NAG Library Function Document

nag_zhegv (f08snc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or Nag_ColMajor.

2:     itype – IntegerInput

On entry: specifies the problem type to be solved.

$\mathbf{itype}=1$

$Az=\lambda Bz$.

$\mathbf{itype}=2$

$ABz=\lambda z$.

$\mathbf{itype}=3$

$BAz=\lambda z$.

Constraint: $\mathbf{itype}=1$, $2$ or $3$.

3:     job – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

$\mathbf{job}=\mathrm{Nag\_EigVals}$

Only eigenvalues are computed.

$\mathbf{job}=\mathrm{Nag\_DoBoth}$

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{job}=\mathrm{Nag\_EigVals}$ or $\mathrm{Nag\_DoBoth}$.

4:     uplo – Nag_UploTypeInput

On entry: if $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the lower triangles of $A$ and $B$ are stored.

Constraint: $\mathbf{uplo}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

5:     n – IntegerInput

On entry: $n$, the order of the matrices $A$ and $B$.

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

6:     a[$\mathit{dim}$] – ComplexInput/Output

Note: the dimension, dim, of the array a must be at least $\max \left(1,\mathbf{pda}\times \mathbf{n}\right)$.

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

If $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $A_{ij}$ is stored in $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$.

If $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $A_{ij}$ is stored in $\mathbf{a}\left[\left(i-1\right)\times \mathbf{pda}+j-1\right]$.

If $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, a contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:

• if $\mathbf{itype}=1$ or $2$, $Z^{\mathrm{H}}BZ=I$;
• if $\mathbf{itype}=3$, $Z^{\mathrm{H}}{B}^{-1}Z=I$.

If $\mathbf{job}=\mathrm{Nag\_EigVals}$, the upper triangle (if $\mathbf{uplo}=\mathrm{Nag\_Upper}$) or the lower triangle (if $\mathbf{uplo}=\mathrm{Nag\_Lower}$) of a, including the diagonal, is overwritten.

7:     pda – IntegerInput

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

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

8:     b[$\mathit{dim}$] – ComplexInput/Output

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

On entry: the $n$ by $n$ Hermitian positive definite matrix $B$.

If $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $B_{ij}$ is stored in $\mathbf{b}\left[\left(j-1\right)\times \mathbf{pdb}+i-1\right]$.

If $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $B_{ij}$ is stored in $\mathbf{b}\left[\left(i-1\right)\times \mathbf{pdb}+j-1\right]$.

If $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the upper triangular part of $B$ must be stored and the elements of the array below the diagonal are not referenced.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the lower triangular part of $B$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$NE_NOERROR or NE_CONVERGENCE, the part of b containing the matrix is overwritten by the triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$.

9:     pdb – IntegerInput

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

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

10:   w[n] – doubleOutput

On exit: the eigenvalues in ascending order.

11:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

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

NE_CONVERGENCE

The algorithm failed to converge; $〈\mathit{\text{value}}〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

NE_INT

On entry, $\mathbf{itype}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itype}=1$, $2$ or $3$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}>0$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}>0$.

NE_INT_2

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}\ge \max \left(1,\mathbf{n}\right)$.

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 contact NAG for assistance.

NE_MAT_NOT_POS_DEF

If $\mathbf{fail}\mathbf{.}\mathbf{errnum}=\mathbf{n}+〈\mathit{\text{value}}〉$, for $1\le 〈\mathit{\text{value}}〉\le \mathbf{n}$, then the leading minor of order $〈\mathit{\text{value}}〉$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08snce.c)

9.2  Program Data

Program Data (f08snce.d)

9.3  Program Results

Program Results (f08snce.r)