NAG Library Routine Document

F07QPF (ZSPSVX)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     FACT – CHARACTER(1)Input

On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.

$\mathbf{FACT}=\text{'F'}$

AFP and IPIV contain the factorized form of the matrix $A$. AFP and IPIV will not be modified.

$\mathbf{FACT}=\text{'N'}$

The matrix $A$ will be copied to AFP and factorized.

Constraint: $\mathbf{FACT}=\text{'F'}$ or $\text{'N'}$.

2:     UPLO – CHARACTER(1)Input

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

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

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

3:     N – INTEGERInput

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

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

4:     NRHS – INTEGERInput

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

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

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

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 by columns.

More precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

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

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

On entry: if $\mathbf{FACT}=\text{'F'}$, AFP contains 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 $A$.

On exit: if $\mathbf{FACT}=\text{'N'}$, AFP contains 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 $A$.

7:     IPIV(N) – INTEGER arrayInput/Output

On entry: if $\mathbf{FACT}=\text{'F'}$, IPIV contains details of the interchanges and the block structure of $D$, as determined by F07QRF (ZSPTRF).

• if $\mathbf{IPIV}\left(i\right)=k>0$, $d_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if $\mathbf{UPLO}=\text{'U'}$ and $\mathbf{IPIV}\left(i-1\right)=\mathbf{IPIV}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if $\mathbf{UPLO}=\text{'L'}$ and $\mathbf{IPIV}\left(i\right)=\mathbf{IPIV}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.

On exit: if $\mathbf{FACT}=\text{'N'}$, IPIV contains details of the interchanges and the block structure of $D$, as determined by F07QRF (ZSPTRF), as described above.

8:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput

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

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

9:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07QPF (ZSPSVX) is called.

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

10:   X(LDX,$*$) – COMPLEX (KIND=nag_wp) arrayOutput

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

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

11:   LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07QPF (ZSPSVX) is called.

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

12:   RCOND – REAL (KIND=nag_wp)Output

On exit: the estimate of the reciprocal condition number of the matrix $A$. If $\mathbf{RCOND}=0.0$, the matrix may be exactly singular. This condition is indicated by $\mathbf{INFO}>\mathbf{0}\text{ and }\mathbf{INFO}\le \mathbf{N}$. Otherwise, if RCOND is less than the machine precision, the matrix is singular to working precision. This condition is indicated by $\mathbf{INFO}=\mathbf{N}+\mathbf{1}$.

13:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{INFO}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\hat{x}}_j-x_j‖}_{\infty }/{‖x_j‖}_{\infty }\le \mathbf{FERR}\left(j\right)$ where ${\hat{x}}_j$ is the $j$th column of the computed solution returned in the array X and $x_j$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.

14:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{INFO}=\mathbf{0}$ or $\mathbf{N}+\mathbf{1}$, an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_j$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_j$ an exact solution).

15:   WORK($2\times \mathbf{N}$) – COMPLEX (KIND=nag_wp) arrayWorkspace

16:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

17:   INFO – INTEGEROutput

On exit: $\mathbf{INFO}=0$ unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

$\mathbf{INFO}<0$

If $\mathbf{INFO}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

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

If $\mathbf{INFO}\le \mathbf{N}$, $d\left(i,i\right)$ is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be computed. $\mathbf{RCOND}=0.0$ is returned.

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

$D$ is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07qpfe.f90)

9.2  Program Data

Program Data (f07qpfe.d)

9.3  Program Results

Program Results (f07qpfe.r)