NAG Library Routine Document

f08ubf  (dsbgvx)

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 triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

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

4:     $\mathbf{n}$ – IntegerInput

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

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

5:     $\mathbf{ka}$ – IntegerInput

On entry: if $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_a$, of the matrix $A$.

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

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

6:     $\mathbf{kb}$ – IntegerInput

On entry: if $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_b$, of the matrix $B$.

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

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

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

On exit: the contents of ab are overwritten.

8:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f08ubf (dsbgvx) is called.

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

9:     $\mathbf{bb}\left(\mathbf{ldbb},*\right)$ – Real (Kind=nag_wp) arrayInput/Output

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

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

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

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

On exit: the factor $S$ from the split Cholesky factorization $B=S^{\mathrm{T}}S$, as returned by f08uff (dpbstf).

10:   $\mathbf{ldbb}$ – IntegerInput

On entry: the first dimension of the array bb as declared in the (sub)program from which f08ubf (dsbgvx) is called.

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

11:   $\mathbf{q}\left(\mathbf{ldq},*\right)$ – 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 $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.

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

12:   $\mathbf{ldq}$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08ubf (dsbgvx) is called.

Constraints:

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

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

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

15:   $\mathbf{il}$ – IntegerInput

16:   $\mathbf{iu}$ – 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 $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

17:   $\mathbf{abstol}$ – 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 $C$ 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 $\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{x02amf}\left( \right)$. See Demmel and Kahan (1990).

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

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

On exit: the eigenvalues in ascending order.

20:   $\mathbf{z}\left(\mathbf{ldz},*\right)$ – 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 eigenvectors corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$. The eigenvectors are normalized so that $Z^{\mathrm{T}}BZ=I$;
• if an eigenvector fails to converge ($\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$), 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.

21:   $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08ubf (dsbgvx) is called.

Constraints:

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

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

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

24:   $\mathbf{jfail}\left(*\right)$ – 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 $\mathbf{info}=\mathbf{1}\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$, jfail contains the indices of the eigenvectors that failed to converge.

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

25:   $\mathbf{info}$ – 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.

$\mathbf{info}=1\hspace{0.17em}\text{to}\hspace{0.17em}\mathbf{n}$

If $\mathbf{info}=i$, $i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.

$\mathbf{info}>\mathbf{n}$

f08uff (dpbstf) returned an error code; i.e., if $\mathbf{info}=\mathbf{n}+i$, for $1\le i\le \mathbf{n}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08ubfe.f90)

10.2

Program Data

Program Data (f08ubfe.d)

10.3

Program Results

Program Results (f08ubfe.r)