NAG Library Function Document

nag_zhbevd (f08hqc)

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:    $\mathbf{job}$ – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{job}=\mathrm{Nag\_DoNothing}$ or $\mathrm{Nag\_EigVecs}$.

3:    $\mathbf{uplo}$ – Nag_UploTypeInput

On entry: indicates whether the upper or lower triangular part of $A$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Upper}$

The upper triangular part of $A$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Lower}$

The lower triangular part of $A$ is stored.

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

4:    $\mathbf{n}$ – IntegerInput

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

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

5:    $\mathbf{kd}$ – IntegerInput

On entry: if $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the number of superdiagonals, $k_d$, of the matrix $A$.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the number of subdiagonals, $k_d$, of the matrix $A$.

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

6:    $\mathbf{ab}\left[\mathit{dim}\right]$ – ComplexInput/Output

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

On entry: the upper or lower triangle of the $n$ by $n$ Hermitian band matrix $A$.

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of $A_{ij}$, depends on the order and uplo arguments as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k_d+i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max \left(1,j-k_d\right),\dots ,j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min \left(n,j+k_d\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min \left(n,i+k_d\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k_d+j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max \left(1,i-k_d\right),\dots ,i$.

On exit: ab is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in ab using the same storage format as described above.

7:    $\mathbf{pdab}$ – IntegerInput

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

Constraint: $\mathbf{pdab}\ge \mathbf{kd}+1$.

8:    $\mathbf{w}\left[\mathit{dim}\right]$ – doubleOutput

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

On exit: the eigenvalues of the matrix $A$ in ascending order.

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

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

• $\max \left(1,\mathbf{pdz}\times \mathbf{n}\right)$ when $\mathbf{job}=\mathrm{Nag\_EigVecs}$;
• $1$ when $\mathbf{job}=\mathrm{Nag\_DoNothing}$.

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: if $\mathbf{job}=\mathrm{Nag\_EigVecs}$, z is overwritten by the unitary matrix $Z$ which contains the eigenvectors of $A$. The $i$th column of $Z$ contains the eigenvector which corresponds to the eigenvalue $\mathbf{w}\left[i-1\right]$.

If $\mathbf{job}=\mathrm{Nag\_DoNothing}$, z is not referenced.

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 $\mathbf{job}=\mathrm{Nag\_EigVecs}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{job}=\mathrm{Nag\_DoNothing}$, $\mathbf{pdz}\ge 1$.

11:  $\mathbf{fail}$ – 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_CONVERGENCE

If $\mathbf{fail}\mathbf{.}\mathbf{errnum}=〈\mathit{\text{value}}〉$ and $\mathbf{job}=\mathrm{Nag\_DoNothing}$, the algorithm failed to converge; $〈\mathit{\text{value}}〉$ elements of an intermediate tridiagonal form did not converge to zero; if $\mathbf{fail}\mathbf{.}\mathbf{errnum}=〈\mathit{\text{value}}〉$ and $\mathbf{job}=\mathrm{Nag\_EigVecs}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $〈\mathit{\text{value}}〉/\left(\mathbf{n}+1\right)$ through $〈\mathit{\text{value}}〉\mathrm{ mod }\left(\mathbf{n}+1\right)$.

NE_ENUM_INT_2

On entry, $\mathbf{job}=〈\mathit{\text{value}}〉$, $\mathbf{pdz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{job}=\mathrm{Nag\_EigVecs}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{n}\right)$;
if $\mathbf{job}=\mathrm{Nag\_DoNothing}$, $\mathbf{pdz}\ge 1$.

NE_INT

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

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

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

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

NE_INT_2

On entry, $\mathbf{pdab}=〈\mathit{\text{value}}〉$ and $\mathbf{kd}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdab}\ge \mathbf{kd}+1$.

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

10.1

Program Text

Program Text (f08hqce.c)

10.2

Program Data

Program Data (f08hqce.d)

10.3

Program Results

Program Results (f08hqce.r)