func TestMinimalSurface(t *testing.T) {
for _, size := range [][2]int{
{20, 30},
{30, 30},
{50, 40},
} {
f := NewMinimalSurface(size[0], size[1])
x0 := f.InitX()
grad := make([]float64, len(x0))
f.Grad(grad, x0)
fdGrad := fd.Gradient(nil, f.Func, x0, &fd.Settings{Formula: fd.Central})
// Test that the numerical and analytical gradients agree.
dist := floats.Distance(grad, fdGrad, math.Inf(1))
if dist > 1e-9 {
t.Errorf("grid %v x %v: numerical and analytical gradient do not match. |fdGrad - grad|_∞ = %v",
size[0], size[1], dist)
}
// Test that the gradient at the minimum is small enough.
// In some sense this test is not completely correct because ExactX
// returns the exact solution to the continuous problem projected on the
// grid, not the exact solution to the discrete problem which we are
// solving. This is the reason why a relatively loose tolerance 1e-4
// must be used.
xSol := f.ExactX()
f.Grad(grad, xSol)
norm := floats.Norm(grad, math.Inf(1))
if norm > 1e-4 {
t.Errorf("grid %v x %v: gradient at the minimum not small enough. |grad|_∞ = %v",
size[0], size[1], norm)
}
}
}
// isLeftEigenvectorOf returns whether the vector yRe+i*yIm, where i is the
// imaginary unit, is the left eigenvector of A corresponding to the eigenvalue
// lambda.
//
// A left eigenvector corresponding to a complex eigenvalue λ is a complex
// non-zero vector y such that
// y^H A = λ y^H,
// which is equivalent for real A to
// A^T y = conj(λ) y,
func isLeftEigenvectorOf(a blas64.General, yRe, yIm []float64, lambda complex128, tol float64) bool {
if a.Rows != a.Cols {
panic("matrix not square")
}
if imag(lambda) != 0 && yIm == nil {
// Complex eigenvalue of a real matrix cannot have a real
// eigenvector.
return false
}
n := a.Rows
// Compute A^T real(y) and store the result into yReAns.
yReAns := make([]float64, n)
blas64.Gemv(blas.Trans, 1, a, blas64.Vector{1, yRe}, 0, blas64.Vector{1, yReAns})
if imag(lambda) == 0 && yIm == nil {
// Real eigenvalue and eigenvector.
// Compute λy and store the result into lambday.
lambday := make([]float64, n)
floats.AddScaled(lambday, real(lambda), yRe)
if floats.Distance(yReAns, lambday, math.Inf(1)) > tol {
return false
}
return true
}
// Complex eigenvector, and real or complex eigenvalue.
// Compute A^T imag(y) and store the result into yImAns.
yImAns := make([]float64, n)
blas64.Gemv(blas.Trans, 1, a, blas64.Vector{1, yIm}, 0, blas64.Vector{1, yImAns})
// Compute conj(λ)y and store the result into lambday.
lambda = cmplx.Conj(lambda)
lambday := make([]complex128, n)
for i := range lambday {
lambday[i] = lambda * complex(yRe[i], yIm[i])
}
for i, v := range lambday {
ay := complex(yReAns[i], yImAns[i])
if cmplx.Abs(v-ay) > tol {
return false
}
}
return true
}
// isRightEigenvectorOf returns whether the vector xRe+i*xIm, where i is the
// imaginary unit, is the right eigenvector of A corresponding to the eigenvalue
// lambda.
//
// A right eigenvector corresponding to a complex eigenvalue λ is a complex
// non-zero vector x such that
// A x = λ x.
func isRightEigenvectorOf(a blas64.General, xRe, xIm []float64, lambda complex128, tol float64) bool {
if a.Rows != a.Cols {
panic("matrix not square")
}
if imag(lambda) != 0 && xIm == nil {
// Complex eigenvalue of a real matrix cannot have a real
// eigenvector.
return false
}
n := a.Rows
// Compute A real(x) and store the result into xReAns.
xReAns := make([]float64, n)
blas64.Gemv(blas.NoTrans, 1, a, blas64.Vector{1, xRe}, 0, blas64.Vector{1, xReAns})
if imag(lambda) == 0 && xIm == nil {
// Real eigenvalue and eigenvector.
// Compute λx and store the result into lambdax.
lambdax := make([]float64, n)
floats.AddScaled(lambdax, real(lambda), xRe)
if floats.Distance(xReAns, lambdax, math.Inf(1)) > tol {
return false
}
return true
}
// Complex eigenvector, and real or complex eigenvalue.
// Compute A imag(x) and store the result into xImAns.
xImAns := make([]float64, n)
blas64.Gemv(blas.NoTrans, 1, a, blas64.Vector{1, xIm}, 0, blas64.Vector{1, xImAns})
// Compute λx and store the result into lambdax.
lambdax := make([]complex128, n)
for i := range lambdax {
lambdax[i] = lambda * complex(xRe[i], xIm[i])
}
for i, v := range lambdax {
ax := complex(xReAns[i], xImAns[i])
if cmplx.Abs(v-ax) > tol {
return false
}
}
return true
}
func testStepper(stepper Stepper, t *testing.T) {
errorStepper, isErrorStepper := stepper.(ErrorStepper)
var (
r *Result
err error
)
for _, test := range ivpTests {
s := DefaultSettings()
tol := 1e-5
if test.tol != 0 {
tol = test.tol
}
var yEnd []float64
if test.exact != nil {
yEnd = make([]float64, len(test.p.InitialValue))
test.exact(test.p.To, yEnd)
} else {
yEnd = test.yEnd
}
if isErrorStepper {
if test.atol != 0 {
s.Absolute = test.atol
}
if test.rtol != 0 {
s.Relative = test.rtol
}
r, err = Integrate(test.p, s, errorStepper)
if err != nil {
t.Errorf("adaptive integration of test '%v' failed (%v)\n", test.name, err)
}
if r.Time != test.p.To {
t.Errorf("adaptive integration of '%v' did not integrate to time To\n", test.name)
}
d := floats.Distance(r.Y, yEnd, math.Inf(1))
if d > tol {
t.Errorf("adaptive solution of '%v' too far from the reference at t = %v: dist = %v", test.name, r.Time, d)
}
}
}
}
func (k *KMeans) Train(inputs *common.RowMatrix) {
nSamples, inputDim := inputs.Dims()
centroids := make([][]float64, k.Clusters)
// Assign the centroids to random data point to start
perm := rand.Perm(nSamples)
for i := 0; i < k.Clusters; i++ {
data := make([]float64, inputDim)
idx := perm[i]
inputs.Row(data, idx)
centroids[i] = data
}
distances := mat64.NewDense(nSamples, k.Clusters, nil)
row := make([]float64, inputDim)
for i := 0; i < nSamples; i++ {
inputs.Row(row, i)
for j := 0; j < k.Clusters; j++ {
d := floats.Distance(row, centroids[j])
distances.Set(i, j, d)
}
}
}
// testFunction checks that the function can evaluate itself (and its gradient)
// correctly.
func testFunction(f function, ftests []funcTest, t *testing.T) {
// Make a copy of tests because we may append to the slice.
tests := make([]funcTest, len(ftests))
copy(tests, ftests)
// Get information about the function.
fMinima, isMinimumer := f.(minimumer)
fGradient, isGradient := f.(gradient)
// If the function is a Minimumer, append its minima to the tests.
if isMinimumer {
for _, minimum := range fMinima.Minima() {
// Allocate gradient only if the function can evaluate it.
var grad []float64
if isGradient {
grad = make([]float64, len(minimum.X))
}
tests = append(tests, funcTest{
X: minimum.X,
F: minimum.F,
Gradient: grad,
})
}
}
for i, test := range tests {
F := f.Func(test.X)
// Check that the function value is as expected.
if math.Abs(F-test.F) > defaultTol {
t.Errorf("Test #%d: function value given by Func is incorrect. Want: %v, Got: %v",
i, test.F, F)
}
if test.Gradient == nil {
continue
}
// Evaluate the finite difference gradient.
fdGrad := fd.Gradient(nil, f.Func, test.X, nil)
// Check that the finite difference and expected gradients match.
if !floats.EqualApprox(fdGrad, test.Gradient, defaultFDGradTol) {
dist := floats.Distance(fdGrad, test.Gradient, math.Inf(1))
t.Errorf("Test #%d: numerical and expected gradients do not match. |fdGrad - WantGrad|_∞ = %v",
i, dist)
}
// If the function is a Gradient, check that it computes the gradient correctly.
if isGradient {
grad := make([]float64, len(test.Gradient))
fGradient.Grad(grad, test.X)
if !floats.EqualApprox(grad, test.Gradient, defaultGradTol) {
dist := floats.Distance(grad, test.Gradient, math.Inf(1))
t.Errorf("Test #%d: gradient given by Grad is incorrect. |grad - WantGrad|_∞ = %v",
i, dist)
}
}
}
}
func Dlaqr1Test(t *testing.T, impl Dlaqr1er) {
rnd := rand.New(rand.NewSource(1))
for _, n := range []int{2, 3} {
for _, ldh := range []int{n, n + 1, n + 10} {
for _, cas := range []int{1, 2} {
for k := 0; k < 100; k++ {
v := make([]float64, n)
for i := range v {
v[i] = math.NaN()
}
h := make([]float64, n*(n-1)*ldh)
for i := range h {
h[i] = math.NaN()
}
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
h[i*ldh+j] = rnd.NormFloat64()
}
}
var sr1, sr2, si1, si2 float64
if cas == 1 {
sr1 = rnd.NormFloat64()
sr2 = sr1
si1 = rnd.NormFloat64()
si2 = -si1
} else {
sr1 = rnd.NormFloat64()
sr2 = rnd.NormFloat64()
si1 = 0
si2 = 0
}
impl.Dlaqr1(n, h, ldh, sr1, si1, sr2, si2, v)
// Matrix H - s1*I.
h1 := make([]complex128, n*n)
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
h1[i*n+j] = complex(h[i*ldh+j], 0)
if i == j {
h1[i*n+j] -= complex(sr1, si1)
}
}
}
// First column of H - s2*I.
h2 := make([]complex128, n)
for i := 0; i < n; i++ {
h2[i] = complex(h[i*ldh], 0)
}
h2[0] -= complex(sr2, si2)
wantv := make([]float64, n)
// Multiply (H-s1*I)*(H-s2*I) to get a tentative
// wantv.
for i := 0; i < n; i++ {
for j := 0; j < n; j++ {
wantv[i] += real(h1[i*n+j] * h2[j])
}
}
// Get the unknown scale.
scale := v[0] / wantv[0]
// Compute the actual wantv.
floats.Scale(scale, wantv)
// The scale must be the same for all elements.
if floats.Distance(wantv, v, math.Inf(1)) > 1e-13 {
t.Errorf("n = %v, ldh = %v, case = %v: Unexpected value of v: got %v, want %v", n, ldh, cas, v, wantv)
}
}
}
}
}
}