NAG Library Routine Document

F04CBF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     N – INTEGERInput

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

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

2:     KL – INTEGERInput

On entry: the number of subdiagonals $k_l$, within the band of $A$.

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

3:     KU – INTEGERInput

On entry: the number of superdiagonals $k_u$, within the band of $A$.

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

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,$*$) – 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 $n$ matrix $A$.

The matrix is stored in rows $k_l+1$ to $2k_l+k_u+1$; the first $k_l$ rows need not be set, more precisely, the element $A_{ij}$ must be stored in

$$\mathbf{AB}\left(k_l+k_u+1+i-j,j\right)=A_{ij}\text{\hspace{1em} for }\max \left(1,j-k_u\right)\le i\le \min \left(n,j+k_l\right)\text{.}$$

See Section 8 for further details.

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, AB is overwritten by details of the factorization.

The upper triangular band matrix $U$, with $k_l+k_u$ superdiagonals, is stored in rows $1$ to $k_l+k_u+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows $k_l+k_u+2$ to $2k_l+k_u+1$.

6:     LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F04CBF is called.

Constraint: $\mathbf{LDAB}\ge 2\times \mathbf{KL}+\mathbf{KU}+1$.

7:     IPIV(N) – INTEGER arrayOutput

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row $\mathbf{IPIV}\left(i\right)$. $\mathbf{IPIV}\left(i\right)=i$ indicates a row interchange was not required.

8:     B(LDB,$*$) – COMPLEX (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 $X$.

9:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04CBF is called.

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

10:   RCOND – REAL (KIND=nag_wp)Output

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

11:   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.

12:   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 real allocatable memory required is N, and the complex allocatable memory required is $2\times \mathbf{N}$. In this case the factorization and the solution $X$ have been computed, but RCOND and ERRBND have not been computed.

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

If $\mathbf{IFAIL}=i$, $u_{ii}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be 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 (f04cbfe.f90)

9.2  Program Data

Program Data (f04cbfe.d)

9.3  Program Results

Program Results (f04cbfe.r)