NAG Library Routine Document

F07GNF (ZPPSV)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F07GNF (ZPPSV) computes the solution to a complex system of linear equations

A X = B,

where

A

is an

n

n

Hermitian positive definite matrix stored in packed format and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07GNF (

UPLO, N, NRHS, AP, B, LDB, INFO)

INTEGER	N, NRHS, LDB, INFO
COMPLEX (KIND=nag_wp)	AP(), B(LDB,)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zppsv.

3 Description

F07GNF (ZPPSV) uses the Cholesky decomposition to factor

A

A = U^{H} U

UPLO ='U'

A = L L^{H}

UPLO ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular matrix. The factored form of

A

is then used to solve the system of equations

A X = B

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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: $UPLO$ – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangle of

A

is stored.

UPLO ='L'

, the lower triangle of

A

is stored.

Constraint:

UPLO ='U'

'L'

2: $N$ – INTEGERInput

On entry:

n

, the number of linear equations, i.e., 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: $AP (*)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, N \times (N + 1) / 2)

On entry: the

n

n

Hermitian matrix

A

, packed by columns.

More precisely,

if $UPLO ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $UPLO ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

On exit: if

INFO = 0

, the factor

U

L

from the Cholesky factorization

A = U^{H} U

A = L L^{H}

, in the same storage format as

A

5: $B (LDB *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, NRHS)

To solve the equations

A x = b

, where

b

is a single right-hand side, B may be supplied as a one-dimensional array with length

LDB = \max (1, N)

On entry: the

n

r

right-hand side matrix

B

On exit: if

INFO = 0

, the

n

r

solution matrix

X

6: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07GNF (ZPPSV) is called.

Constraint:

LDB \geq \max (1, N)

7: $INFO$ – 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.

$INFO > 0$: The leading minor of order $〈value〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

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.

F07GPF (ZPPSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04CEF solves

A x = b

and returns a forward error bound and condition estimate. F04CEF calls F07GNF (ZPPSV) to solve the equations.

8 Parallelism and Performance

F07GNF (ZPPSV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F07GNF (ZPPSV) 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 is approximately

\frac{4}{3} n^{3} + 8 n^{2} r

, where

r

is the number of right-hand sides.

The real analogue of this routine is F07GAF (DPPSV).

10 Example

This example solves the equations

A x = b,

where

A

is the Hermitian positive definite matrix

A = (\begin{array}{r} 3.23 & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 \end{array})

and

b = (\begin{array}{r} 3.93 - 6.14 i \\ 6.17 + 9.42 i \\ - 7.17 - 21.83 i \\ 1.99 - 14.38 i \end{array}) .

Details of the Cholesky factorization of

A

are also output.

NAG Library Routine DocumentF07GNF (ZPPSV)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

F07GNF (ZPPSV)