F08GBF (DSPEVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GBF (DSPEVX)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08GBF (DSPEVX) computes selected eigenvalues and, optionally, eigenvectors of a real $n$ by $n$ symmetric matrix $A$ in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## 2  Specification

 SUBROUTINE F08GBF ( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, JFAIL, INFO)
 INTEGER N, IL, IU, M, LDZ, IWORK(5*N), JFAIL(*), INFO REAL (KIND=nag_wp) AP(*), VL, VU, ABSTOL, W(N), Z(LDZ,*), WORK(8*N) CHARACTER(1) JOBZ, RANGE, UPLO
The routine may be called by its LAPACK name dspevx.

## 3  Description

The symmetric matrix $A$ is first reduced to tridiagonal form, using orthogonal similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     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:     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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     AP($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: the upper or lower triangle of the $n$ by $n$ symmetric matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
On exit: AP is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of $A$.
6:     VL – REAL (KIND=nag_wp)Input
7:     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 ${\mathbf{RANGE}}=\text{'V'}$, ${\mathbf{VL}}<{\mathbf{VU}}$.
8:     IL – INTEGERInput
9:     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 ${\mathbf{N}}>0$, $1\le {\mathbf{IL}}\le {\mathbf{IU}}\le {\mathbf{N}}$.
10:   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
 $ABSTOL+ε maxa,b ,$
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 , not zero. If this routine returns with ${\mathbf{INFO}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting ABSTOL to . See Demmel and Kahan (1990).
11:   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 ${\mathbf{RANGE}}=\text{'I'}$, ${\mathbf{M}}={\mathbf{IU}}-{\mathbf{IL}}+1$.
12:   W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the selected eigenvalues in ascending order.
13:   Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ 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.
14:   LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08GBF (DSPEVX) is called.
Constraints:
• if ${\mathbf{JOBZ}}=\text{'V'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• otherwise ${\mathbf{LDZ}}\ge 1$.
15:   WORK($8×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
16:   IWORK($5×{\mathbf{N}}$) – INTEGER arrayWorkspace
17:   JFAIL($*$) – INTEGER arrayOutput
Note: the dimension of the array JFAIL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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.
18:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, then $i$ eigenvectors failed to converge. Their indices are stored in array JFAIL. Please see ABSTOL.

## 7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

## 8  Further Comments

The total number of floating point operations is proportional to ${n}^{3}$.
The complex analogue of this routine is F08GPF (ZHPEVX).

## 9  Example

This example finds the eigenvalues in the half-open interval $\left(-1,1\right]$, and the corresponding eigenvectors, of the symmetric matrix
 $A = 1 2 3 4 2 2 3 4 3 3 3 4 4 4 4 4 .$

### 9.1  Program Text

Program Text (f08gbfe.f90)

### 9.2  Program Data

Program Data (f08gbfe.d)

### 9.3  Program Results

Program Results (f08gbfe.r)

F08GBF (DSPEVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual