NAG Library Routine Document

f08hbf (dsbevx)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{jobz}$ – Character(1)Input

On entry: indicates whether eigenvectors are computed.

$\mathbf{jobz}=\text{'N'}$

Only eigenvalues are computed.

$\mathbf{jobz}=\text{'V'}$

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{jobz}=\text{'N'}$ or $\text{'V'}$.

2:     $\mathbf{range}$ – Character(1)Input

On entry: if $\mathbf{range}=\text{'A'}$, all eigenvalues will be found.

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

If $\mathbf{range}=\text{'I'}$, the ilth to iuth eigenvalues will be found.

Constraint: $\mathbf{range}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

3:     $\mathbf{uplo}$ – Character(1)Input

On entry: if $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $A$ is stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $A$ is stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

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 $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_d$, of the matrix $A$.

If $\mathbf{uplo}=\text{'L'}$, the number of subdiagonals, $k_d$, of the matrix $A$.

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

6:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

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

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

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

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:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f08hbf (dsbevx) is called.

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

8:     $\mathbf{q}\left(\mathbf{ldq},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array q must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobz}=\text{'V'}$, the $n$ by $n$ orthogonal matrix used in the reduction to tridiagonal form.

If $\mathbf{jobz}=\text{'N'}$, q is not referenced.

9:     $\mathbf{ldq}$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08hbf (dsbevx) is called.

Constraints:

• if $\mathbf{jobz}=\text{'V'}$, $\mathbf{ldq}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{ldq}\ge 1$.

10:   $\mathbf{vl}$ – Real (Kind=nag_wp)Input

11:   $\mathbf{vu}$ – Real (Kind=nag_wp)Input

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

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

12:   $\mathbf{il}$ – IntegerInput

13:   $\mathbf{iu}$ – IntegerInput

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

If $\mathbf{range}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

14:   $\mathbf{abstol}$ – Real (Kind=nag_wp)Input

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 $A$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold $2\times \mathbf{x02amf}\left( \right)$, not zero. If this routine returns with $\mathbf{info}>\mathbf{0}$, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{x02amf}\left( \right)$. See Demmel and Kahan (1990).

15:   $\mathbf{m}$ – IntegerOutput

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

If $\mathbf{range}=\text{'A'}$, $\mathbf{m}=\mathbf{n}$.

If $\mathbf{range}=\text{'I'}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

16:   $\mathbf{w}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the first m elements contain the selected eigenvalues in ascending order.

17:   $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array z must be at least $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobz}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$;
• if an eigenvector fails to converge ($\mathbf{info}>\mathbf{0}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

If $\mathbf{jobz}=\text{'N'}$, z is not referenced.

Note:  you must ensure that at least $\max \left(1,\mathbf{m}\right)$ columns are supplied in the array z; if $\mathbf{range}=\text{'V'}$, the exact value of m is not known in advance and an upper bound of at least n must be used.

18:   $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08hbf (dsbevx) is called.

Constraints:

• if $\mathbf{jobz}=\text{'V'}$, $\mathbf{ldz}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{ldz}\ge 1$.

19:   $\mathbf{work}\left(7\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayWorkspace

20:   $\mathbf{iwork}\left(5\times \mathbf{n}\right)$ – Integer arrayWorkspace

21:   $\mathbf{jfail}\left(*\right)$ – Integer arrayOutput

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

On exit: if $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m elements of jfail are zero;
• if $\mathbf{info}>\mathbf{0}$, jfail contains the indices of the eigenvectors that failed to converge.

If $\mathbf{jobz}=\text{'N'}$, jfail is not referenced.

22:   $\mathbf{info}$ – IntegerOutput

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$\mathbf{info}>0$

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

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08hbfe.f90)

10.2

Program Data

Program Data (f08hbfe.d)

10.3

Program Results

Program Results (f08hbfe.r)