NAG FL Interfacef07jsf (zpttrs)

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1Purpose

f07jsf computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n×n$ Hermitian positive definite tridiagonal matrix and $X$ and $B$ are $n×r$ matrices, using the $LD{L}^{\mathrm{H}}$ factorization returned by f07jrf.

2Specification

Fortran Interface
 Subroutine f07jsf ( uplo, n, nrhs, d, e, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*) Complex (Kind=nag_wp), Intent (In) :: e(*) Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07jsf_ (const char *uplo, const Integer *n, const Integer *nrhs, const double d[], const Complex e[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07jsf, nagf_lapacklin_zpttrs or its LAPACK name zpttrs.

3Description

f07jsf should be preceded by a call to f07jrf, which computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLH ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. f07jsf then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form ${U}^{\mathrm{H}}DU$, where $U$ is a unit upper bidiagonal matrix.

4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies the form of the factorization as follows:
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}DU$.
${\mathbf{uplo}}=\text{'L'}$
$A=LD{L}^{\mathrm{H}}$.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{H}}$ or ${U}^{\mathrm{H}}DU$ factorization of $A$.
5: $\mathbf{e}\left(*\right)$Complex (Kind=nag_wp) array Input
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: if ${\mathbf{uplo}}=\text{'U'}$, e must contain the $\left(n-1\right)$ superdiagonal elements of the unit upper bidiagonal matrix $U$ from the ${U}^{\mathrm{H}}DU$ factorization of $A$.
If ${\mathbf{uplo}}=\text{'L'}$, e must contain the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ matrix of right-hand sides $B$.
On exit: the $n×r$ solution matrix $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07jsf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $(A+E) x^=b ,$
where
 $‖E‖1 =O(ε)‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of this routine f07juf can be used to estimate the condition number of $A$ and f07jvf can be used to obtain approximate error bounds.

8Parallelism and Performance

f07jsf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$.
The real analogue of this routine is f07jef.

10Example

This example solves the equations
 $AX=B ,$
where $A$ is the Hermitian positive definite tridiagonal matrix
 $A = ( 16.0i+00.0 16.0-16.0i 0.0i+0.0 0.0i+0.0 16.0+16.0i 41.0i+00.0 18.0+9.0i 0.0i+0.0 0.0i+00.0 18.0-09.0i 46.0i+0.0 1.0+4.0i 0.0i+00.0 0.0i+00.0 1.0-4.0i 21.0i+0.0 )$
and
 $B = ( 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i ) .$

10.1Program Text

Program Text (f07jsfe.f90)

10.2Program Data

Program Data (f07jsfe.d)

10.3Program Results

Program Results (f07jsfe.r)