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dlahqr.go
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/
dlahqr.go
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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"
"testing"
"golang.org/x/exp/rand"
"gonum.org/v1/gonum/blas"
"gonum.org/v1/gonum/blas/blas64"
)
type Dlahqrer interface {
Dlahqr(wantt, wantz bool, n, ilo, ihi int, h []float64, ldh int, wr, wi []float64, iloz, ihiz int, z []float64, ldz int) int
}
type dlahqrTest struct {
h blas64.General
ilo, ihi int
iloz, ihiz int
wantt, wantz bool
evWant []complex128 // Optional slice holding known eigenvalues.
}
func DlahqrTest(t *testing.T, impl Dlahqrer) {
rnd := rand.New(rand.NewSource(1))
// Tests that choose the [ilo:ihi+1,ilo:ihi+1] and
// [iloz:ihiz+1,ilo:ihi+1] blocks randomly.
for _, wantt := range []bool{true, false} {
for _, wantz := range []bool{true, false} {
for _, n := range []int{1, 2, 3, 4, 5, 6, 10, 18, 31, 53} {
for _, extra := range []int{0, 1, 11} {
for cas := 0; cas < 100; cas++ {
ilo := rnd.Intn(n)
ihi := rnd.Intn(n)
if ilo > ihi {
ilo, ihi = ihi, ilo
}
iloz := rnd.Intn(ilo + 1)
ihiz := ihi + rnd.Intn(n-ihi)
h := randomHessenberg(n, n+extra, rnd)
if ilo-1 >= 0 {
h.Data[ilo*h.Stride+ilo-1] = 0
}
if ihi+1 < n {
h.Data[(ihi+1)*h.Stride+ihi] = 0
}
test := dlahqrTest{
h: h,
ilo: ilo,
ihi: ihi,
iloz: iloz,
ihiz: ihiz,
wantt: wantt,
wantz: wantz,
}
testDlahqr(t, impl, test)
}
}
}
}
}
// Tests that make sure that some potentially problematic corner cases,
// like zero-sized matrix, are covered.
for _, wantt := range []bool{true, false} {
for _, wantz := range []bool{true, false} {
for _, extra := range []int{0, 1, 11} {
for _, test := range []dlahqrTest{
{
h: randomHessenberg(0, extra, rnd),
ilo: 0,
ihi: -1,
iloz: 0,
ihiz: -1,
},
{
h: randomHessenberg(1, 1+extra, rnd),
ilo: 0,
ihi: 0,
iloz: 0,
ihiz: 0,
},
{
h: randomHessenberg(2, 2+extra, rnd),
ilo: 1,
ihi: 1,
iloz: 1,
ihiz: 1,
},
{
h: randomHessenberg(2, 2+extra, rnd),
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 1,
},
{
h: randomHessenberg(10, 10+extra, rnd),
ilo: 0,
ihi: 0,
iloz: 0,
ihiz: 0,
},
{
h: randomHessenberg(10, 10+extra, rnd),
ilo: 0,
ihi: 9,
iloz: 0,
ihiz: 9,
},
{
h: randomHessenberg(10, 10+extra, rnd),
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 1,
},
{
h: randomHessenberg(10, 10+extra, rnd),
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 9,
},
{
h: randomHessenberg(10, 10+extra, rnd),
ilo: 9,
ihi: 9,
iloz: 0,
ihiz: 9,
},
} {
if test.ilo-1 >= 0 {
test.h.Data[test.ilo*test.h.Stride+test.ilo-1] = 0
}
if test.ihi+1 < test.h.Rows {
test.h.Data[(test.ihi+1)*test.h.Stride+test.ihi] = 0
}
test.wantt = wantt
test.wantz = wantz
testDlahqr(t, impl, test)
}
}
}
}
// Tests with explicit eigenvalues computed by Octave.
for _, test := range []dlahqrTest{
{
h: blas64.General{
Rows: 1,
Cols: 1,
Stride: 1,
Data: []float64{7.09965484086874e-1},
},
ilo: 0,
ihi: 0,
iloz: 0,
ihiz: 0,
evWant: []complex128{7.09965484086874e-1},
},
{
h: blas64.General{
Rows: 2,
Cols: 2,
Stride: 2,
Data: []float64{
0, -1,
1, 0,
},
},
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 1,
evWant: []complex128{1i, -1i},
},
{
h: blas64.General{
Rows: 2,
Cols: 2,
Stride: 2,
Data: []float64{
6.25219991450918e-1, 8.17510791994361e-1,
3.31218891622294e-1, 1.24103744878131e-1,
},
},
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 1,
evWant: []complex128{9.52203547663447e-1, -2.02879811334398e-1},
},
{
h: blas64.General{
Rows: 4,
Cols: 4,
Stride: 4,
Data: []float64{
1, 0, 0, 0,
0, 6.25219991450918e-1, 8.17510791994361e-1, 0,
0, 3.31218891622294e-1, 1.24103744878131e-1, 0,
0, 0, 0, 1,
},
},
ilo: 1,
ihi: 2,
iloz: 0,
ihiz: 3,
evWant: []complex128{9.52203547663447e-1, -2.02879811334398e-1},
},
{
h: blas64.General{
Rows: 2,
Cols: 2,
Stride: 2,
Data: []float64{
-1.1219562276608, 6.85473513349362e-1,
-8.19951061145131e-1, 1.93728523178888e-1,
},
},
ilo: 0,
ihi: 1,
iloz: 0,
ihiz: 1,
evWant: []complex128{
-4.64113852240958e-1 + 3.59580510817350e-1i,
-4.64113852240958e-1 - 3.59580510817350e-1i,
},
},
{
h: blas64.General{
Rows: 5,
Cols: 5,
Stride: 5,
Data: []float64{
9.57590178533658e-1, -5.10651295522708e-1, 9.24974510015869e-1, -1.30016306879522e-1, 2.92601986926954e-2,
-1.08084756637964, 1.77529701001213, -1.36480197632509, 2.23196371219601e-1, 1.12912853063308e-1,
0, -8.44075612174676e-1, 1.067867614486, -2.55782915176399e-1, -2.00598563137468e-1,
0, 0, -5.67097237165410e-1, 2.07205057427341e-1, 6.54998340743380e-1,
0, 0, 0, -1.89441413886041e-1, -4.18125416021786e-1,
},
},
ilo: 0,
ihi: 4,
iloz: 0,
ihiz: 4,
evWant: []complex128{
2.94393309555622,
4.97029793606701e-1 + 3.63041654992384e-1i,
4.97029793606701e-1 - 3.63041654992384e-1i,
-1.74079119166145e-1 + 2.01570009462092e-1i,
-1.74079119166145e-1 - 2.01570009462092e-1i,
},
},
} {
test.wantt = true
test.wantz = true
testDlahqr(t, impl, test)
}
}
func testDlahqr(t *testing.T, impl Dlahqrer, test dlahqrTest) {
const tol = 1e-14
h := cloneGeneral(test.h)
n := h.Cols
extra := h.Stride - h.Cols
wantt := test.wantt
wantz := test.wantz
ilo := test.ilo
ihi := test.ihi
iloz := test.iloz
ihiz := test.ihiz
var z, zCopy blas64.General
if wantz {
z = eye(n, n+extra)
zCopy = cloneGeneral(z)
}
wr := nanSlice(ihi + 1)
wi := nanSlice(ihi + 1)
unconverged := impl.Dlahqr(wantt, wantz, n, ilo, ihi, h.Data, h.Stride, wr, wi, iloz, ihiz, z.Data, max(1, z.Stride))
prefix := fmt.Sprintf("Case wantt=%v, wantz=%v, n=%v, ilo=%v, ihi=%v, iloz=%v, ihiz=%v, extra=%v",
wantt, wantz, n, ilo, ihi, iloz, ihiz, extra)
if !generalOutsideAllNaN(h) {
t.Errorf("%v: out-of-range write to H\n%v", prefix, h.Data)
}
if !generalOutsideAllNaN(z) {
t.Errorf("%v: out-of-range write to Z\n%v", prefix, z.Data)
}
if !isUpperHessenberg(h) {
t.Logf("%v: H is not Hessenberg", prefix)
}
start := ilo // Index of the first computed eigenvalue.
if unconverged != 0 {
start = unconverged
if start == ihi+1 {
t.Logf("%v: no eigenvalue has converged", prefix)
}
}
// Check that wr and wi have not been modified in [:start].
if !isAllNaN(wr[:start]) {
t.Errorf("%v: unexpected modification of wr", prefix)
}
if !isAllNaN(wi[:start]) {
t.Errorf("%v: unexpected modification of wi", prefix)
}
var hasReal bool
for i := start; i <= ihi; {
if wi[i] == 0 { // Real eigenvalue.
hasReal = true
// Check that the eigenvalue corresponds to a 1×1 block
// on the diagonal of H.
if wantt {
if wr[i] != h.Data[i*h.Stride+i] {
t.Errorf("%v: wr[%v] != H[%v,%v]", prefix, i, i, i)
}
for _, index := range []struct{ r, c int }{
{i, i - 1}, // h h h
{i + 1, i - 1}, // 0 wr[i] h
{i + 1, i}, // 0 0 h
} {
if index.r >= n || index.c < 0 {
continue
}
if h.Data[index.r*h.Stride+index.c] != 0 {
t.Errorf("%v: H[%v,%v] != 0", prefix, index.r, index.c)
}
}
}
i++
continue
}
// Complex eigenvalue.
// In the conjugate pair the real parts must be equal.
if wr[i] != wr[i+1] {
t.Errorf("%v: real part of conjugate pair not equal, i=%v", prefix, i)
}
// The first imaginary part must be positive.
if wi[i] < 0 {
t.Errorf("%v: wi[%v] not positive", prefix, i)
}
// The second imaginary part must be negative with the same
// magnitude.
if wi[i] != -wi[i+1] {
t.Errorf("%v: wi[%v] != -wi[%v]", prefix, i, i+1)
}
if wantt {
// Check that wi[i] has the correct value.
if wr[i] != h.Data[i*h.Stride+i] {
t.Errorf("%v: wr[%v] != H[%v,%v]", prefix, i, i, i)
}
if wr[i] != h.Data[(i+1)*h.Stride+i+1] {
t.Errorf("%v: wr[%v] != H[%v,%v]", prefix, i, i+1, i+1)
}
prod := math.Abs(h.Data[(i+1)*h.Stride+i] * h.Data[i*h.Stride+i+1])
if math.Abs(math.Sqrt(prod)-wi[i]) > tol {
t.Errorf("%v: unexpected value of wi[%v]: want %v, got %v", prefix, i, math.Sqrt(prod), wi[i])
}
// Check that the corresponding diagonal block is 2×2.
for _, index := range []struct{ r, c int }{
{i, i - 1}, // i
{i + 1, i - 1}, // h h h h
{i + 2, i - 1}, // 0 wr[i] b h i
{i + 2, i}, // 0 c wr[i+1] h
{i + 2, i + 1}, // 0 0 0 h
} {
if index.r >= n || index.c < 0 {
continue
}
if h.Data[index.r*h.Stride+index.c] != 0 {
t.Errorf("%v: H[%v,%v] != 0", prefix, index.r, index.c)
}
}
}
i += 2
}
// If the number of found eigenvalues is odd, at least one must be real.
if (ihi+1-start)%2 != 0 && !hasReal {
t.Errorf("%v: expected at least one real eigenvalue", prefix)
}
// Compare found eigenvalues to the reference, if known.
if test.evWant != nil {
for i := start; i <= ihi; i++ {
ev := complex(wr[i], wi[i])
found, _ := containsComplex(test.evWant, ev, tol)
if !found {
t.Errorf("%v: unexpected eigenvalue %v", prefix, ev)
}
}
}
if !wantz {
return
}
// Z should contain the orthogonal matrix U.
if resid := residualOrthogonal(z, false); resid > tol*float64(n) {
t.Errorf("Case %v: Z is not orthogonal; resid=%v, want<=%v", prefix, resid, tol*float64(n))
}
// Z should have been modified only in the
// [iloz:ihiz+1,ilo:ihi+1] block.
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
if iloz <= i && i <= ihiz && ilo <= j && j <= ihi {
continue
}
if z.Data[i*z.Stride+j] != zCopy.Data[i*zCopy.Stride+j] {
t.Errorf("%v: Z modified outside of [iloz:ihiz+1,ilo:ihi+1] block", prefix)
}
}
}
if wantt {
hu := eye(n, n)
blas64.Gemm(blas.NoTrans, blas.NoTrans, 1, test.h, z, 0, hu)
uhu := eye(n, n)
blas64.Gemm(blas.Trans, blas.NoTrans, 1, z, hu, 0, uhu)
if !equalApproxGeneral(uhu, h, 10*tol) {
t.Errorf("%v: Zᵀ*(initial H)*Z and (final H) are not equal", prefix)
}
}
}