NAG Library Routine Document

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

C Header Interface

#include <nagmk26.h>

void	f04cjf_ (const char uplo, const Integer n, const Integer nrhs, Complex ap[], Integer ipiv[], Complex b[], const Integer ldb, double rcond, double errbnd, Integer *ifail, const Charlen length_uplo)

3

Description

The diagonal pivoting method is used to factor

A

A = U D U^{H}

, if

uplo ='U'

, or

A = L D L^{H}

, if

uplo ='L'

, where

U

(or

L

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

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

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 the matrix

A

is stored.

uplo ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo ='U'

'L'

2: $n$ – IntegerInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – IntegerInput

On entry: the number of right-hand sides

r

, 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 column-wise in a linear array. The

j

th column of the matrix

A

is stored in the array ap as follows:

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

ifail \geq 0

, 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 (zhptrf), stored as a packed triangular matrix in the same storage format as

A

5: $ipiv (n)$ – Integer arrayOutput

On exit: if

ifail \geq 0

, details of the interchanges and the block structure of

D

, as determined by f07prf (zhptrf).

If $ipiv (k) > 0$ , then rows and columns $k$ and $ipiv (k)$ were interchanged, and $d_{k k}$ is a $1$ by $1$ diagonal block;
if $uplo ='U'$ and $ipiv (k) = ipiv (k - 1) < 0$ , then rows and columns $k - 1$ and $- ipiv (k)$ were interchanged and $d_{k - 1 : k, k - 1 : k}$ is a $2$ by $2$ diagonal block;
if $uplo ='L'$ and $ipiv (k) = ipiv (k + 1) < 0$ , then rows and columns $k + 1$ and $- ipiv (k)$ were interchanged and $d_{k : k + 1, k : k + 1}$ is a $2$ by $2$ diagonal block.

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: if

ifail = 0

n + 1

, 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 f04cjf is called.

Constraint:

ldb \geq \max (1, n)

8: $rcond$ – Real (Kind=nag_wp)Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

9: $errbnd$ – Real (Kind=nag_wp)Output

On exit: if

ifail = 0

n + 1

, an estimate of the forward error bound for a computed solution

\hat{x}

, such that

{‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq errbnd

, where

\hat{x}

is a column of the computed solution returned in the array b and

x

is the corresponding column of the exact solution

X

. If rcond is less than machine precision, errbnd is returned as unity.

10: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail > 0 and ifail \leq n$: Diagonal block $〈value〉$ of the block diagonal matrix is zero. The factorization has been completed, but the solution could not be computed.

$ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: On entry, $uplo \neq'U'$ or $'L'$ : $uplo = 〈value〉$ .

$ifail = - 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = - 3$: On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

$ifail = - 7$: On entry, $ldb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
The real allocatable memory required is n, and the complex allocatable memory required is $2 \times n$ . Allocation failed before the solution could be computed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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. f04cjf uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

f04cjf 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 packed storage scheme is illustrated by the following example when

n = 4

and

uplo ='U'

. Two-dimensional storage of the Hermitian matrix

A

\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{22} & a_{23} & a_{24} \\ a_{33} & a_{34} \\ a_{44} \end{matrix} (a_{i j} = {\bar{a}}_{j i}) .

Packed storage of the upper triangle of

A

ap = [\begin{matrix} a_{11}, & a_{12}, & a_{22}, & a_{13}, & a_{23}, & a_{33}, & a_{14}, & a_{24}, & a_{34}, & a_{44} \end{matrix}] .

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

A X = B

is proportional to

(\frac{1}{3} n^{3} + 2 n^{2} r)

. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

Routine f04djf is for complex symmetric matrices, and the real analogue of f04cjf is f04bjf.

10

Example

This example solves the equations

A X = B,

where

A

is the Hermitian indefinite 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 & 28.68 - 39.89 i \\ - 9.58 + 3.88 i & - 24.79 - 8.40 i \\ - 0.77 - 16.05 i & 4.23 - 70.02 i \\ 7.79 + 5.48 i & - 35.39 + 18.01 i \end{array}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

NAG Library Routine Document

f04cjf (complex_herm_packed_solve)

▸▿ 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