NAG Library Routine Document

F04BCF

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:     NRHS – INTEGERInput

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

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

3:     DL($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array DL must be at least $\max \left(1,\mathbf{N}-1\right)$.

On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, DL is overwritten by the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.

4:     D($*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: must contain the $n$ diagonal elements of the matrix $A$.

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, D is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.

5:     DU($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array DU must be at least $\max \left(1,\mathbf{N}-1\right)$.

On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, DU is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.

6:     DU2($\mathbf{N}-2$) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, DU2 returns the $\left(n-2\right)$ elements of the second superdiagonal of $U$.

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 $\mathbf{IPIV}\left(i\right)$. $\mathbf{IPIV}\left(i\right)$ will always be either $i$ or $\left(i+1\right)$; $\mathbf{IPIV}\left(i\right)=i$ indicates a row interchange was not required.

8:     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 $X$.

9:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04BCF is called.

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

10:   RCOND – REAL (KIND=nag_wp)Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as $\mathbf{RCOND}=1/\left({‖A‖}_1{‖A^{-1}‖}_1\right)$.

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 integer allocatable memory required is N, and the real 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 (f04bcfe.f90)

9.2  Program Data

Program Data (f04bcfe.d)

9.3  Program Results

Program Results (f04bcfe.r)