NAG Library Function Document

nag_dsbgvx (f08ubc)

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.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

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

3:    $\mathbf{range}$ – Nag_RangeTypeInput

On entry: if $\mathbf{range}=\mathrm{Nag\_AllValues}$, all eigenvalues will be found.

If $\mathbf{range}=\mathrm{Nag\_Interval}$, all eigenvalues in the half-open interval $\left(\mathbf{vl},\mathbf{vu}\right]$ will be found.

If $\mathbf{range}=\mathrm{Nag\_Indices}$, the ilth to iuth eigenvalues will be found.

Constraint: $\mathbf{range}=\mathrm{Nag\_AllValues}$, $\mathrm{Nag\_Interval}$ or $\mathrm{Nag\_Indices}$.

4:    $\mathbf{uplo}$ – 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:    $\mathbf{n}$ – IntegerInput

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

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

6:    $\mathbf{ka}$ – IntegerInput

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

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

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

7:    $\mathbf{kb}$ – IntegerInput

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

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

Constraint: $\mathbf{ka}\ge \mathbf{kb}\ge 0$.

8:    $\mathbf{ab}\left[\mathit{dim}\right]$ – doubleInput/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 $n$ 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 $\mathbf{ab}\left[k_a+i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max \left(1,j-k_a\right),\dots ,j$;
• 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 $i=j,\dots ,\min \left(n,j+k_a\right)$;
• 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 $j=i,\dots ,\min \left(n,i+k_a\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k_a+j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max \left(1,i-k_a\right),\dots ,i$.

On exit: the contents of ab are overwritten.

9:    $\mathbf{pdab}$ – 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: $\mathbf{pdab}\ge \mathbf{ka}+1$.

10:  $\mathbf{bb}\left[\mathit{dim}\right]$ – doubleInput/Output

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

On entry: the upper or lower triangle of the $n$ by $n$ symmetric positive definite band matrix $B$.

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

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

On exit: the factor $S$ from the split Cholesky factorization $B=S^{\mathrm{T}}S$, as returned by nag_dpbstf (f08ufc).

11:  $\mathbf{pdbb}$ – IntegerInput

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

Constraint: $\mathbf{pdbb}\ge \mathbf{kb}+1$.

12:  $\mathbf{q}\left[\mathit{dim}\right]$ – doubleOutput

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

• $\max \left(1,\mathbf{pdq}\times \mathbf{n}\right)$ when $\mathbf{job}=\mathrm{Nag\_DoBoth}$;
• $1$ otherwise.

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

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

On exit: if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.

If $\mathbf{job}=\mathrm{Nag\_EigVals}$, q is not referenced.

13:  $\mathbf{pdq}$ – IntegerInput

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

Constraints:

• if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, $\mathbf{pdq}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{pdq}\ge 1$.

14:  $\mathbf{vl}$ – doubleInput

15:  $\mathbf{vu}$ – doubleInput

On entry: if $\mathbf{range}=\mathrm{Nag\_Interval}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $\mathbf{range}=\mathrm{Nag\_AllValues}$ or $\mathrm{Nag\_Indices}$, vl and vu are not referenced.

Constraint: if $\mathbf{range}=\mathrm{Nag\_Interval}$, $\mathbf{vl}<\mathbf{vu}$.

16:  $\mathbf{il}$ – IntegerInput

17:  $\mathbf{iu}$ – IntegerInput

On entry: if $\mathbf{range}=\mathrm{Nag\_Indices}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

If $\mathbf{range}=\mathrm{Nag\_AllValues}$ or $\mathrm{Nag\_Interval}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\mathrm{Nag\_Indices}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\mathrm{Nag\_Indices}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

18:  $\mathbf{abstol}$ – doubleInput

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to

$$\mathbf{abstol}+\epsilon \max \left(\left|a\right|,\left|b\right|\right)\text{,}$$

where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_1$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold $2\times \mathbf{nag\_real\_safe\_small\_number}\left( \right)$, not zero. If this function returns with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{nag\_real\_safe\_small\_number}\left( \right)$. See Demmel and Kahan (1990).

19:  $\mathbf{m}$ – Integer *Output

On exit: the total number of eigenvalues found. $0\le \mathbf{m}\le \mathbf{n}$.

If $\mathbf{range}=\mathrm{Nag\_AllValues}$, $\mathbf{m}=\mathbf{n}$.

If $\mathbf{range}=\mathrm{Nag\_Indices}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

20:  $\mathbf{w}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the eigenvalues in ascending order.

21:  $\mathbf{z}\left[\mathit{dim}\right]$ – doubleOutput

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

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

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

• if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the first m columns of $Z$ contain the eigenvectors corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left[i-1\right]$. The eigenvectors are normalized so that $Z^{\mathrm{T}}BZ=I$;
• if an eigenvector fails to converge ($\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

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

22:  $\mathbf{pdz}$ – IntegerInput

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

Constraints:

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

23:  $\mathbf{jfail}\left[\mathit{dim}\right]$ – IntegerOutput

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

On exit: if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, then

• if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the first m elements of jfail are zero;
• if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.

If $\mathbf{job}=\mathrm{Nag\_EigVals}$, jfail is not referenced.

24:  $\mathbf{fail}$ – 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

The algorithm failed to converge; $〈\mathit{\text{value}}〉$ eigenvectors did not converge. Their indices are stored in array jfail.

NE_ENUM_INT_2

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

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

NE_ENUM_INT_3

On entry, $\mathbf{range}=〈\mathit{\text{value}}〉$, $\mathbf{il}=〈\mathit{\text{value}}〉$, $\mathbf{iu}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{range}=\mathrm{Nag\_Indices}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
if $\mathbf{range}=\mathrm{Nag\_Indices}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

NE_ENUM_REAL_2

On entry, $\mathbf{range}=〈\mathit{\text{value}}〉$, $\mathbf{vl}=〈\mathit{\text{value}}〉$ and $\mathbf{vu}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{range}=\mathrm{Nag\_Interval}$, $\mathbf{vl}<\mathbf{vu}$.

NE_INT

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

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

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

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

NE_INT_2

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

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

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

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_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 nag_dpbstf (f08ufc) returned $\mathbf{fail}\mathbf{.}\mathbf{errnum}=〈\mathit{\text{value}}〉$: $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

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 (f08ubce.c)

10.2

Program Data

Program Data (f08ubce.d)

10.3

Program Results

Program Results (f08ubce.r)