NAG Library Routine Document

f07krf (zpstrf)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{uplo}$ – Character(1)Input

On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

$\mathbf{uplo}=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $U^{\mathrm{H}}U$, where $U$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

3:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

On entry: the $n$ by $n$ Hermitian positive semidefinite matrix $A$.

• If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if $\mathbf{uplo}=\text{'U'}$, the first rank rows of the upper triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $U$, and the remaining rows of the triangle are destroyed.

If $\mathbf{uplo}=\text{'L'}$, the first rank columns of the lower triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $L$, and the remaining columns of the triangle are destroyed.

4:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07krf (zpstrf) is called.

Constraint: $\mathbf{lda}\ge \max \left(1,\mathbf{n}\right)$.

5:     $\mathbf{piv}\left(\mathbf{n}\right)$ – Integer arrayOutput

On exit: piv is such that the nonzero entries of $P$ are $P\left(\mathbf{piv}\left(\mathit{k}\right),\mathit{k}\right)=1$, for $\mathit{k}=1,2,\dots ,n$.

6:     $\mathbf{rank}$ – IntegerOutput

On exit: the computed rank of $A$ given by the number of steps the algorithm completed.

7:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input

On entry: user defined tolerance. If $\mathbf{tol}<0$, $n\times {\displaystyle \underset{k=1,n}{\text{max}}}\left|A_{kk}\right|\times \mathit{machine precision}$ will be used. The algorithm terminates at the $r$th step if the $\left(r+1\right)$th step pivot $<\mathbf{tol}$.

8:     $\mathbf{work}\left(2*\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayWorkspace

9:     $\mathbf{info}$ – IntegerOutput

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{info}=1$

The matrix $A$ is not positive definite. It is either positive semidefinite with computed rank as returned in rank and less than $n$, or it may be indefinite, see Section 9.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07krfe.f90)

10.2

Program Data

Program Data (f07krfe.d)

10.3

Program Results

Program Results (f07krfe.r)