NAG FL Interface
f07paf (dspsv)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{uplo}$ – 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 $\text{'L'}$.

2: $\mathbf{n}$ – Integer Input

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

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

3: $\mathbf{nrhs}$ – Integer Input

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

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

4: $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) array Input/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 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 $i\ge j$.

On exit: 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 f07pdf, stored as a packed triangular matrix in the same storage format as $A$.

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

On exit: details of the interchanges and the block structure of $D$. More precisely,

• 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 $\mathbf{ipiv}\left(i-1\right)=\mathbf{ipiv}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\overline{d}}_{i,i-1}\\ {\overline{d}}_{i,i-1}& d_{ii}\end{array}\right)$ 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.

6: $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Real (Kind=nag_wp) array Input/Output

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

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

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

7: $\mathbf{ldb}$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07paf is called.

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

8: $\mathbf{info}$ – 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.

$\mathbf{info}>0$

Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results