NAG Library Routine Document

F04BFF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     UPLO – CHARACTER(1)Input

On entry: if $\mathbf{UPLO}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.

If $\mathbf{UPLO}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.

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

2:     N – INTEGERInput

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.

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

3:     KD – INTEGERInput

On entry: the number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.

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

4:     NRHS – INTEGERInput

On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.

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

5:     AB(LDAB,$*$) – REAL (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 $n$ symmetric band matrix $A$. The upper or lower triangular part of the symmetric matrix is stored in the first $\mathbf{KD}+1$ rows of the array. The $j$th column of $A$ is stored in the $j$th column of the array AB as follows:

• if $\mathbf{UPLO}=\text{'U'}$, $\mathbf{AB}\left(k+1+i-j,j\right)=a_{ij}$ for $\max \left(1,j-k\right)\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, $\mathbf{AB}\left(1+i-j,j\right)=a_{ij}$ for $j\le i\le \min \left(n,j+k\right)$.

See Section 8 below for further details.

On exit: if $\mathbf{IFAIL}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, the factor $U$ or $L$ from the Cholesky factorization $A=U^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.

6:     LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F04BFF is called.

Constraint: $\mathbf{LDAB}\ge \mathbf{KD}+1$.

7:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $r$ matrix of right-hand sides $B$.

On exit: if $\mathbf{IFAIL}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, the $n$ by $r$ solution matrix $X$.

8:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04BFF is called.

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

9:     RCOND – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{IFAIL}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $\mathbf{RCOND}=1/\left({‖A‖}_1{‖A^{-1}‖}_1\right)$.

10:   ERRBND – REAL (KIND=nag_wp)Output

On exit: if $\mathbf{IFAIL}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${‖\hat{x}-x‖}_1/{‖x‖}_1\le \mathbf{ERRBND}$, where $\hat{x}$ is a column of the computed solution returned in the array B and $x$ is the corresponding column of the exact solution $X$. If RCOND is less than machine precision, then ERRBND is returned as unity.

11:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}<0\text{ and }\mathbf{IFAIL}\ne -999$

If $\mathbf{IFAIL}=-i$, the $i$th argument had an illegal value.

$\mathbf{IFAIL}=-999$

Allocation of memory failed. The integer allocatable memory required is N, and the real allocatable memory required is $3\times \mathbf{N}$. Allocation failed before the solution could be computed.

$\mathbf{IFAIL}>0\text{ and }\mathbf{IFAIL}\le \mathbf{N}$

If $\mathbf{IFAIL}=i$, the leading minor of order $i$ of $A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.

$\mathbf{IFAIL}=\mathbf{N}+1$

RCOND is less than machine precision, so that the matrix $A$ is numerically singular. A solution to the equations $AX=B$ has nevertheless been computed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f04bffe.f90)

9.2  Program Data

Program Data (f04bffe.d)

9.3  Program Results

Program Results (f04bffe.r)