p?trevc

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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p?trevc

Computes right and/or left eigenvectors of a complex upper triangular matrix in parallel.

Syntax

call pctrevc (side, howmny, select, n, t, desct, vl, descvl, vr, descvr, mm, m, work, rwork, info )

call pztrevc (side, howmny, select, n, t, desct, vl, descvl, vr, descvr, mm, m, work, rwork, info )

call pdtrevc (side, howmny, select, n, t, desct, vl, descvl, vr, descvr, mm, m, work, info )

call pstrevc (side, howmny, select, n, t, desct, vl, descvl, vr, descvr, mm, m, work, info )

Description

p?trevc computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T in parallel.

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

T*x = w*x,

y'*T = w*y'

where y' denotes the conjugate transpose of the vector y.

If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input unitary matrix. If T was obtained from the Schur factorization of an original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or left eigenvectors of A.

Input Parameters

side

(global)

CHARACTER*1.

= 'R': compute right eigenvectors only;

= 'L': compute left eigenvectors only;

= 'B': compute both right and left eigenvectors.

howmny

(global)

CHARACTER*1.

= 'A': compute all right and/or left eigenvectors;

= 'B': compute all right and/or left eigenvectors, and backtransform them using the input matrices supplied in vr and/or vl;

= 'S': compute selected right and/or left eigenvectors, specified by the logical array select.

select

(global)

LOGICAL.

Array, size (n)

If howmny = 'S', select specifies the eigenvectors to be computed.

If howmny = 'A' or 'B', select is not referenced. To select the eigenvector corresponding to the j-th eigenvalue, select(j) must be set to .TRUE..

n

(global)

INTEGER.

The order of the matrix T. n >= 0.

t

(local)

COMPLEX for pctrevc

DOUBLE COMPLEX for pztrevc

DOUBLE PRECISION for pdtrevc

REAL for pstrevc

Array, size (lld_t,LOCc(n)).

The upper triangular matrix T. T is modified, but restored on exit.

desct

(global and local)

INTEGER.

Array of size dlen_.

The array descriptor for the distributed matrix T.

vl

(local)

COMPLEX for pctrevc

DOUBLE COMPLEX for pztrevc

DOUBLE PRECISION for pdtrevc

REAL for pstrevc

Array, size (descvl(lld_),mm)

On entry, if side = 'L' or 'B' and howmny = 'B', vl must contain an n-by-n matrix Q (usually the unitary matrix Q of Schur vectors returned by ?hseqr).

descvl

(global and local)

INTEGER.

Array of size dlen_.

The array descriptor for the distributed matrix VL.

vr

(local)

COMPLEX for pctrevc

DOUBLE COMPLEX for pztrevc

DOUBLE PRECISION for pdtrevc

REAL for pstrevc

Array, size (descvr(lld_),mm).

On entry, if side = 'R' or 'B' and howmny = 'B', vr must contain an n-by-n matrix Q (usually the unitary matrix Q of Schur vectors returned by ?hseqr).

descvr

(global and local)

INTEGER.

Array of size dlen_.

The array descriptor for the distributed matrix VR.

mm

(global)

INTEGER.

The number of columns in the arrays vl and/or vr. mm >= m.

work

(local)

COMPLEX for pctrevc

DOUBLE COMPLEX for pztrevc

DOUBLE PRECISION for pdtrevc

REAL for pstrevc

Array, size ( 2*desct(lld_) )

Additional workspace may be required if p?lattrs is updated to use work.

rwork

REAL for pctrevc

DOUBLE PRECISION for pztrevc

Array, size ( desct(lld_) )

Output Parameters

t	The upper triangular matrix `T`. `T` is modified, but restored on exit.
vl	On exit, if side = 'L' or 'B', vl contains: if howmny = 'A', the matrix `Y` of left eigenvectors of `T`; if howmny = 'B', the matrix `Q`*`Y`; if howmny = 'S', the left eigenvectors of `T` specified by select, stored consecutively in the columns of vl, in the same order as their eigenvalues. If side = 'R', vl is not referenced.
vr	On exit, if side = 'R' or 'B', vr contains: if howmny = 'A', the matrix `X` of right eigenvectors of `T`; if howmny = 'B', the matrix `Q`*`X`; if howmny = 'S', the right eigenvectors of `T` specified by select, stored consecutively in the columns of vr, in the same order as their eigenvalues. If side = 'L', vr is not referenced.
m	(global) INTEGER. The number of columns in the arrays vl and/or vr actually used to store the eigenvectors. If howmny = 'A' or 'B', m is set to n. Each selected eigenvector occupies one column.
info	(global) INTEGER. = 0: successful exit < 0: if info = -i, the i-th argument had an illegal value

Application Notes

The algorithm used in this program is basically backward (forward) substitution. Scaling should be used to make the code robust against possible overflow. But scaling has not yet been implemented in p?lattrs which is called by this routine to solve the triangular systems. p?lattrs just calls p?trsv.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

Parent topic: Nonsymmetric Eigenvalue Problems: ScaLAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

p?trevc