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// Copyright (c) 2017-2023, University of Tennessee. All rights reserved.
// SPDX-License-Identifier: BSD-3-Clause
// This program is free software: you can redistribute it and/or modify it under
// the terms of the BSD 3-Clause license. See the accompanying LICENSE file.
#include "blas.hh"
#include "lapack.hh"
#include <vector>
// -----------------------------------------------------------------------------
// work = V*W^{trans}; used in check_geev
// version for real
template< typename real_t >
void check_geev_multiply_VW(
blas::Op trans, int64_t n,
std::complex<real_t> const* W,
real_t const* V, int64_t ldv,
real_t* work, int64_t ldwork )
{
#define V(i_, j_) V[ (i_) + (j_)*ldv ]
#define work(i_, j_) work[ (i_) + (j_)*ldwork ]
// W is block diagonal, with 2x2 blocks for each complex eigenvalue pair.
lapack::laset( lapack::MatrixType::General, n, n, 0., 0., &work(0,0), n );
bool ipair = false;
for (int64_t j = 0; j < n; ++j) {
if (ipair) {
// 2nd part of complex eigenvector, already handled
ipair = false;
}
else if (imag( W[j] ) != 0 && j+1 < n) {
// complex eigenvector
ipair = true;
// Wmat = [ Wr Wi ] (stored columnwise)
// [ -Wi Wr ]
real_t Wmat[4] = { real( W[j] ), -imag( W[j] ),
imag( W[j] ), real( W[j] ) };
// work(:,j:j+1) = V(:,j:j+1) * Wmat^{trans}
blas::gemm( blas::Layout::ColMajor,
blas::Op::NoTrans, trans, n, 2, 2,
1.0, &V(0,j), ldv,
Wmat, 2,
0.0, &work(0,j), n );
}
else {
// work(:,j) = V(:,j) * W[j]
blas::axpy( n, real( W[j] ), &V(0,j), 1, &work(0,j), 1 );
}
}
#undef V
#undef work
}
// -----------------------------------------------------------------------------
// work = V*W; used in check_geev
// version for complex
template< typename real_t >
void check_geev_multiply_VW(
blas::Op trans, int64_t n,
std::complex<real_t> const* W,
std::complex<real_t> const* V, int64_t ldv,
std::complex<real_t>* work, int64_t ldwork )
{
#define V(i_, j_) V[ (i_) + (j_)*ldv ]
#define work(i_, j_) work[ (i_) + (j_)*ldwork ]
// W is diagonal.
lapack::laset( lapack::MatrixType::General, n, n, 0., 0., &work(0,0), n );
for (int64_t j = 0; j < n; ++j) {
std::complex<real_t> Wtmp;
if (trans == blas::Op::ConjTrans)
Wtmp = conj( W[j] );
else
Wtmp = W[j];
// work(:,j) = V(:,j) * W[j]
blas::axpy( n, Wtmp, &V(0,j), 1, &work(0,j), 1 );
}
#undef V
#undef work
}
// -----------------------------------------------------------------------------
// checks | ||V||_2 - 1 |
// 2-norm as used in drvev, not max-norm as used in get22
// also checks that conjugate pairs are together correctly
// version for real
template< typename real_t >
real_t check_geev_Vnormalization(
int64_t n,
std::complex<real_t> const* W,
real_t const* V, int64_t ldv )
{
using namespace blas;
#define V(i_, j_) V[ (i_) + (j_)*ldv ]
real_t eps = std::numeric_limits< real_t >::epsilon();
real_t error = 0;
bool ipair = false;
for (int64_t j = 0; j < n; ++j) {
if (imag( W[j] ) < 0) {
if (ipair != true) {
error = std::numeric_limits< real_t >::infinity();
}
ipair = false;
}
else {
if (ipair == true) {
error = std::numeric_limits< real_t >::infinity();
}
real_t nrm;
if (imag( W[j] ) > 0) {
// complex eigenvector
require( j < n-1 );
require( real( W[j] ) == real( W[j+1] ) );
require( imag( W[j] ) == -imag( W[j+1] ) );
ipair = true;
nrm = std::hypot( blas::nrm2( n, &V( 0, j ), 1 ),
blas::nrm2( n, &V( 0, j+1 ), 1 ) );
}
else {
// real eigenvector
nrm = blas::nrm2( n, &V(0,j), 1 );
}
error = max( error, std::abs( nrm - 1 ) );
// check largest component is real; set to inf if not
if (imag( W[j] ) > 0) {
real_t Vmax = 0;
real_t Vrmax = 0;
for (int64_t i = 0; i < n; ++i) {
real_t tmp = std::hypot( V( i, j ), V( i, j+1 ) );
if (tmp > Vmax)
Vmax = tmp;
if (V(i,j+1) == 0 && std::abs( V(i,j) ) > Vrmax)
Vrmax = std::abs( V(i,j) );
}
if (Vrmax / Vmax < 1 - 2*eps) {
//printf( "Vrmax %.2e, Vmax %.2e\n", Vrmax, Vmax );
error = std::numeric_limits< real_t >::infinity();
}
}
}
}
return error;
#undef V
}
// -----------------------------------------------------------------------------
// checks | ||V||_2 - 1 |
// 2-norm as used in drvev, not max-norm as used in get22
// also checks that conjugate pairs are together correctly
// version for complex
template< typename real_t >
real_t check_geev_Vnormalization(
int64_t n,
std::complex<real_t> const* W,
std::complex<real_t> const* V, int64_t ldv )
{
using namespace blas;
#define V(i_, j_) V[ (i_) + (j_)*ldv ]
real_t eps = std::numeric_limits< real_t >::epsilon();
real_t error = 0;
for (int64_t j = 0; j < n; ++j) {
real_t nrm = blas::nrm2( n, &V(0,j), 1 );
error = max( error, std::abs( nrm - 1 ) );
// check largest component is real; set to inf if not
if (imag( W[j] ) > 0) {
real_t Vmax = 0;
real_t Vrmax = 0;
for (int64_t i = 0; i < n; ++i) {
real_t tmp = std::abs( V(i,j) );
if (tmp > Vmax)
Vmax = tmp;
if (imag( V(i,j) ) == 0 && std::abs( real( V(i,j) )) > Vrmax)
Vrmax = std::abs( real( V(i,j) ));
}
if (Vrmax / Vmax < 1 - 2*eps) {
//printf( "Vrmax %.2e, Vmax %.2e\n", Vrmax, Vmax );
error = std::numeric_limits< real_t >::infinity();
}
}
}
return error;
#undef V
}
// -----------------------------------------------------------------------------
// || op(A) V - V W ||_1 / (||A||_1 ||V||_1)
// max_j | || E(j) ||_2 - 1 | / n
//
// For right eigenvectors, use trans = NoTrans
// to compute || A VR - VR W || / (||A|| ||VR||)
//
// For left eigenvectors, use trans = ConjTrans
// to compute || A^H VL - VL W^H || / (||A|| ||V||)
//
// TODO: doesn't quite match LAWN 41, which normalizes by (n ||A||)
// See LAPACK testing drvev and get22.
// version for real
template< typename scalar_t >
void check_geev(
blas::Op trans, int64_t n,
scalar_t const* A, int64_t lda,
blas::complex_type< scalar_t > const* W,
scalar_t const* V, int64_t ldv,
int64_t verbose,
blas::real_type< scalar_t > results[2] )
{
using real_t = blas::real_type< scalar_t >;
// work = op(A) V - work = op(A) V - (V W)
std::vector< scalar_t > work( n * n );
check_geev_multiply_VW( trans, n, W, V, ldv, &work[0], n );
if (verbose >= 2) {
printf( "VW = " ); print_matrix( n, n, &work[0], n );
}
blas::gemm( blas::Layout::ColMajor,
trans, blas::Op::NoTrans, n, n, n,
1.0, A, lda,
V, ldv,
-1.0, &work[0], n );
if (verbose >= 2) {
printf( "R = " ); print_matrix( n, n, &work[0], n );
}
// || op(A) V - V W || / (n ||A||)
// instead of n, get22 uses ||V||, which generally gives a larger error:
// || op(A) V - V W || / (||V|| ||A||)
real_t error = lapack::lange( lapack::Norm::One, n, n, &work[0], n );
real_t Anorm = lapack::lange( lapack::Norm::One, n, n, A, lda );
real_t Vnorm = lapack::lange( lapack::Norm::One, n, n, V, ldv );
results[0] = error / Vnorm / Anorm;
if (verbose >= 1) {
printf( "error: { ||A^{%c} V - V W||=%.2e / (||V||=%.2e ||A||=%.2e) } = %.2e; n=%.0f\n",
to_char( trans ), error, Vnorm, Anorm, results[0], real_t( n ) );
}
results[1] = check_geev_Vnormalization( n, W, V, ldv );
}
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