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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"
// Dlaev2 computes the Eigen decomposition of a symmetric 2×2 matrix.
// The matrix is given by
// [a b]
// [b c]
// Dlaev2 returns rt1 and rt2, the eigenvalues of the matrix where |RT1| > |RT2|,
// and [cs1, sn1] which is the unit right eigenvalue for RT1.
// [ cs1 sn1] [a b] [cs1 -sn1] = [rt1 0]
// [-sn1 cs1] [b c] [sn1 cs1] [ 0 rt2]
//
// Dlaev2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlaev2(a, b, c float64) (rt1, rt2, cs1, sn1 float64) {
sm := a + c
df := a - c
adf := math.Abs(df)
tb := b + b
ab := math.Abs(tb)
acmx := c
acmn := a
if math.Abs(a) > math.Abs(c) {
acmx = a
acmn = c
}
var rt float64
if adf > ab {
rt = adf * math.Sqrt(1+(ab/adf)*(ab/adf))
} else if adf < ab {
rt = ab * math.Sqrt(1+(adf/ab)*(adf/ab))
} else {
rt = ab * math.Sqrt(2)
}
var sgn1 float64
if sm < 0 {
rt1 = 0.5 * (sm - rt)
sgn1 = -1
rt2 = (acmx/rt1)*acmn - (b/rt1)*b
} else if sm > 0 {
rt1 = 0.5 * (sm + rt)
sgn1 = 1
rt2 = (acmx/rt1)*acmn - (b/rt1)*b
} else {
rt1 = 0.5 * rt
rt2 = -0.5 * rt
sgn1 = 1
}
var cs, sgn2 float64
if df >= 0 {
cs = df + rt
sgn2 = 1
} else {
cs = df - rt
sgn2 = -1
}
acs := math.Abs(cs)
if acs > ab {
ct := -tb / cs
sn1 = 1 / math.Sqrt(1+ct*ct)
cs1 = ct * sn1
} else {
if ab == 0 {
cs1 = 1
sn1 = 0
} else {
tn := -cs / tb
cs1 = 1 / math.Sqrt(1+tn*tn)
sn1 = tn * cs1
}
}
if sgn1 == sgn2 {
tn := cs1
cs1 = -sn1
sn1 = tn
}
return rt1, rt2, cs1, sn1
}