NAG Library Function Document

nag_zstein (f08jxc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

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

2:    $\mathbf{n}$ – IntegerInput

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

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

3:    $\mathbf{d}\left[\mathit{dim}\right]$ – const doubleInput

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

On entry: the diagonal elements of the tridiagonal matrix $T$.

4:    $\mathbf{e}\left[\mathit{dim}\right]$ – const doubleInput

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

On entry: the off-diagonal elements of the tridiagonal matrix $T$.

5:    $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of eigenvectors to be returned.

Constraint: $0\le \mathbf{m}\le \mathbf{n}$.

6:    $\mathbf{w}\left[\mathit{dim}\right]$ – const doubleInput

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

On entry: the eigenvalues of the tridiagonal matrix $T$ stored in $\mathbf{w}\left[0\right]$ to $\mathbf{w}\left[m-1\right]$, as returned by nag_dstebz (f08jjc) with $\mathbf{rank}=\mathrm{Nag\_ByBlock}$. Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.

Constraint: if $\mathbf{iblock}\left[\mathit{i}\right]=\mathbf{iblock}\left[\mathit{i}+1\right]$, $\mathbf{w}\left[\mathit{i}\right]\le \mathbf{w}\left[\mathit{i}+1\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{m}-2$.

7:    $\mathbf{iblock}\left[\mathit{dim}\right]$ – const IntegerInput

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

On entry: the first $m$ elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by nag_dstebz (f08jjc) with $\mathbf{rank}=\mathrm{Nag\_ByBlock}$. If the eigenvalues were not computed by nag_dstebz (f08jjc) with $\mathbf{rank}=\mathrm{Nag\_ByBlock}$, set $\mathbf{iblock}\left[\mathit{i}-1\right]$ to $1$, for $\mathit{i}=1,2,\dots ,m$.

Constraint: $\mathbf{iblock}\left[\mathit{i}\right]\le \mathbf{iblock}\left[\mathit{i}+1\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{m}-2$.

8:    $\mathbf{isplit}\left[\mathit{dim}\right]$ – const IntegerInput

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

On entry: the points at which $T$ breaks up into sub-matrices, as returned by nag_dstebz (f08jjc) with $\mathbf{rank}=\mathrm{Nag\_ByBlock}$. If the eigenvalues were not computed by nag_dstebz (f08jjc) with $\mathbf{rank}=\mathrm{Nag\_ByBlock}$, set $\mathbf{isplit}\left[0\right]$ to n.

9:    $\mathbf{z}\left[\mathit{dim}\right]$ – ComplexOutput

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

• $\max \left(1,\mathbf{pdz}\times \mathbf{m}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{n}\times \mathbf{pdz}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $Z$ is stored in

• $\mathbf{z}\left[\left(j-1\right)\times \mathbf{pdz}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{z}\left[\left(i-1\right)\times \mathbf{pdz}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: the $m$ eigenvectors, stored as columns of $Z$; the $i$th column corresponds to the $i$th specified eigenvalue, unless $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE (in which case see Section 6).

10:  $\mathbf{pdz}$ – IntegerInput

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

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{m}\right)$.

11:  $\mathbf{ifailv}\left[\mathbf{m}\right]$ – IntegerOutput

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{errnum}=i>0$, the first $i$ elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to $0$.

12:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

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

NE_CONSTRAINT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\mathbf{iblock}\left[\mathit{i}\right]\mathbf{iblock}\left[\mathit{i}+1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{w}\left[\mathit{i}\right]\mathbf{w}\left[\mathit{i}+1\right]=〈\mathit{\text{value}}〉$.
Constraint: , for $\mathit{i}=0,1,\dots ,\mathbf{m}-2$

NE_CONVERGENCE

$〈\mathit{\text{value}}〉$ eigenvectors (as indicated by argument ifailv) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.

NE_INT

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

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

NE_INT_2

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

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

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

NE_INT_ARRAY

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{iblock}\left[\mathit{i}\right]\mathbf{iblock}\left[\mathit{i}+1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{iblock}\left[\mathit{i}\right]\le \mathbf{iblock}\left[\mathit{i}+1\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{m}-2$

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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example