NAG Library Routine Document

F04DJF

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:     AP($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array AP must be at least $\max \left(1,\mathbf{N}\times \left(\mathbf{N}+1\right)/2\right)$.

On entry: the $n$ by $n$ symmetric matrix $A$, packed column-wise in a linear array. The $j$th column of the matrix $A$ is stored in the array AP as follows:

• if $\mathbf{UPLO}=\text{'U'}$, $\mathbf{AP}\left(i+\left(j-1\right)j/2\right)=a_{ij}$ for $1\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, $\mathbf{AP}\left(i+\left(j-1\right)\left(2n-j\right)/2\right)=a_{ij}$ for $j\le i\le n$.

See Section 8 below for further details.

On exit: if no constraints are violated, 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 F07QRF (ZSPTRF), stored as a packed triangular matrix in the same storage format as $A$.

5:     IPIV(N) – INTEGER arrayOutput

On exit: if no constraints are violated, details of the interchanges and the block structure of $D$, as determined by F07QRF (ZSPTRF).

• 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.

6:     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 $X$.

7:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04DJF is called.

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

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

9:     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.

10:   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 $2\times \mathbf{N}$. Allocation failed before the solution could be computed.

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

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 (f04djfe.f90)

9.2  Program Data

Program Data (f04djfe.d)

9.3  Program Results

Program Results (f04djfe.r)