NAG Library Routine Document

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

C Header Interface

#include <nagmk26.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 its LAPACK name zpttrs.

3

Description

f07jsf (zpttrs) should be preceded by a call to f07jrf (zpttrf), which computes a modified Cholesky factorization of the matrix

A

A = L D L^{H},

where

L

is a unit lower bidiagonal matrix and

D

is a diagonal matrix, with positive diagonal elements. f07jsf (zpttrs) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form

U^{H} D U

, where

U

is a unit upper bidiagonal matrix.

4

References

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 http://www.netlib.org/lapack/lug

5

Arguments

1: $uplo$ – Character(1)Input

On entry: specifies the form of the factorization as follows:

$uplo ='U'$: $A = U^{H} D U$ .
$uplo ='L'$: $A = L D L^{H}$ .

Constraint:

uplo ='U'

'L'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

4: $d (*)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array d must be at least

\max (1, n)

On entry: must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{H}

U^{H} D U

factorization of

A

5: $e (*)$ – Complex (Kind=nag_wp) arrayInput

Note: the dimension of the array e must be at least

\max (1, n - 1)

On entry: if

uplo ='U'

, e must contain the

(n - 1)

superdiagonal elements of the unit upper bidiagonal matrix

U

from the

U^{H} D U

factorization of

A

uplo ='L'

, e must contain the

(n - 1)

subdiagonal elements of the unit lower bidiagonal matrix

L

from the

L D L^{H}

factorization of

A

6: $b (ldb *)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least

\max (1, nrhs)

On entry: the

n

r

matrix of right-hand sides

B

On exit: the

n

r

solution matrix

X

7: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07jsf (zpttrs) is called.

Constraint:

ldb \geq \max (1, n)

8: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

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

7

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖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 (zptcon) can be used to estimate the condition number of

A

and f07jvf (zptrfs) can be used to obtain approximate error bounds.

8

Parallelism and Performance

f07jsf (zpttrs) 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.

9

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

The real analogue of this routine is f07jef (dpttrs).

10

Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0.0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array})

and

B = (\begin{array}{r} 64.0 + 16.0 i & - 16.0 - 32.0 i \\ 93.0 + 62.0 i & 61.0 - 66.0 i \\ 78.0 - 80.0 i & 71.0 - 74.0 i \\ 14.0 - 27.0 i & 35.0 + 15.0 i \end{array}) .

NAG Library Routine Document

f07jsf (zpttrs)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results