NAG Library Routine Document

F07HRF (ZPBTRF)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     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:     N – INTEGERInput

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

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

3:     KD – INTEGERInput

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

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

4:     AB(LDAB,$*$) – 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:     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:     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 $\mathbf{INFO}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{INFO}>0$

If $\mathbf{INFO}=i$, the leading minor of order $i$ 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 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 matrix, by calling F07MRF (ZHETRF).

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07hrfe.f90)

9.2  Program Data

Program Data (f07hrfe.d)

9.3  Program Results

Program Results (f07hrfe.r)