f08jef computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.

2 Specification

Fortran Interface

Subroutine f08jef (

compz, n, d, e, z, ldz, work, info)

Integer, Intent (In)	::	n, ldz
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	d(), e(), z(ldz,), work()
Character (1), Intent (In)	::	compz

C Header Interface

#include <nag.h>

void	f08jef_ (const char compz, const Integer n, double d[], double e[], double z[], const Integer ldz, double work[], Integer info, const Charlen length_compz)

The routine may be called by the names f08jef, nagf_lapackeig_dsteqr or its LAPACK name dsteqr.

3 Description

f08jef computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix

T

. In other words, it can compute the spectral factorization of

T

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the orthogonal matrix whose columns are the eigenvectors

z_{i}

. Thus

T z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

The routine may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix

A

which has been reduced to tridiagonal form

T

\begin{array}{l} A & = Q T Q^{T}, where ​ Q ​ is orthogonal \\ = (Q Z) Λ {(Q Z)}^{T} . \end{array}

In this case, the matrix

Q

must be formed explicitly and passed to f08jef, which must be called with

compz ='V'

. The routines which must be called to perform the reduction to tridiagonal form and form

Q

are:

full matrix	f08fef and f08fff
full matrix, packed storage	f08gef and f08gff
band matrix	f08hef with $vect ='V'$ .

f08jef uses the implicitly shifted

Q R

algorithm, switching between the

Q R

and

Q L

variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that

{‖ z_{i} ‖}_{2} = 1

, but are determined only to within a factor

\pm 1

If only the eigenvalues of

T

are required, it is more efficient to call f08jff instead. If

T

is positive definite, small eigenvalues can be computed more accurately by f08jgf.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn17.pdf

Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

5 Arguments

1: $compz$ – Character(1) Input

On entry: indicates whether the eigenvectors are to be computed.

$compz ='N'$: Only the eigenvalues are computed (and the array z is not referenced).
$compz ='V'$: The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz ='I'$: The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the routine).

Constraint:

compz ='N'

'V'

'I'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

3: $d (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension of the array d must be at least

\max (1, n)

On entry: the diagonal elements of the tridiagonal matrix

T

On exit: the

n

eigenvalues in ascending order, unless

info > 0

(in which case see Section 6).

4: $e (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension of the array e must be at least

\max (1, n - 1)

On entry: the off-diagonal elements of the tridiagonal matrix

T

On exit: e is overwritten.

5: $z (ldz, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array z must be at least

\max (1, n)

compz ='V'

'I'

On entry: if

compz ='V'

, z must contain the orthogonal matrix

Q

from the reduction to tridiagonal form.

compz ='I'

, z need not be set.

On exit: if

compz ='V'

'I'

, the

n

required orthonormal eigenvectors stored as columns of

Z

; the

i

th column corresponds to the

i

th eigenvalue, where

i = 1, 2, \dots, n

, unless

info > 0

compz ='N'

, z is not referenced.

6: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08jef is called.

Constraints:

if $compz ='V'$ or $'I'$ , $ldz \geq \max (1, n)$ ;
if $compz ='N'$ , $ldz \geq 1$ .

7: $work (*)$ – Real (Kind=nag_wp) array Workspace

Note: the dimension of the array work must be at least

\max (1, 2 \times (n - 1))

compz ='V'

'I'

and at least

1

compz ='N'

compz ='N'

, work is not referenced.

8: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: The algorithm has failed to find all the eigenvalues after a total of $30 \times n$ iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix orthogonally similar to $T$ . $⟨ value ⟩$ off-diagonal elements have not converged to zero.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(T + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ T ‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue and

{\tilde{λ}}_{i}

is the corresponding computed value, then

| {\tilde{λ}}_{i} - λ_{i} | \leq c (n) ε {‖ T ‖}_{2},

where

c (n)

is a modestly increasing function of

n

z_{i}

is the corresponding exact eigenvector, and

{\tilde{z}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{z}}_{i}, z_{i})

between them is bounded as follows:

θ ({\tilde{z}}_{i}, z_{i}) \leq \frac{c (n) ε {‖ T ‖}_{2}}{\min_{i \neq j} | λ_{i} - λ_{j} |} .

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08jef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is typically about

24 n^{2}

compz ='N'

and about

7 n^{3}

compz ='V'

'I'

, but depends on how rapidly the algorithm converges. When

compz ='N'

, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when

compz ='V'

'I'

can be vectorized and on some machines may be performed much faster.

The complex analogue of this routine is f08jsf.

10 Example

This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix

T

, where

T = (\begin{array}{r} - 6.99 & - 0.44 & 0.00 & 0.00 \\ - 0.44 & 7.92 & - 2.63 & 0.00 \\ 0.00 & - 2.63 & 2.34 & - 1.18 \\ 0.00 & 0.00 & - 1.18 & 0.32 \end{array}) .

See also the examples for f08fff, f08gff or f08hef, which illustrate the use of this routine to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.

f08je: FL CL CPP AD PY MB

NAG FL Interfacef08jef (dsteqr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f08jef (dsteqr)