f08juf computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian positive definite matrix which has been reduced to tridiagonal form.

2 Specification

Fortran Interface

Subroutine f08juf (

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()
Real (Kind=nag_wp), Intent (Out)	::	work(4*n)
Complex (Kind=nag_wp), Intent (Inout)	::	z(ldz,*)
Character (1), Intent (In)	::	compz

C Header Interface

#include <nag.h>

void	f08juf_ (const char compz, const Integer n, double d[], double e[], Complex z[], const Integer ldz, double work[], Integer info, const Charlen length_compz)

The routine may be called by the names f08juf, nagf_lapackeig_zpteqr or its LAPACK name zpteqr.

3 Description

f08juf computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite 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 stores the real orthogonal matrix

Z

in a complex array, so that it may be used to compute all the eigenvalues and eigenvectors of a complex Hermitian positive definite matrix

A

which has been reduced to tridiagonal form

T

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

In this case, the matrix

Q

must be formed explicitly and passed to f08juf, 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	f08fsf and f08ftf
full matrix, packed storage	f08gsf and f08gtf
band matrix	f08hsf with $vect ='V'$ .

f08juf first factorizes

T

L D L^{H}

where

L

is unit lower bidiagonal and

D

is diagonal. It forms the bidiagonal matrix

B = L D^{\frac{1}{2}}

, and then calls f08msf to compute the singular values of

B

which are the same as the eigenvalues of

T

. The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of

T

. The eigenvectors are normalized so that

{‖ z_{i} ‖}_{2} = 1

, but are determined only to within a complex factor of absolute value

1

4 References

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

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 descending order, unless

info > 0

, in which case d is overwritten.

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

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

\max (1, n)

compz ='V'

'I'

and at least

1

compz ='N'

On entry: if

compz ='V'

, z must contain the unitary 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 f08juf is called.

Constraints:

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

7: $work (4 \times n)$ – Real (Kind=nag_wp) array Workspace

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.

$info = 1, \dots, n$: The leading minor of order $⟨ value ⟩$ is not positive definite and the Cholesky factorization of $T$ could not be completed. Hence $T$ itself is not positive definite.

$info > n$: The algorithm to compute the singular values of the Cholesky factor $B$ failed to converge; $⟨ value ⟩$ off-diagonal elements did not converge to zero.

7 Accuracy

The eigenvalues and eigenvectors of

T

are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard

Q R

method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.

To be more precise, let

H

be the tridiagonal matrix defined by

H = D T D

, where

D

is diagonal with

d_{i i} = t_{i i}^{- \frac{1}{2}}

, and

h_{i i} = 1

for all

i

. If

λ_{i}

is an exact eigenvalue of

T

and

{\tilde{λ}}_{i}

is the corresponding computed value, then

| {\tilde{λ}}_{i} - λ_{i} | \leq c (n) ε κ_{2} (H) λ_{i}

where

c (n)

is a modestly increasing function of

n

ε

is the machine precision, and

κ_{2} (H)

is the condition number of

H

with respect to inversion defined by:

κ_{2} (H) = ‖ H ‖ \cdot ‖ H^{- 1} ‖

z_{i}

is the corresponding exact eigenvector of

T

, 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 κ_{2} (H)}{{relgap}_{i}}

where

{relgap}_{i}

is the relative gap between

λ_{i}

and the other eigenvalues, defined by

{relgap}_{i} = \min_{i \neq j} \frac{| λ_{i} - λ_{j} |}{(λ_{i} + λ_{j})} .

8 Parallelism and Performance

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

f08juf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08juf 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 real floating-point operations is typically about

30 n^{2}

compz ='N'

and about

12 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 real analogue of this routine is f08jgf.

10 Example

This example computes all the eigenvalues and eigenvectors of the complex Hermitian positive definite matrix

A

, where

A = (\begin{array}{r} 6.02 + 0.00 i & - 0.45 + 0.25 i & - 1.30 + 1.74 i & 1.45 - 0.66 i \\ - 0.45 - 0.25 i & 2.91 + 0.00 i & 0.05 + 1.56 i & - 1.04 + 1.27 i \\ - 1.30 - 1.74 i & 0.05 - 1.56 i & 3.29 + 0.00 i & 0.14 + 1.70 i \\ 1.45 + 0.66 i & - 1.04 - 1.27 i & 0.14 - 1.70 i & 4.18 + 0.00 i \end{array}) .

f08ju: FL CL CPP AD PY MB

NAG FL Interfacef08juf (zpteqr)

▸▿ 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
f08juf (zpteqr)