NAG FL Interface
f07qpf (zspsvx)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{fact}$ – 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: $\mathbf{uplo}$ – 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: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{nrhs}$ – Integer Input

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: $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) array Input

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

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

6: $\mathbf{afp}\left(*\right)$ – Complex (Kind=nag_wp) array Input/Output

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

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, 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, stored as a packed triangular matrix in the same storage format as $A$.

7: $\mathbf{ipiv}\left(\mathbf{n}\right)$ – Integer array Input/Output

On entry: if $\mathbf{fact}=\text{'F'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07qrf.

• if $\mathbf{ipiv}\left(i\right)=k>0$, $d_{ii}$ is a $1\times 1$ 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 $\mathbf{ipiv}\left(i-1\right)=\mathbf{ipiv}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2\times 2$ pivot block and the $(i-1)$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 $2\times 2$ pivot block and the $(i+1)$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, as described above.

8: $\mathbf{b}(\mathbf{ldb},*)$ – Complex (Kind=nag_wp) array Input

Note: the second dimension of the array b must be at least $\max (1,\mathbf{nrhs})$.

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

9: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07qpf is called.

Constraint: $\mathbf{ldb}\ge \max (1,\mathbf{n})$.

10: $\mathbf{x}(\mathbf{ldx},*)$ – Complex (Kind=nag_wp) array Output

Note: the second dimension of the array x must be at least $\max (1,\mathbf{nrhs})$.

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

11: $\mathbf{ldx}$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f07qpf is called.

Constraint: $\mathbf{ldx}\ge \max (1,\mathbf{n})$.

12: $\mathbf{rcond}$ – 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 $\mathbf{info}>\mathbf{0}\hspace{0.17em}\text{and}\hspace{0.17em}\mathbf{info}\le \mathbf{n}$. 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: $\mathbf{ferr}\left(\mathbf{nrhs}\right)$ – Real (Kind=nag_wp) array Output

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 ${\Vert {\hat{x}}_j-x_j\Vert }_{\infty }/{\Vert x_j\Vert }_{\infty }\le \mathbf{ferr}\left(j\right)$ 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: $\mathbf{berr}\left(\mathbf{nrhs}\right)$ – Real (Kind=nag_wp) array Output

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: $\mathbf{work}\left(2\times \mathbf{n}\right)$ – Complex (Kind=nag_wp) array Workspace

16: $\mathbf{rwork}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Workspace

17: $\mathbf{info}$ – Integer Output

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

6 Error Indicators and Warnings

$\mathbf{info}<0$

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

$\mathbf{info}>0\hspace{0.17em}\text{and}\hspace{0.17em}\mathbf{info}\le \mathbf{n}$

Element $⟨\mathit{\text{value}}⟩$ of the diagonal 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 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results