--- /dev/null
+// 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, 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 !isOrthonormal(z) {
+ t.Errorf("%v: Z is not orthogonal", prefix)
+ }
+ // 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: Z^T*(initial H)*Z and (final H) are not equal", prefix)
+ }
+ }
+}
OSDN Git Service
