f08ut: FL CL CPP AD

NAG FL Interface
f08utf (zpbstf)

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NAG Library Manual, Mark 28.3

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08ut: FL CL CPP AD

▸▿ 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

1 Purpose

f08utf computes a split Cholesky factorization of a complex Hermitian positive definite band matrix.

2 Specification

Fortran Interface

Subroutine f08utf (

uplo, n, kb, bb, ldbb, info)

Integer, Intent (In)	::	n, kb, ldbb
Integer, Intent (Out)	::	info
Complex (Kind=nag_wp), Intent (Inout)	::	bb(ldbb,*)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f08utf_ (const char uplo, const Integer n, const Integer kb, Complex bb[], const Integer ldbb, Integer *info, const Charlen length_uplo)

The routine may be called by the names f08utf, nagf_lapackeig_zpbstf or its LAPACK name zpbstf.

3 Description

f08utf computes a split Cholesky factorization of a complex Hermitian positive definite band matrix

B

. It is designed to be used in conjunction with f08usf.

The factorization has the form

B = S^{H} S

, where

S

is a band matrix of the same bandwidth as

B

and the following structure:

S

is upper triangular in the first

(n + k) / 2

rows, and transposed — hence, lower triangular — in the remaining rows. For example, if

n = 9

and

k = 2

, then

S = (\begin{array}{l} s_{11} & s_{12} & s_{13} \\ s_{22} & s_{23} & s_{24} \\ s_{33} & s_{34} & s_{35} \\ s_{44} & s_{45} \\ s_{55} \\ s_{64} & s_{65} & s_{66} \\ s_{75} & s_{76} & s_{77} \\ s_{86} & s_{87} & s_{88} \\ s_{97} & s_{98} & s_{99} \end{array}) .

4 References

None.

5 Arguments

1: $uplo$ – Character(1) Input

On entry: indicates whether the upper or lower triangular part of

B

is stored.

$uplo ='U'$: The upper triangular part of $B$ is stored.
$uplo ='L'$: The lower triangular part of $B$ is stored.

Constraint:

uplo ='U'

'L'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

B

Constraint:

n \geq 0

3: $kb$ – Integer Input

On entry: if

uplo ='U'

, the number of superdiagonals,

k_{b}

, of the matrix

B

uplo ='L'

, the number of subdiagonals,

k_{b}

, of the matrix

B

Constraint:

kb \geq 0

4: $bb (ldbb, *)$ – Complex (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the

n \times n

Hermitian positive definite band matrix

B

The matrix is stored in rows

1

k_{b} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $B$ within the band must be stored with element $B_{i j}$ in $bb (k_{b} + 1 + i - j, j) for \max (1, j - k_{b}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $B$ within the band must be stored with element $B_{i j}$ in $bb (1 + i - j, j) for j \leq i \leq \min (n, j + k_{b}) .$

On exit:

B

is overwritten by the elements of its split Cholesky factor

S

5: $ldbb$ – Integer Input

On entry: the first dimension of the array bb as declared in the (sub)program from which f08utf is called.

Constraint:

ldbb \geq kb + 1

6: $info$ – 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.

$info > 0$: The factorization could not be completed, because the updated element $b (⟨ value ⟩, ⟨ value ⟩)$ would be the square root of a negative number. Hence $B$ is not positive definite. This may indicate an error in forming the matrix $B$ .

7 Accuracy

The computed factor

S

is the exact factor of a perturbed matrix

(B + E)

, where

| E | \leq c (k + 1) ε | S^{H} | | S |,

c (k + 1)

is a modest linear function of

k + 1

, and

ε

is the machine precision. It follows that

| e_{i j} | \leq c (k + 1) ε \sqrt{(b_{i i} b_{j j})}

8 Parallelism and Performance

f08utf 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.