void	f08jlc (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, Integer m, double w[], double z[], Integer pdz, Integer isuppz[], NagError fail)

The function may be called by the names: f08jlc, nag_lapackeig_dstegr or nag_dstegr.

3 Description

f08jlc computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix

T

. That is, the function computes the (partial) spectral factorization of

T

given by

Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the selected eigenvalues,

λ_{i}

, of

T

and

Z

is an orthogonal matrix whose columns are the corresponding eigenvectors,

z_{i}

, of

T

. Thus

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

where

m

is the number of selected eigenvalues computed.

The function may also be used to compute selected eigenvalues and eigenvectors of a real symmetric matrix

A

which has been reduced to tridiagonal form

T

(Q Z) Λ {(Q Z)}^{T}, where ​ Q ​ is orthogonal.

In this case, the matrix

Q

must be explicitly applied to the output matrix

Z

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

Q

are:

full matrix	f08fec and f08fgc
full matrix, packed storage	f08gec and f08ggc
band matrix	f08hec with $vect = Nag_FormQ$ and f16yac.

This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. f08jlc can usually compute all the eigenvalues and eigenvectors in

O (n^{2})

floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the

Q R

algorithm.

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 https://www.netlib.org/lapack/lug

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps SIAM J. Appl. Math. 25 858–899

Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

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: $job$ – Nag_JobType Input

On entry: indicates whether eigenvectors are computed.

$job = Nag_EigVals$: Only eigenvalues are computed.
$job = Nag_DoBoth$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_EigVals

Nag_DoBoth

3: $range$ – Nag_RangeType Input

On entry: indicates which eigenvalues should be returned.

$range = Nag_AllValues$: All eigenvalues will be found.
$range = Nag_Interval$: All eigenvalues in the half-open interval $(vl, vu]$ will be found.
$range = Nag_Indices$: The ilth through iuth eigenvectors will be found.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

5: $d [\dim]$ – double Input/Output

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

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

T

On exit: d is overwritten.

6: $e [\dim]$ – double Input/Output

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

\max (1, n)

On entry:

e [0]

e [n - 2]

are the subdiagonal elements of the tridiagonal matrix

T

e [n - 1]

need not be set.

On exit: e is overwritten.

7: $vl$ – double Input

8: $vu$ – double Input

On entry: if

range = Nag_Interval

, vl and vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.

range = Nag_AllValues

Nag_Indices

, vl and vu are not referenced.

Constraint: if

range = Nag_Interval

vl < vu

9: $il$ – Integer Input

10: $iu$ – Integer Input

On entry: if

range = Nag_Indices

, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ ;
if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ .

11: $m$ – Integer * Output

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range = Nag_AllValues

m = n

range = Nag_Indices

m = iu - il + 1

12: $w [\dim]$ – double Output

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

\max (1, n)

On exit: the eigenvalues in ascending order.

13: $z [\dim]$ – double Output

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

$\max (1, pdz \times n)$ when $job = Nag_DoBoth$ ;
$1$ otherwise.

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 exit: if

job = Nag_DoBoth

, then if

fail . code =

NE_NOERROR, the columns of z contain the orthonormal eigenvectors of the matrix

T

, with the

i

th column of

Z

holding the eigenvector associated with

w [i - 1]

job = Nag_EigVals

, z is not referenced.

14: $pdz$ – Integer Input

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

Constraints:

if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

15: $isuppz [\dim]$ – Integer Output

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

\max (1, 2 \times m)

On exit: the support of the eigenvectors in

Z

, i.e., the indices indicating the nonzero elements in

Z

. The

i

th eigenvector is nonzero only in elements

isuppz [2 \times i - 2]

through

isuppz [2 \times i - 1]

16: $fail$ – 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: Inverse iteration failed to converge.

The $dqds$ algorithm failed to converge.
NE_ENUM_INT_2: On entry, $job = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_ENUM_INT_3: On entry, $range = ⟨ value ⟩$ , $il = ⟨ value ⟩$ , $iu = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ ;
if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ .
NE_ENUM_REAL_2: On entry, $range = ⟨ value ⟩$ , $vl = ⟨ value ⟩$ and $vu = ⟨ value ⟩$ .
Constraint: if $range = Nag_Interval$ , $vl < vu$ .
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

See Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

8 Parallelism and Performance

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

f08jlc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jlc 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 required to compute all the eigenvalues and eigenvectors is approximately proportional to

n^{2}

The complex analogue of this function is f08jyc.

10 Example

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

T = (\begin{array}{r} 1.0 & 1.0 & 0 & 0 \\ 1.0 & 4.0 & 2.0 & 0 \\ 0 & 2.0 & 9.0 & 3.0 \\ 0 & 0 & 3.0 & 16.0 \end{array}) .

f08jl: FL CL CPP AD PY MB

NAG CL Interfacef08jlc (dstegr)

▸▿ 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
f08jlc (dstegr)