NAG Library Routine Document

f07pnf  (zhpsv)

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:     $\mathbf{n}$ – IntegerInput

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

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

3:     $\mathbf{nrhs}$ – IntegerInput

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

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

4:     $\mathbf{ap}\left(*\right)$ – 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 $n$ Hermitian 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 $A=UD{U}^{\mathrm{H}}$ or $A=LD{L}^{\mathrm{H}}$ as computed by f07prf (zhptrf), stored as a packed triangular matrix in the same storage format as $A$.

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

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

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

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

To solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathbf{ldb}=\max \left(1,\mathbf{n}\right)$.

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:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07pnf (zhpsv) is called.

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

8:     $\mathbf{info}$ – IntegerOutput

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

6

Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, 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

Program Text (f07pnfe.f90)

10.2

Program Data

Program Data (f07pnfe.d)

10.3

Program Results

Program Results (f07pnfe.r)