?pteqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 10/31/2024

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?pteqr

Computes all eigenvalues and (optionally) all eigenvectors of a real symmetric positive-definite tridiagonal matrix.

Syntax

call spteqr(compz, n, d, e, z, ldz, work, info)

call dpteqr(compz, n, d, e, z, ldz, work, info)

call cpteqr(compz, n, d, e, z, ldz, work, info)

call zpteqr(compz, n, d, e, z, ldz, work, info)

call rpteqr(d, e [,z] [,compz] [,info])

call pteqr(d, e [,z] [,compz] [,info])

Include Files

mkl.fi, mkl_lapack.f90

Description

The routine computes all the eigenvalues and (optionally) all the eigenvectors of a real symmetric positive-definite tridiagonal matrix T. In other words, the routine can compute the spectral factorization: T = Z*Λ*Z^T.

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i; Z is an orthogonal matrix whose columns are eigenvectors. Thus,

T*z_i = λ_i*z_i for i = 1, 2, ..., n.

(The routine normalizes the eigenvectors so that ||z_i||₂ = 1.)

You can also use the routine for computing the eigenvalues and eigenvectors of real symmetric (or complex Hermitian) positive-definite matrices A reduced to tridiagonal form T: A = Q*T*Q^H. In this case, the spectral factorization is as follows: A = Q*T*Q^H = (QZ)*Λ*(QZ)^H. Before calling ?pteqr, you must reduce A to tridiagonal form and generate the explicit matrix Q by calling the following routines:

	for real matrices:	for complex matrices:
full storage	?sytrd, ?orgtr	?hetrd, ?ungtr
packed storage	?sptrd, ?opgtr	?hptrd, ?upgtr
band storage	?sbtrd`(vect`='V')	?hbtrd`(vect`='V')

The routine first factorizes T as L*D*L^H where L is a unit lower bidiagonal matrix, and D is a diagonal matrix. Then it forms the bidiagonal matrix B = L*D^1/2 and calls ?bdsqr to compute the singular values of B, which are the square roots of the eigenvalues of T.

Input Parameters

compz

CHARACTER*1. Must be 'N' or 'I' or 'V'.

If compz = 'N', the routine computes eigenvalues only.

If compz = 'I', the routine computes the eigenvalues and eigenvectors of the tridiagonal matrix T.

If compz = 'V', the routine computes the eigenvalues and eigenvectors of A (and the array z must contain the matrix Q on entry).

n

INTEGER. The order of the matrix T (n≥ 0).

d, e, work

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Arrays:

d(*) contains the diagonal elements of T.

The size of d must be at least max(1, n).

e(*) contains the off-diagonal elements of T.

The size of e must be at least max(1, n-1).

work(*) is a workspace array.

The dimension of work must be:

at least 1 if compz = 'N';

at least max(1, 4*n-4) if compz = 'V' or 'I'.

z

REAL for spteqr

DOUBLE PRECISION for dpteqr

COMPLEX for cpteqr

DOUBLE COMPLEX for zpteqr.

Array, size (ldz,*)

If compz = 'N' or 'I', z need not be set.

If compz = 'V', z must contain the orthogonal matrix used in the reduction to tridiagonal form..

The second dimension of z must be:

at least 1 if compz = 'N';

at least max(1, n) if compz = 'V' or 'I'.

ldz

INTEGER. The leading dimension of z. Constraints:

ldz≥ 1 if compz = 'N';

ldz≥ max(1, n) if compz = 'V' or 'I'.

Output Parameters

d

The n eigenvalues in descending order, unless info > 0.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine pteqr interface are the following:

d

Holds the vector of length n.

e

Holds the vector of length (n-1).

z

Holds the matrix Z of size (n,n).

compz

If omitted, this argument is restored based on the presence of argument z as follows:

compz = 'I', if z is present,

compz = 'N', if z is omitted.

If present, compz must be equal to 'I' or 'V' and the argument z must also be present. Note that there will be an error condition if compz is present and z omitted.

Note that two variants of Fortran 95 interface for pteqr routine are needed because of an ambiguous choice between real and complex cases appear when z is omitted. Thus, the name rpteqr is used in real cases (single or double precision), and the name pteqr is used in complex cases (single or double precision).

Application Notes

If λ_i is an exact eigenvalue, and μ_i is the corresponding computed value, then

|μ_i - λ_i| ≤c(n)*ε*K*λ_i

where c(n) is a modestly increasing function of n, ε is the machine precision, and K = ||DTD||₂ *||(DTD)^-1||₂, D is diagonal with d_ii = t_ii^-1/2.

If z_i is the corresponding exact eigenvector, and w_i is the corresponding computed vector, then the angle θ(z_i, w_i) between them is bounded as follows:

θ(u_i, w_i) ≤c(n)εK / min_i≠j(|λ_i - λ_j|/|λ_i + λ_j|).

Here min_i≠j(|λ_i - λ_j|/|λ_i + λ_j|) is the relative gap between λ_i and the other eigenvalues.

The total number of floating-point operations depends on how rapidly the algorithm converges.

Typically, it is about

30n² if compz = 'N';

6n³ (for complex flavors, 12n³) if compz = 'V' or 'I'.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?pteqr

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes