NAG CL Interface
f08hcc (dsbevd)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{order}$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface 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_JobType Input

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_UploType Input

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: $\mathbf{n}$ – Integer Input

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

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

5: $\mathbf{kd}$ – Integer Input

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

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

On entry: the upper or lower triangle of the $n\times n$ symmetric 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 $\mathbf{ab}\left[k_d+i-j+(j-1)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max (1,j-k_d),\dots ,j$;

if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[i-j+(j-1)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min (n,j+k_d)$;

if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[j-i+(i-1)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min (n,i+k_d)$;

if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[k_d+j-i+(i-1)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max (1,i-k_d),\dots ,i$.

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: $\mathbf{pdab}$ – Integer Input

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: $\mathbf{w}\left[\mathit{dim}\right]$ – double Output

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

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

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

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

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

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

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

On exit: if $\mathbf{job}=\mathrm{Nag\_EigVecs}$, z is overwritten by the orthogonal 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: $\mathbf{pdz}$ – Integer Input

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

Constraints:

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

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

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 $⟨\mathit{\text{value}}⟩/(\mathbf{n}+1)$ 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 $\mathbf{job}=\mathrm{Nag\_EigVecs}$, $\mathbf{pdz}\ge \max (1,\mathbf{n})$;
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 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results