f08jec 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

#include <nag.h>

void	f08jec (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)

The function may be called by the names: f08jec, nag_lapackeig_dsteqr or nag_dsteqr.

3 Description

f08jec 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 function 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 f08jec, which must be called with

compz = Nag_UpdateZ

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

Q

are:

full matrix	f08fec and f08ffc
full matrix, packed storage	f08gec and f08gfc
band matrix	f08hec with $vect = Nag_FormQ$ .

f08jec 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 f08jfc instead. If

T

is positive definite, small eigenvalues can be computed more accurately by f08jgc.

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: $order$ – Nag_OrderType Input

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

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $compz$ – Nag_ComputeZType Input

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

$compz = Nag_NotZ$: Only the eigenvalues are computed (and the array z is not referenced).
$compz = Nag_UpdateZ$: The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).
$compz = Nag_InitZ$: The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

4: $d [\dim]$ – double Input/Output

Note: the dimension, dim, 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

fail . code =

NE_CONVERGENCE (in which case see Section 6).

5: $e [\dim]$ – double Input/Output

Note: the dimension, dim, 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.

6: $z [\dim]$ – double Input/Output

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

pdz \times n

when

compz = Nag_UpdateZ

Nag_InitZ

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On entry: if

compz = Nag_UpdateZ

, z must contain the orthogonal matrix

Q

from the reduction to tridiagonal form.

compz = Nag_InitZ

, z must be allocated, but its contents need not be set.

compz = Nag_NotZ

, z is not referenced and may be NULL.

On exit: if

compz = Nag_UpdateZ

Nag_InitZ

, 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

fail . errnum > 0

z is not changed if

compz = Nag_NotZ

7: $pdz$ – Integer Input

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

Constraints:

if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq n$ ;
if $compz = Nag_NotZ$ , z may be NULL.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: 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.
NE_ENUM_INT_2: On entry, $compz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq n$ .

On entry, $compz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ , $n = ⟨ value ⟩$ .
Constraint: $pdz \geq n$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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.

f08jec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jec 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 function. 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 = Nag_NotZ

and about

7 n^{3}

compz = Nag_UpdateZ

Nag_InitZ

, but depends on how rapidly the algorithm converges. When

compz = Nag_NotZ

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

compz = Nag_UpdateZ

Nag_InitZ

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

The complex analogue of this function is f08jsc.

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 f08ffc, f08gfc or f08hec, which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.

f08je: FL CL CPP AD PY MB

NAG CL Interfacef08jec (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 CL Interface
f08jec (dsteqr)