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// Copyright ©2015 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package gonum
import "math"
// Dlartg generates a plane rotation so that
//
// [ cs sn] * [f] = [r]
// [-sn cs] [g] = [0]
//
// where cs*cs + sn*sn = 1.
//
// This is a more accurate version of BLAS Drotg that uses scaling to avoid
// overflow or underflow, with the other differences that
// - cs >= 0
// - if g = 0, then cs = 1 and sn = 0
// - if f = 0 and g != 0, then cs = 0 and sn = sign(1,g)
//
// Dlartg is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlartg(f, g float64) (cs, sn, r float64) {
// Implementation based on Supplemental Material to:
//
// Edward Anderson
// Algorithm 978: Safe Scaling in the Level 1 BLAS
// ACM Trans. Math. Softw. 44, 1, Article 12 (2017)
// DOI: https://doi.org/10.1145/3061665
//
// For further details see:
//
// W. Pereira, A. Lotfi, J. Langou
// Numerical analysis of Givens rotation
// DOI: https://doi.org/10.48550/arXiv.2211.04010
if g == 0 {
return 1, 0, f
}
g1 := math.Abs(g)
if f == 0 {
return 0, math.Copysign(1, g), g1
}
const safmin = dlamchS
const safmax = 1 / safmin
rtmin := math.Sqrt(safmin)
rtmax := math.Sqrt(safmax / 2)
f1 := math.Abs(f)
if rtmin < f1 && f1 < rtmax && rtmin < g1 && g1 < rtmax {
d := math.Sqrt(f*f + g*g)
cs = f1 / d
r = math.Copysign(d, f)
sn = g / r
return cs, sn, r
}
u := math.Min(math.Max(safmin, math.Max(f1, g1)), safmax)
fs := f / u
gs := g / u
d := math.Sqrt(fs*fs + gs*gs)
cs = math.Abs(fs) / d
r = math.Copysign(d, f)
sn = gs / r
r *= u
return cs, sn, r
}
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