NAG C Library Function Document

nag_zhetrf (f07mrc)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     order – Nag_OrderTypeInput

On entry: the order parameter specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order $=$ Nag_RowMajor. See Section 2.2.1.4 of the Essential Introduction for a more detailed explanation of the use of this parameter.

Constraint: order $=$ Nag_RowMajor or Nag_ColMajor.

2:     uplo – Nag_UploType Input

On entry: indicates whether the upper or lower triangular part of $A$ is stored and how $A$ has been factorized, as follows:

• if  $=$ Nag_Upper, the upper triangular part of $A$ is stored and $A$ is factorized as $PUD{U}^{\mathit{H}}{P}^{\mathit{T}}$, where $U$ is upper triangular;
• if  $=$ Nag_Lower, the lower triangular part of $A$ is stored and $A$ is factorized as $PLD{L}^{\mathit{H}}{P}^{\mathit{T}}$, where $L$ is lower triangular.

Constraint:  $=$ Nag_Upper or Nag_Lower.

3:     n – Integer Input

On entry: $n$, the order of the matrix $A$.

Constraint: .

4:     a[$\mathit{dim}$] – Complex Input/Output

Note: the dimension, dim, of the array a must be at least .

If order $=$ Nag_ColMajor, the $\left(i,j\right)$th element of the matrix $A$ is stored in  and if order $=$ Nag_RowMajor, the $\left(i,j\right)$th element of the matrix $A$ is stored in .

On entry: the $n$ by $n$ Hermitian matrix $A$. If  $=$ Nag_Upper, the upper triangle of $A$ must be stored and the elements of the array below the diagonal are not referenced; if  $=$ Nag_Lower, the lower triangle of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the upper or lower triangle of $A$ is overwritten by details of the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ as specified by uplo.

5:     pda – Integer Input

On entry: the stride separating row or column elements (depending on the value of order ) of the matrix $A$ in the array a .

Constraint: .

6:     ipiv[$\mathit{dim}$] – Integer Output

Note: the dimension, dim, of the array ipiv must be at least .

On exit: details of the interchanges and the block structure of $D$. More precisely, if , $d_{ii}$ 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  $=$ Nag_Upper and , $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& d_{ii}\end{array}\right)$ 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  $=$ Nag_Lower and , $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ 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:     fail – NagError *Input/Output

The NAG error parameter (see the Essential Introduction).

6  Error Indicators and Warnings

NE_INT

On entry, .
Constraint: .

On entry, .
Constraint: .

NE_INT_2

On entry, , .
Constraint: .

NE_SINGULAR

The block diagonal matrix $D$ is exactly singular.

NE_ALLOC_FAIL

Memory allocation failed.

NE_BAD_PARAM

On entry, parameter $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

7  Accuracy

8  Further Comments

• nag_zhetrs (f07msc) to solve $AX=B$;
• nag_zhecon (f07muc) to estimate the condition number of $A$;
• nag_zhetri (f07mwc) to compute the inverse of $A$.

9  Example

9.1  Program Text

9.2  Program Data

9.3  Program Results