Integer, Intent (In)	::	n, nrhs, ldb
Integer, Intent (Out)	::	ipiv(n), info
Complex (Kind=nag_wp), Intent (Inout)	::	ap(), b(ldb,)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f07pnf_ (const char uplo, const Integer n, const Integer nrhs, Complex ap[], Integer ipiv[], Complex b[], const Integer ldb, Integer *info, const Charlen length_uplo)

The routine may be called by the names f07pnf, nagf_lapacklin_zhpsv or its LAPACK name zhpsv.

3 Description

f07pnf uses the diagonal pivoting method to factor

A

A = U D U^{H}

uplo ='U'

A = L D L^{H}

uplo ='L'

, where

U

(or

L

) is a product of permutation and unit upper (lower) triangular matrices,

D

is Hermitian and block diagonal with

1

1

and

2

2

diagonal blocks. 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 https://www.netlib.org/lapack/lug

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Arguments

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: $n$ – Integer Input

On entry:

n

, the number of linear equations, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – Integer Input

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) array Input/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: the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

from the factorization

A = U D U^{H}

A = L D L^{H}

as computed by f07prf, stored as a packed triangular matrix in the same storage format as

A

5: $ipiv (n)$ – Integer array Output

On exit: details of the interchanges and the block structure of

D

. More precisely,

if $ipiv (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $uplo ='U'$ and $ipiv (i - 1) = ipiv (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $uplo ='L'$ and $ipiv (i) = ipiv (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

6: $b (ldb *)$ – Complex (Kind=nag_wp) array Input/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

7: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07pnf is called.

Constraint:

ldb \geq \max (1, n)

8: $info$ – Integer Output

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.

$info > 0$: Element $〈value〉$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be 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) and Chapter 11 of Higham (2002) for further details.

f07ppf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04cjf solves

A X = B

and returns a forward error bound and condition estimate. f04cjf calls f07pnf to solve the equations.

8 Parallelism and Performance

f07pnf 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 f07paf. The complex symmetric analogue of this routine is f07qnf.

10 Example

This example solves the equations

A x = b,

where

A

is the Hermitian matrix

A = (\begin{array}{r} - 1.84 & 0.11 - 0.11 i & - 1.78 - 1.18 i & 3.91 - 1.50 i \\ 0.11 + 0.11 i & - 4.63 & - 1.84 + 0.03 i & 2.21 + 0.21 i \\ - 1.78 + 1.18 i & - 1.84 - 0.03 i & - 8.87 & 1.58 - 0.90 i \\ 3.91 + 1.50 i & 2.21 - 0.21 i & 1.58 + 0.90 i & - 1.36 \end{array})

and

b = (\begin{array}{r} 2.98 - 10.18 i \\ - 9.58 + 3.88 i \\ - 0.77 - 16.05 i \\ 7.79 + 5.48 i \end{array}) .

Details of the factorization of

A

are also output.

f07pn: FL CL CPP AD

NAG FL Interfacef07pnf (zhpsv)

▸▿ 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

NAG FL Interface
f07pnf (zhpsv)