SPTEQR

Purpose:


SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using SPTTRF, and then calling SBDSQR 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 SSYTRD, SSPTRD, or SSBTRD 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 REAL 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 REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.

Z


Z is REAL 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 REAL 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

Univ. of Colorado Denver

NAG Ltd.

subroutine sptrfs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) DF, real, dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)

SPTRFS

Purpose:


SPTRFS 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 REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.

E


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

DF


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

EF


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

B


B is REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SPTTRS.
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 REAL 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 REAL 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 REAL 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

Univ. of Colorado Denver

NAG Ltd.

subroutine spttrs (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPTTRS

Purpose:


SPTTRS solves a tridiagonal system of the form
A * X = B
using the L*D*L**T factorization of A computed by SPTTRF. 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 REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
L*D*L**T factorization of A.

E


E is REAL 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 REAL 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

Univ. of Colorado Denver

NAG Ltd.

subroutine sptts2 (integer N, integer NRHS, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldb, * ) B, integer LDB)

SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:


SPTTS2 solves a tridiagonal system of the form
A * X = B
using the L*D*L**T factorization of A computed by SPTTRF. 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 REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
L*D*L**T factorization of A.

E


E is REAL 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 REAL 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

Univ. of Colorado Denver

NAG Ltd.

Author

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