NAG Library Routine Document

F07MNF (ZHESV)

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 $A$ is stored.

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

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

2:     N – INTEGERInput

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

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

3:     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:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ Hermitian matrix $A$.

• If $\mathbf{UPLO}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{UPLO}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if $\mathbf{INFO}=\mathbf{0}$, 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 F07MRF (ZHETRF).

5:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07MNF (ZHESV) is called.

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

6:     IPIV($*$) – INTEGER arrayOutput

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

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.

7:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

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 $X$.

8:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07MNF (ZHESV) is called.

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

9:     WORK($\max \left(1,\mathbf{LWORK}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{WORK}\left(1\right)$ returns the optimal LWORK.

10:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F07MNF (ZHESV) is called.

$\mathbf{LWORK}\ge 1$, and for best performance $\mathbf{LWORK}\ge \max \left(1,\mathbf{N}\times \mathit{nb}\right)$, where $\mathit{nb}$ is the optimal block size for F07MRF (ZHETRF).

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

11:   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.

$\mathbf{INFO}>0$

If $\mathbf{INFO}=i$, $d_{ii}$ is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07mnfe.f90)

9.2  Program Data

Program Data (f07mnfe.d)

9.3  Program Results

Program Results (f07mnfe.r)