| // 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) |
| |
| const eps = dlamchE |
| anorm := math.Max(dlange(lapack.MaxColumnSum, n, n, a.Data, a.Stride), safmin) |
| enorm := math.Max(dlange(lapack.MaxColumnSum, n, n, e.Data, e.Stride), eps) |
| 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) |
| |
| const eps = dlamchE |
| anorm := math.Max(dlange(lapack.MaxRowSum, n, n, a.Data, a.Stride), safmin) |
| enorm := math.Max(dlange(lapack.MaxColumnSum, n, n, e.Data, e.Stride), eps) |
| 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) |
| } |
