NAG Library Routine Document

f07grf  (zpptrf)

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:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array ap must be at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$.

On entry: the $n$ by $n$ Hermitian matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

On exit: if $\mathbf{info}=\mathbf{0}$, the factor $U$ or $L$ from the Cholesky factorization $A=U^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$, in the same storage format as $A$.

4:     $\mathbf{info}$ – 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.

$\mathbf{info}>0$

The leading minor of order $〈\mathit{\text{value}}〉$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. To factorize a Hermitian matrix which is not positive definite, call f07prf (zhptrf) instead.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07grfe.f90)

10.2

Program Data

Program Data (f07grfe.d)

10.3

Program Results

Program Results (f07grfe.r)