void	f04chc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, Complex a[], Integer pda, Integer ipiv[], Complex b[], Integer pdb, double rcond, double errbnd, NagError *fail)

The function may be called by the names: f04chc, nag_linsys_complex_herm_solve or nag_herm_lin_solve.

3 Description

The diagonal pivoting method is used to factor

A

A = U D U^{H}

, if

uplo = Nag_Upper

, or

A = L D L^{H}

, if

uplo = Nag_Lower

, 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 https://www.netlib.org/lapack/lug

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

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploType Input

On entry: if

uplo = Nag_Upper

, the upper triangle of the matrix

A

is stored.

uplo = Nag_Lower

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

4: $nrhs$ – Integer Input

On entry: the number of right-hand sides

r

, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

5: $a [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array a must be at least

\max (1, pda \times n)

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

Hermitian matrix

A

uplo = Nag_Upper

, the leading n by n upper triangular part of the array a contains the upper triangular part of the matrix

A

, and the strictly lower triangular part of a is not referenced.

uplo = Nag_Lower

, the leading n by n lower triangular part of the array a contains the lower triangular part of the matrix

A

, and the strictly upper triangular part of a is not referenced.

On exit: if

fail . code =

NE_NOERROR, 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 f07mrc.

6: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint:

pda \geq \max (1, n)

7: $ipiv [n]$ – Integer Output

On exit: if

fail . code =

NE_NOERROR, details of the interchanges and the block structure of

D

, as determined by f07mrc.

If $ipiv [k - 1] > 0$ , then rows and columns $k$ and $ipiv [k - 1]$ were interchanged, and $d_{k k}$ is a $1$ by $1$ diagonal block;
if $uplo = Nag_Upper$ and $ipiv [k - 1] = ipiv [k - 2] < 0$ , then rows and columns $k - 1$ and $- ipiv [k - 1]$ were interchanged and $d_{k - 1 : k, k - 1 : k}$ is a $2$ by $2$ diagonal block;
if $uplo = Nag_Lower$ and $ipiv [k - 1] = ipiv [k] < 0$ , then rows and columns $k + 1$ and $- ipiv [k - 1]$ were interchanged and $d_{k : k + 1, k : k + 1}$ is a $2$ by $2$ diagonal block.

8: $b [\dim]$ – Complex Input/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

matrix of right-hand sides

B

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, the

n

r

solution matrix

X

9: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $rcond$ – double * 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})

11: $errbnd$ – double * Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, 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.

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
The double allocatable memory required is n, and the Complex allocatable memory required is $\max (2 \times n, lwork)$ , where lwork is the optimum workspace required by f07mnc. If this failure occurs it may be possible to solve the equations by calling the packed storage version of f04chc, f04cjc, or by calling f07mnc directly with less than the optimum workspace (see Chapter F07).
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

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

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR: Diagonal block $〈value〉$ of the block diagonal matrix is zero. The factorization has been completed, but 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. f04chc 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

f04chc 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 function. 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

(\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.

Function f04dhc is for complex symmetric matrices, and the real analogue of f04chc is f04bhc.

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.

f04ch: FL CL CPP AD

NAG CL Interfacef04chc (complex_​herm_​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

NAG CL Interface
f04chc (complex_herm_solve)