NAG Library Routine Document

f07hrf  (zpbtrf)

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:     $\mathbf{kd}$ – IntegerInput

On entry: $k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

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

4:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

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

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

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

On exit: the upper or lower triangle of $A$ is overwritten by the Cholesky factor $U$ or $L$ as specified by uplo, using the same storage format as described above.

5:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f07hrf (zpbtrf) is called.

Constraint: $\mathbf{ldab}\ge \mathbf{kd}+1$.

6:     $\mathbf{info}$ – 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.

If $\mathbf{info}=-999$, dynamic memory allocation failed. See Section 3.7 in How to Use the NAG Library and its Documentation for further information. 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 $A$. There is no routine specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07brf (zgbtrf) or as a full Hermitian matrix, by calling f07mrf (zhetrf).

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07hrfe.f90)

10.2

Program Data

Program Data (f07hrfe.d)

10.3

Program Results

Program Results (f07hrfe.r)