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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.
#ifndef BLAS_ROTG_HH
#define BLAS_ROTG_HH
#include "blas/util.hh"
#include <limits>
namespace blas {
// =============================================================================
/// Construct plane rotation that eliminates b, such that:
/// \[
/// \begin{bmatrix} r \\ 0 \end{bmatrix}
/// = \begin{bmatrix} c & s \\ -s & c \end{bmatrix}
/// \begin{bmatrix} a \\ b \end{bmatrix}.
/// \]
///
/// @see rot to apply the rotation.
///
/// Generic implementation for arbitrary data types.
///
/// @param[in, out] a
/// On entry, scalar a. On exit, set to r.
///
/// @param[in, out] b
/// On entry, scalar b. On exit, set to s, 1/c, or 0.
///
/// @param[out] c
/// Cosine of rotation; real.
///
/// @param[out] s
/// Sine of rotation; real.
///
/// __Further details__
///
/// Anderson E (2017) Algorithm 978: Safe scaling in the level 1 BLAS.
/// ACM Trans Math Softw 44:. https://doi.org/10.1145/3061665
///
/// @ingroup rotg
template <typename TA, typename TB>
void rotg(
TA *a,
TB *b,
blas::real_type<TA, TB> *c,
blas::real_type<TA, TB> *s )
{
typedef real_type<TA, TB> real_t;
using std::abs;
// Constants
const real_t one = 1;
const real_t zero = 0;
// Scaling constants
const real_t safmin = safe_min<real_t>();
const real_t safmax = safe_max<real_t>();
// Norms
const real_t anorm = abs(*a);
const real_t bnorm = abs(*b);
// quick return
if (bnorm == zero) {
*c = one;
*s = zero;
*b = TB( 0.0 );
}
else if (anorm == zero) {
*c = zero;
*s = one;
*a = *b;
*b = TB( 1.0 );
}
else {
real_t scl = min( safmax, max(safmin, anorm, bnorm) );
real_t sigma = (anorm > bnorm)
? sgn(*a)
: sgn(*b);
real_t r = sigma * scl * sqrt( (*a/scl) * (*a/scl) + (*b/scl) * (*b/scl) );
*c = *a / r;
*s = *b / r;
*a = r;
if (anorm > bnorm)
*b = *s;
else if (*c != zero)
*b = one / *c;
else
*b = one;
}
}
// =============================================================================
/// Construct plane rotation that eliminates b, such that:
/// \[
/// \begin{bmatrix} r \\ 0 \end{bmatrix}
/// = \begin{bmatrix} c & s \\ -conjg(s) & c \end{bmatrix}
/// \begin{bmatrix} a \\ b \end{bmatrix}.
/// \]
///
/// @see rot to apply the rotation.
///
/// Generic implementation for arbitrary data types.
///
/// @param[in, out] a
/// On entry, scalar a. On exit, set to r.
///
/// @param[in, out] b
/// On entry, scalar b. On exit, set to s, 1/c, or 0.
///
/// @param[out] c
/// Cosine of rotation; real.
///
/// @param[out] s
/// Sine of rotation; complex.
///
/// __Further details__
///
/// Anderson E (2017) Algorithm 978: Safe scaling in the level 1 BLAS.
/// ACM Trans Math Softw 44:. https://doi.org/10.1145/3061665
///
/// @ingroup rotg
template <typename TA, typename TB>
void rotg(
std::complex<TA> *a,
std::complex<TB> *b,
blas::real_type<TA, TB> *c,
blas::complex_type<TA, TB> *s )
{
typedef real_type<TA, TB> real_t;
typedef complex_type<TA, TB> scalar_t;
using std::abs;
#define BLAS_ABSSQ(t_) real(t_)*real(t_) + imag(t_)*imag(t_)
// Constants
const real_t r_one = 1;
const real_t r_zero = 0;
const scalar_t zero = 0;
const std::complex<TA> zero_ta = 0;
const std::complex<TB> zero_tb = 0;
// Scaling constants
const real_t safmin = safe_min<real_t>();
const real_t safmax = safe_max<real_t>();
const real_t rtmin = root_min<real_t>();
const real_t rtmax = root_max<real_t>();
// quick return
if (*b == zero_tb) {
*c = r_one;
*s = zero;
return;
}
if (*a == zero_ta) {
*c = r_zero;
real_t g1 = max( abs(real(*b)), abs(imag(*b)) );
if (g1 > rtmin && g1 < rtmax) {
// Use unscaled algorithm
real_t g2 = BLAS_ABSSQ( *b );
real_t d = sqrt( g2 );
*s = conj( *b ) / d;
*a = d;
}
else {
// Use scaled algorithm
real_t u = min( safmax, max( safmin, g1 ) );
real_t uu = r_one / u;
scalar_t gs = (*b)*uu;
real_t g2 = BLAS_ABSSQ( gs );
real_t d = sqrt( g2 );
*s = conj( gs ) / d;
*a = d*u;
}
}
else {
real_t f1 = max( abs(real(*a)), abs(imag(*a)) );
real_t g1 = max( abs(real(*b)), abs(imag(*b)) );
if (f1 > rtmin && f1 < rtmax
&& g1 > rtmin && g1 < rtmax) {
// Use unscaled algorithm
real_t f2 = BLAS_ABSSQ( *a );
real_t g2 = BLAS_ABSSQ( *b );
real_t h2 = f2 + g2;
real_t d = ( f2 > rtmin && h2 < rtmax )
? sqrt( f2*h2 )
: sqrt( f2 )*sqrt( h2 );
real_t p = r_one / d;
*c = f2*p;
*s = conj( *b )*( (*a)*p );
*a *= h2*p;
}
else {
// Use scaled algorithm
real_t u = min( safmax, max( safmin, f1, g1 ) );
real_t uu = r_one / u;
scalar_t gs = (*b)*uu;
real_t g2 = BLAS_ABSSQ( gs );
real_t f2, h2, w;
scalar_t fs;
if (f1*uu < rtmin) {
// a is not well-scaled when scaled by g1.
real_t v = min( safmax, max( safmin, f1 ) );
real_t vv = r_one / v;
w = v * uu;
fs = (*a)*vv;
f2 = BLAS_ABSSQ( fs );
h2 = f2*w*w + g2;
}
else {
// Otherwise use the same scaling for a and b.
w = r_one;
fs = (*a)*uu;
f2 = BLAS_ABSSQ( fs );
h2 = f2 + g2;
}
real_t d = ( f2 > rtmin && h2 < rtmax )
? sqrt( f2*h2 )
: sqrt( f2 )*sqrt( h2 );
real_t p = r_one / d;
*c = ( f2*p )*w;
*s = conj( gs )*( fs*p );
*a = ( fs*( h2*p ) )*u;
}
}
#undef BLAS_ABSSQ
}
} // namespace blas
#endif // #ifndef BLAS_ROTG_HH
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