NAG Library Routine Document

F04DHF

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:     NRHS – INTEGERInput

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

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

4:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ complex symmetric matrix $A$.

If $\mathbf{UPLO}=\text{'U'}$, the leading N by N upper triangular part of the array A contains the upper triangular part of the matrix $A$, and the strictly lower triangular part of A is not referenced.

If $\mathbf{UPLO}=\text{'L'}$, the leading N by N lower triangular part of the array A contains the lower triangular part of the matrix $A$, and the strictly upper triangular part of A is not referenced.

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by F07NRF (ZSYTRF).

5:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F04DHF is called.

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

6:     IPIV(N) – INTEGER arrayOutput

On exit: if $\mathbf{IFAIL}\ge \mathbf{0}$, details of the interchanges and the block structure of $D$, as determined by F07NRF (ZSYTRF).

• If $\mathbf{IPIV}\left(k\right)>0$, then rows and columns $k$ and $\mathbf{IPIV}\left(k\right)$ were interchanged, and $d_{kk}$ is a $1$ by $1$ diagonal block;
• if $\mathbf{UPLO}=\text{'U'}$ and $\mathbf{IPIV}\left(k\right)=\mathbf{IPIV}\left(k-1\right)<0$, then rows and columns $k-1$ and $-\mathbf{IPIV}\left(k\right)$ were interchanged and $d_{k-1:k,k-1:k}$ is a $2$ by $2$ diagonal block;
• if $\mathbf{UPLO}=\text{'L'}$ and $\mathbf{IPIV}\left(k\right)=\mathbf{IPIV}\left(k+1\right)<0$, then rows and columns $k+1$ and $-\mathbf{IPIV}\left(k\right)$ were interchanged and $d_{k:k+1,k:k+1}$ is a $2$ by $2$ diagonal block.

7:     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 $X$.

8:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04DHF is called.

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

9:     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 $\mathbf{RCOND}=1/\left({‖A‖}_1{‖A^{-1}‖}_1\right)$.

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 real allocatable memory required is N, and the complex allocatable memory required is $\max \left(2\times \mathbf{N},\mathbf{LWORK}\right)$, where LWORK is the optimum workspace required by F07NNF (ZSYSV). If this failure occurs it may be possible to solve the equations by calling the packed storage version of F04DHF, F04DJF, or by calling F07NNF (ZSYSV) directly with less than the optimum workspace (see Chapter F07).

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

If $\mathbf{IFAIL}=i$, $d_{ii}$ is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ 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 (f04dhfe.f90)

9.2  Program Data

Program Data (f04dhfe.d)

9.3  Program Results

Program Results (f04dhfe.r)