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// Copyright ©2016 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 testlapack
import (
"fmt"
"math"
"math/cmplx"
"strconv"
"testing"
"golang.org/x/exp/rand"
"gonum.org/v1/gonum/blas"
"gonum.org/v1/gonum/blas/blas64"
"gonum.org/v1/gonum/lapack"
)
type Dgeever interface {
Dgeev(jobvl lapack.LeftEVJob, jobvr lapack.RightEVJob, n int, a []float64, lda int,
wr, wi []float64, vl []float64, ldvl int, vr []float64, ldvr int, work []float64, lwork int) int
}
type dgeevTest struct {
a blas64.General
evWant []complex128 // If nil, the eigenvalues are not known.
valTol float64 // Tolerance for eigenvalue checks.
vecTol float64 // Tolerance for eigenvector checks.
}
func DgeevTest(t *testing.T, impl Dgeever) {
rnd := rand.New(rand.NewSource(1))
for i, test := range []dgeevTest{
{
a: A123{}.Matrix(),
evWant: A123{}.Eigenvalues(),
},
dgeevTestForAntisymRandom(10, rnd),
dgeevTestForAntisymRandom(11, rnd),
dgeevTestForAntisymRandom(50, rnd),
dgeevTestForAntisymRandom(51, rnd),
dgeevTestForAntisymRandom(100, rnd),
dgeevTestForAntisymRandom(101, rnd),
{
a: Circulant(2).Matrix(),
evWant: Circulant(2).Eigenvalues(),
},
{
a: Circulant(3).Matrix(),
evWant: Circulant(3).Eigenvalues(),
},
{
a: Circulant(4).Matrix(),
evWant: Circulant(4).Eigenvalues(),
},
{
a: Circulant(5).Matrix(),
evWant: Circulant(5).Eigenvalues(),
},
{
a: Circulant(10).Matrix(),
evWant: Circulant(10).Eigenvalues(),
},
{
a: Circulant(15).Matrix(),
evWant: Circulant(15).Eigenvalues(),
valTol: 1e-12,
},
{
a: Circulant(30).Matrix(),
evWant: Circulant(30).Eigenvalues(),
valTol: 1e-11,
},
{
a: Circulant(50).Matrix(),
evWant: Circulant(50).Eigenvalues(),
valTol: 1e-11,
},
{
a: Circulant(101).Matrix(),
evWant: Circulant(101).Eigenvalues(),
valTol: 1e-10,
},
{
a: Circulant(150).Matrix(),
evWant: Circulant(150).Eigenvalues(),
valTol: 1e-9,
},
{
a: Clement(2).Matrix(),
evWant: Clement(2).Eigenvalues(),
},
{
a: Clement(3).Matrix(),
evWant: Clement(3).Eigenvalues(),
},
{
a: Clement(4).Matrix(),
evWant: Clement(4).Eigenvalues(),
},
{
a: Clement(5).Matrix(),
evWant: Clement(5).Eigenvalues(),
},
{
a: Clement(10).Matrix(),
evWant: Clement(10).Eigenvalues(),
},
{
a: Clement(15).Matrix(),
evWant: Clement(15).Eigenvalues(),
},
{
a: Clement(30).Matrix(),
evWant: Clement(30).Eigenvalues(),
valTol: 1e-11,
},
{
a: Clement(50).Matrix(),
evWant: Clement(50).Eigenvalues(),
valTol: 1e-8,
},
{
a: Creation(2).Matrix(),
evWant: Creation(2).Eigenvalues(),
},
{
a: Creation(3).Matrix(),
evWant: Creation(3).Eigenvalues(),
},
{
a: Creation(4).Matrix(),
evWant: Creation(4).Eigenvalues(),
},
{
a: Creation(5).Matrix(),
evWant: Creation(5).Eigenvalues(),
},
{
a: Creation(10).Matrix(),
evWant: Creation(10).Eigenvalues(),
},
{
a: Creation(15).Matrix(),
evWant: Creation(15).Eigenvalues(),
},
{
a: Creation(30).Matrix(),
evWant: Creation(30).Eigenvalues(),
},
{
a: Creation(50).Matrix(),
evWant: Creation(50).Eigenvalues(),
},
{
a: Creation(101).Matrix(),
evWant: Creation(101).Eigenvalues(),
},
{
a: Creation(150).Matrix(),
evWant: Creation(150).Eigenvalues(),
},
{
a: Diagonal(0).Matrix(),
evWant: Diagonal(0).Eigenvalues(),
},
{
a: Diagonal(10).Matrix(),
evWant: Diagonal(10).Eigenvalues(),
},
{
a: Diagonal(50).Matrix(),
evWant: Diagonal(50).Eigenvalues(),
},
{
a: Diagonal(151).Matrix(),
evWant: Diagonal(151).Eigenvalues(),
},
{
a: Downshift(2).Matrix(),
evWant: Downshift(2).Eigenvalues(),
},
{
a: Downshift(3).Matrix(),
evWant: Downshift(3).Eigenvalues(),
},
{
a: Downshift(4).Matrix(),
evWant: Downshift(4).Eigenvalues(),
},
{
a: Downshift(5).Matrix(),
evWant: Downshift(5).Eigenvalues(),
},
{
a: Downshift(10).Matrix(),
evWant: Downshift(10).Eigenvalues(),
},
{
a: Downshift(15).Matrix(),
evWant: Downshift(15).Eigenvalues(),
},
{
a: Downshift(30).Matrix(),
evWant: Downshift(30).Eigenvalues(),
},
{
a: Downshift(50).Matrix(),
evWant: Downshift(50).Eigenvalues(),
},
{
a: Downshift(101).Matrix(),
evWant: Downshift(101).Eigenvalues(),
},
{
a: Downshift(150).Matrix(),
evWant: Downshift(150).Eigenvalues(),
},
{
a: Fibonacci(2).Matrix(),
evWant: Fibonacci(2).Eigenvalues(),
},
{
a: Fibonacci(3).Matrix(),
evWant: Fibonacci(3).Eigenvalues(),
},
{
a: Fibonacci(4).Matrix(),
evWant: Fibonacci(4).Eigenvalues(),
},
{
a: Fibonacci(5).Matrix(),
evWant: Fibonacci(5).Eigenvalues(),
},
{
a: Fibonacci(10).Matrix(),
evWant: Fibonacci(10).Eigenvalues(),
},
{
a: Fibonacci(15).Matrix(),
evWant: Fibonacci(15).Eigenvalues(),
},
{
a: Fibonacci(30).Matrix(),
evWant: Fibonacci(30).Eigenvalues(),
},
{
a: Fibonacci(50).Matrix(),
evWant: Fibonacci(50).Eigenvalues(),
},
{
a: Fibonacci(101).Matrix(),
evWant: Fibonacci(101).Eigenvalues(),
},
{
a: Fibonacci(150).Matrix(),
evWant: Fibonacci(150).Eigenvalues(),
},
{
a: Gear(2).Matrix(),
evWant: Gear(2).Eigenvalues(),
},
{
a: Gear(3).Matrix(),
evWant: Gear(3).Eigenvalues(),
},
{
a: Gear(4).Matrix(),
evWant: Gear(4).Eigenvalues(),
valTol: 1e-7,
vecTol: 1e-8,
},
{
a: Gear(5).Matrix(),
evWant: Gear(5).Eigenvalues(),
},
{
a: Gear(10).Matrix(),
evWant: Gear(10).Eigenvalues(),
valTol: 1e-8,
},
{
a: Gear(15).Matrix(),
evWant: Gear(15).Eigenvalues(),
},
{
a: Gear(30).Matrix(),
evWant: Gear(30).Eigenvalues(),
valTol: 1e-8,
},
{
a: Gear(50).Matrix(),
evWant: Gear(50).Eigenvalues(),
valTol: 1e-8,
},
{
a: Gear(101).Matrix(),
evWant: Gear(101).Eigenvalues(),
},
{
a: Gear(150).Matrix(),
evWant: Gear(150).Eigenvalues(),
valTol: 1e-8,
},
{
a: Grcar{N: 10, K: 3}.Matrix(),
evWant: Grcar{N: 10, K: 3}.Eigenvalues(),
},
{
a: Grcar{N: 10, K: 7}.Matrix(),
evWant: Grcar{N: 10, K: 7}.Eigenvalues(),
},
{
a: Grcar{N: 11, K: 7}.Matrix(),
evWant: Grcar{N: 11, K: 7}.Eigenvalues(),
},
{
a: Grcar{N: 50, K: 3}.Matrix(),
evWant: Grcar{N: 50, K: 3}.Eigenvalues(),
},
{
a: Grcar{N: 51, K: 3}.Matrix(),
evWant: Grcar{N: 51, K: 3}.Eigenvalues(),
},
{
a: Grcar{N: 50, K: 10}.Matrix(),
evWant: Grcar{N: 50, K: 10}.Eigenvalues(),
},
{
a: Grcar{N: 51, K: 10}.Matrix(),
evWant: Grcar{N: 51, K: 10}.Eigenvalues(),
},
{
a: Grcar{N: 50, K: 30}.Matrix(),
evWant: Grcar{N: 50, K: 30}.Eigenvalues(),
},
{
a: Grcar{N: 150, K: 2}.Matrix(),
evWant: Grcar{N: 150, K: 2}.Eigenvalues(),
},
{
a: Grcar{N: 150, K: 148}.Matrix(),
evWant: Grcar{N: 150, K: 148}.Eigenvalues(),
},
{
a: Hanowa{N: 6, Alpha: 17}.Matrix(),
evWant: Hanowa{N: 6, Alpha: 17}.Eigenvalues(),
},
{
a: Hanowa{N: 50, Alpha: -1}.Matrix(),
evWant: Hanowa{N: 50, Alpha: -1}.Eigenvalues(),
},
{
a: Hanowa{N: 100, Alpha: -1}.Matrix(),
evWant: Hanowa{N: 100, Alpha: -1}.Eigenvalues(),
},
{
a: Lesp(2).Matrix(),
evWant: Lesp(2).Eigenvalues(),
},
{
a: Lesp(3).Matrix(),
evWant: Lesp(3).Eigenvalues(),
},
{
a: Lesp(4).Matrix(),
evWant: Lesp(4).Eigenvalues(),
},
{
a: Lesp(5).Matrix(),
evWant: Lesp(5).Eigenvalues(),
},
{
a: Lesp(10).Matrix(),
evWant: Lesp(10).Eigenvalues(),
},
{
a: Lesp(15).Matrix(),
evWant: Lesp(15).Eigenvalues(),
},
{
a: Lesp(30).Matrix(),
evWant: Lesp(30).Eigenvalues(),
},
{
a: Lesp(50).Matrix(),
evWant: Lesp(50).Eigenvalues(),
valTol: 1e-12,
},
{
a: Lesp(101).Matrix(),
evWant: Lesp(101).Eigenvalues(),
valTol: 1e-12,
},
{
a: Lesp(150).Matrix(),
evWant: Lesp(150).Eigenvalues(),
valTol: 1e-12,
},
{
a: Rutis{}.Matrix(),
evWant: Rutis{}.Eigenvalues(),
},
{
a: Tris{N: 74, X: 1, Y: -2, Z: 1}.Matrix(),
evWant: Tris{N: 74, X: 1, Y: -2, Z: 1}.Eigenvalues(),
},
{
a: Tris{N: 74, X: 1, Y: 2, Z: -3}.Matrix(),
evWant: Tris{N: 74, X: 1, Y: 2, Z: -3}.Eigenvalues(),
},
{
a: Tris{N: 75, X: 1, Y: 2, Z: -3}.Matrix(),
evWant: Tris{N: 75, X: 1, Y: 2, Z: -3}.Eigenvalues(),
},
{
a: Wilk4{}.Matrix(),
evWant: Wilk4{}.Eigenvalues(),
},
{
a: Wilk12{}.Matrix(),
evWant: Wilk12{}.Eigenvalues(),
valTol: 1e-7,
},
{
a: Wilk20(0).Matrix(),
evWant: Wilk20(0).Eigenvalues(),
},
{
a: Wilk20(1e-10).Matrix(),
evWant: Wilk20(1e-10).Eigenvalues(),
valTol: 1e-12,
},
{
a: Zero(1).Matrix(),
evWant: Zero(1).Eigenvalues(),
},
{
a: Zero(10).Matrix(),
evWant: Zero(10).Eigenvalues(),
},
{
a: Zero(50).Matrix(),
evWant: Zero(50).Eigenvalues(),
},
{
a: Zero(100).Matrix(),
evWant: Zero(100).Eigenvalues(),
},
} {
for _, jobvl := range []lapack.LeftEVJob{lapack.LeftEVCompute, lapack.LeftEVNone} {
for _, jobvr := range []lapack.RightEVJob{lapack.RightEVCompute, lapack.RightEVNone} {
for _, extra := range []int{0, 11} {
for _, wl := range []worklen{minimumWork, mediumWork, optimumWork} {
testDgeev(t, impl, strconv.Itoa(i), test, jobvl, jobvr, extra, wl)
}
}
}
}
}
for _, n := range []int{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 50, 51, 100, 101} {
for _, jobvl := range []lapack.LeftEVJob{lapack.LeftEVCompute, lapack.LeftEVNone} {
for _, jobvr := range []lapack.RightEVJob{lapack.RightEVCompute, lapack.RightEVNone} {
for cas := 0; cas < 10; cas++ {
// Create a block diagonal matrix with
// random eigenvalues of random multiplicity.
ev := make([]complex128, n)
tmat := zeros(n, n, n)
for i := 0; i < n; {
re := rnd.NormFloat64()
if i == n-1 || rnd.Float64() < 0.5 {
// Real eigenvalue.
nb := rnd.Intn(min(4, n-i)) + 1
for k := 0; k < nb; k++ {
tmat.Data[i*tmat.Stride+i] = re
ev[i] = complex(re, 0)
i++
}
continue
}
// Complex eigenvalue.
im := rnd.NormFloat64()
nb := rnd.Intn(min(4, (n-i)/2)) + 1
for k := 0; k < nb; k++ {
// 2×2 block for the complex eigenvalue.
tmat.Data[i*tmat.Stride+i] = re
tmat.Data[(i+1)*tmat.Stride+i+1] = re
tmat.Data[(i+1)*tmat.Stride+i] = -im
tmat.Data[i*tmat.Stride+i+1] = im
ev[i] = complex(re, im)
ev[i+1] = complex(re, -im)
i += 2
}
}
// Compute A = Q T Qᵀ where Q is an
// orthogonal matrix.
q := randomOrthogonal(n, rnd)
tq := zeros(n, n, n)
blas64.Gemm(blas.NoTrans, blas.Trans, 1, tmat, q, 0, tq)
a := zeros(n, n, n)
blas64.Gemm(blas.NoTrans, blas.NoTrans, 1, q, tq, 0, a)
test := dgeevTest{
a: a,
evWant: ev,
vecTol: 1e-7,
}
testDgeev(t, impl, "random", test, jobvl, jobvr, 0, optimumWork)
}
}
}
}
}
func testDgeev(t *testing.T, impl Dgeever, tc string, test dgeevTest, jobvl lapack.LeftEVJob, jobvr lapack.RightEVJob, extra int, wl worklen) {
const defaultTol = 1e-13
valTol := test.valTol
if valTol == 0 {
valTol = defaultTol
}
vecTol := test.vecTol
if vecTol == 0 {
vecTol = defaultTol
}
a := cloneGeneral(test.a)
n := a.Rows
var vl blas64.General
if jobvl == lapack.LeftEVCompute {
vl = nanGeneral(n, n, n)
} else {
vl.Stride = 1
}
var vr blas64.General
if jobvr == lapack.RightEVCompute {
vr = nanGeneral(n, n, n)
} else {
vr.Stride = 1
}
wr := make([]float64, n)
wi := make([]float64, n)
var lwork int
switch wl {
case minimumWork:
if jobvl == lapack.LeftEVCompute || jobvr == lapack.RightEVCompute {
lwork = max(1, 4*n)
} else {
lwork = max(1, 3*n)
}
case mediumWork:
work := make([]float64, 1)
impl.Dgeev(jobvl, jobvr, n, a.Data, a.Stride, wr, wi, vl.Data, vl.Stride, vr.Data, vr.Stride, work, -1)
if jobvl == lapack.LeftEVCompute || jobvr == lapack.RightEVCompute {
lwork = (int(work[0]) + 4*n) / 2
} else {
lwork = (int(work[0]) + 3*n) / 2
}
lwork = max(1, lwork)
case optimumWork:
work := make([]float64, 1)
impl.Dgeev(jobvl, jobvr, n, a.Data, a.Stride, wr, wi, vl.Data, vl.Stride, vr.Data, vr.Stride, work, -1)
lwork = int(work[0])
}
work := make([]float64, lwork)
first := impl.Dgeev(jobvl, jobvr, n, a.Data, a.Stride, wr, wi,
vl.Data, vl.Stride, vr.Data, vr.Stride, work, len(work))
prefix := fmt.Sprintf("Case #%v: n=%v, jobvl=%c, jobvr=%c, extra=%v, work=%v",
tc, n, jobvl, jobvr, extra, wl)
if !generalOutsideAllNaN(vl) {
t.Errorf("%v: out-of-range write to VL", prefix)
}
if !generalOutsideAllNaN(vr) {
t.Errorf("%v: out-of-range write to VR", prefix)
}
if first > 0 {
t.Logf("%v: all eigenvalues haven't been computed, first=%v", prefix, first)
}
// Check that conjugate pair eigenvalues are ordered correctly.
for i := first; i < n; {
if wi[i] == 0 {
i++
continue
}
if wr[i] != wr[i+1] {
t.Errorf("%v: real parts of %vth conjugate pair not equal", prefix, i)
}
if wi[i] < 0 || wi[i+1] >= 0 {
t.Errorf("%v: unexpected ordering of %vth conjugate pair", prefix, i)
}
i += 2
}
// Check the computed eigenvalues against provided known eigenvalues.
if test.evWant != nil {
used := make([]bool, n)
for i := first; i < n; i++ {
evGot := complex(wr[i], wi[i])
idx := -1
for k, evWant := range test.evWant {
if !used[k] && cmplx.Abs(evWant-evGot) < valTol {
idx = k
used[k] = true
break
}
}
if idx == -1 {
t.Errorf("%v: unexpected eigenvalue %v", prefix, evGot)
}
}
}
if first > 0 || (jobvl == lapack.LeftEVNone && jobvr == lapack.RightEVNone) {
// No eigenvectors have been computed.
return
}
// Check that the columns of VL and VR are eigenvectors that:
// - correspond to the computed eigenvalues
// - have Euclidean norm equal to 1
// - have the largest component real
bi := blas64.Implementation()
if jobvr == lapack.RightEVCompute {
resid := residualRightEV(test.a, vr, wr, wi)
if resid > vecTol {
t.Errorf("%v: unexpected right eigenvectors; residual=%v, want<=%v", prefix, resid, vecTol)
}
for j := 0; j < n; j++ {
nrm := 1.0
if wi[j] == 0 {
nrm = bi.Dnrm2(n, vr.Data[j:], vr.Stride)
} else if wi[j] > 0 {
nrm = math.Hypot(bi.Dnrm2(n, vr.Data[j:], vr.Stride), bi.Dnrm2(n, vr.Data[j+1:], vr.Stride))
}
diff := math.Abs(nrm - 1)
if diff > defaultTol {
t.Errorf("%v: unexpected Euclidean norm of right eigenvector; |VR[%v]-1|=%v, want<=%v",
prefix, j, diff, defaultTol)
}
if wi[j] > 0 {
var vmax float64 // Largest component in the column
var vrmax float64 // Largest real component in the column
for i := 0; i < n; i++ {
vtest := math.Hypot(vr.Data[i*vr.Stride+j], vr.Data[i*vr.Stride+j+1])
vmax = math.Max(vmax, vtest)
if vr.Data[i*vr.Stride+j+1] == 0 {
vrmax = math.Max(vrmax, math.Abs(vr.Data[i*vr.Stride+j]))
}
}
if vrmax/vmax < 1-defaultTol {
t.Errorf("%v: largest component of %vth right eigenvector is not real", prefix, j)
}
}
}
}
if jobvl == lapack.LeftEVCompute {
resid := residualLeftEV(test.a, vl, wr, wi)
if resid > vecTol {
t.Errorf("%v: unexpected left eigenvectors; residual=%v, want<=%v", prefix, resid, vecTol)
}
for j := 0; j < n; j++ {
nrm := 1.0
if wi[j] == 0 {
nrm = bi.Dnrm2(n, vl.Data[j:], vl.Stride)
} else if wi[j] > 0 {
nrm = math.Hypot(bi.Dnrm2(n, vl.Data[j:], vl.Stride), bi.Dnrm2(n, vl.Data[j+1:], vl.Stride))
}
diff := math.Abs(nrm - 1)
if diff > defaultTol {
t.Errorf("%v: unexpected Euclidean norm of left eigenvector; |VL[%v]-1|=%v, want<=%v",
prefix, j, diff, defaultTol)
}
if wi[j] > 0 {
var vmax float64 // Largest component in the column
var vrmax float64 // Largest real component in the column
for i := 0; i < n; i++ {
vtest := math.Hypot(vl.Data[i*vl.Stride+j], vl.Data[i*vl.Stride+j+1])
vmax = math.Max(vmax, vtest)
if vl.Data[i*vl.Stride+j+1] == 0 {
vrmax = math.Max(vrmax, math.Abs(vl.Data[i*vl.Stride+j]))
}
}
if vrmax/vmax < 1-defaultTol {
t.Errorf("%v: largest component of %vth left eigenvector is not real", prefix, j)
}
}
}
}
}
func dgeevTestForAntisymRandom(n int, rnd *rand.Rand) dgeevTest {
a := NewAntisymRandom(n, rnd)
return dgeevTest{
a: a.Matrix(),
evWant: a.Eigenvalues(),
}
}
// residualRightEV returns the residual
// | A E - E W|_1 / ( |A|_1 |E|_1 )
// where the columns of E contain the right eigenvectors of A and W is a block diagonal matrix with
// a 1×1 block for each real eigenvalue and a 2×2 block for each complex conjugate pair.
func residualRightEV(a, e blas64.General, wr, wi []float64) float64 {
// The implementation follows DGET22 routine from the Reference LAPACK's
// testing suite.
n := a.Rows
if n == 0 {
return 0
}
bi := blas64.Implementation()
ldr := n
r := make([]float64, n*ldr)
var (
wmat [4]float64
ipair int
)
for j := 0; j < n; j++ {
if ipair == 0 && wi[j] != 0 {
ipair = 1
}
switch ipair {
case 0:
// Real eigenvalue, multiply j-th column of E with it.
bi.Daxpy(n, wr[j], e.Data[j:], e.Stride, r[j:], ldr)
case 1:
// First of complex conjugate pair of eigenvalues
wmat[0], wmat[1] = wr[j], wi[j]
wmat[2], wmat[3] = -wi[j], wr[j]
bi.Dgemm(blas.NoTrans, blas.NoTrans, n, 2, 2, 1, e.Data[j:], e.Stride, wmat[:], 2, 0, r[j:], ldr)
ipair = 2
case 2:
// Second of complex conjugate pair of eigenvalues
ipair = 0
}
}
bi.Dgemm(blas.NoTrans, blas.NoTrans, n, n, n, 1, a.Data, a.Stride, e.Data, e.Stride, -1, r, ldr)
unfl := dlamchS
ulp := dlamchE
anorm := math.Max(dlange(lapack.MaxColumnSum, n, n, a.Data, a.Stride), unfl)
enorm := math.Max(dlange(lapack.MaxColumnSum, n, n, e.Data, e.Stride), ulp)
errnorm := dlange(lapack.MaxColumnSum, n, n, r, ldr) / enorm
if anorm > errnorm {
return errnorm / anorm
}
if anorm < 1 {
return math.Min(errnorm, anorm) / anorm
}
return math.Min(errnorm/anorm, 1)
}
// residualLeftEV returns the residual
// | Aᵀ E - E Wᵀ|_1 / ( |Aᵀ|_1 |E|_1 )
// where the columns of E contain the left eigenvectors of A and W is a block diagonal matrix with
// a 1×1 block for each real eigenvalue and a 2×2 block for each complex conjugate pair.
func residualLeftEV(a, e blas64.General, wr, wi []float64) float64 {
// The implementation follows DGET22 routine from the Reference LAPACK's
// testing suite.
n := a.Rows
if n == 0 {
return 0
}
bi := blas64.Implementation()
ldr := n
r := make([]float64, n*ldr)
var (
wmat [4]float64
ipair int
)
for j := 0; j < n; j++ {
if ipair == 0 && wi[j] != 0 {
ipair = 1
}
switch ipair {
case 0:
// Real eigenvalue, multiply j-th column of E with it.
bi.Daxpy(n, wr[j], e.Data[j:], e.Stride, r[j:], ldr)
case 1:
// First of complex conjugate pair of eigenvalues
wmat[0], wmat[1] = wr[j], wi[j]
wmat[2], wmat[3] = -wi[j], wr[j]
bi.Dgemm(blas.NoTrans, blas.Trans, n, 2, 2, 1, e.Data[j:], e.Stride, wmat[:], 2, 0, r[j:], ldr)
ipair = 2
case 2:
// Second of complex conjugate pair of eigenvalues
ipair = 0
}
}
bi.Dgemm(blas.Trans, blas.NoTrans, n, n, n, 1, a.Data, a.Stride, e.Data, e.Stride, -1, r, ldr)
unfl := dlamchS
ulp := dlamchE
anorm := math.Max(dlange(lapack.MaxRowSum, n, n, a.Data, a.Stride), unfl)
enorm := math.Max(dlange(lapack.MaxColumnSum, n, n, e.Data, e.Stride), ulp)
errnorm := dlange(lapack.MaxColumnSum, n, n, r, ldr) / enorm
if anorm > errnorm {
return errnorm / anorm
}
if anorm < 1 {
return math.Min(errnorm, anorm) / anorm
}
return math.Min(errnorm/anorm, 1)
}