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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) } } }