func TestEigSymm(t *testing.T) {
n := 100
a := randMat(n, n)
a = mat.Plus(a, mat.T(a))
// Take eigen decomposition.
v, d, err := EigSymm(a)
if err != nil {
t.Fatal(err)
}
got := mat.Mul(mat.Mul(v, mat.NewDiag(d)), mat.T(v))
testMatEq(t, a, got)
}
func ExampleCholFact_Solve() {
// A = V' V, with V = [1, 1; 2, 1]
v := mat.NewRows([][]float64{
{1, 1},
{2, 1},
})
a := mat.Mul(mat.T(v), v)
// x = [1; 2]
// b = V' V x
// = V' [1, 1; 2, 1] [1; 2]
// = [1, 2; 1, 1] [3; 4]
// = [11; 7]
b := []float64{11, 7}
chol, err := Chol(a)
if err != nil {
fmt.Println(err)
return
}
x, err := chol.Solve(b)
if err != nil {
fmt.Println(err)
return
}
fmt.Printf("%.6g", x)
// Output:
// [1 2]
}
func ExampleInvertPosDef() {
// A = V' V, with V = [1, 1; 2, 1]
v := mat.NewRows([][]float64{
{1, 1},
{2, 1},
})
a := mat.Mul(mat.T(v), v)
b, err := InvertPosDef(a)
if err != nil {
fmt.Println(err)
return
}
for j := 0; j < 2; j++ {
for i := 0; i < 2; i++ {
if i > 0 {
fmt.Printf(" ")
}
fmt.Printf("%.3g", b.At(i, j))
}
fmt.Println()
}
// Output:
// 2 -3
// -3 5
}
func TestSVD_vsEig(t *testing.T) {
m, n := 150, 100
a := randMat(m, n)
g := mat.Mul(mat.T(a), a)
// Take eigen decomposition of Gram matrix.
_, eigs, err := EigSymm(g)
if err != nil {
t.Fatal(err)
}
// Sort in descending order.
sort.Sort(sort.Reverse(sort.Float64Slice(eigs)))
// Take square root of eigenvalues.
for i := range eigs {
// Clip small negative values to zero.
eigs[i] = math.Sqrt(math.Max(0, eigs[i]))
}
// Take singular value decomposition.
_, svals, _, err := SVD(a)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, eigs, svals)
}
func overDetProb(m, n int) (a *mat.Mat, b, x []float64, err error) {
if m < n {
panic("expect m >= n")
}
a = randMat(m, n)
b = randVec(m)
// Compute pseudo-inverse explicitly.
// y <- (A' A) \ b
x, err = SolveSymm(mat.Mul(mat.T(a), a), mat.MulVec(mat.T(a), b))
if err != nil {
return nil, nil, nil, err
}
return
}
func TestLUFact_Solve_t(t *testing.T) {
n := 100
// Random square matrix.
a := randMat(n, n)
// Random vector.
want := randVec(n)
// Factorize.
lu, err := LU(a)
if err != nil {
t.Fatal(err)
}
// Solve un-transposed system.
b := mat.MulVec(a, want)
got, err := lu.Solve(false, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
// Then solve transposed system.
b = mat.MulVec(mat.T(a), want)
got, err = lu.Solve(true, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
func underDetProb(m, n int) (a *mat.Mat, b, x []float64, err error) {
if m > n {
panic("expect m <= n")
}
a = randMat(m, n)
b = randVec(m)
// Compute pseudo-inverse explicitly.
// y <- (A A') \ b
y, err := SolveSymm(mat.Mul(a, mat.T(a)), b)
if err != nil {
return nil, nil, nil, err
}
// x <- A' y
x = mat.MulVec(mat.T(a), y)
return
}
func TestSolvePosDef(t *testing.T) {
n := 100
// Random symmetric positive definite matrix.
a := randMat(2*n, n)
a = mat.Mul(mat.T(a), a)
// Random vector.
want := randVec(n)
b := mat.MulVec(a, want)
got, err := SolvePosDef(a, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
// Minimum-norm solution to under-constrained system by QR decomposition.
func TestQRFact_Solve_underdetermined(t *testing.T) {
m, n := 100, 150
a, b, want, err := underDetProb(m, n)
if err != nil {
t.Fatal(err)
}
// Take QR factorization of transpose.
qr, err := QR(mat.T(a))
if err != nil {
t.Fatal(err)
}
got, err := qr.Solve(true, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
func TestLDLFact_Solve(t *testing.T) {
n := 100
// Random symmetric matrix.
a := randMat(n, n)
a = mat.Plus(a, mat.T(a))
// Random vector.
want := randVec(n)
b := mat.MulVec(a, want)
ldl, err := LDL(a)
if err != nil {
t.Fatal(err)
}
got, err := ldl.Solve(b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
func TestCholFact_Solve(t *testing.T) {
n := 100
// Random symmetric positive definite matrix.
a := randMat(2*n, n)
a = mat.Mul(mat.T(a), a)
// Random vector.
want := randVec(n)
b := mat.MulVec(a, want)
// Factorize matrix.
chol, err := Chol(a)
if err != nil {
t.Fatal(err)
}
got, err := chol.Solve(b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
func TestEigSymm_vsTrace(t *testing.T) {
n := 100
a := randMat(n, n)
a = mat.Plus(a, mat.T(a))
// Compute matrix trace.
tr := mat.Tr(a)
// Take eigen decomposition.
_, d, err := EigSymm(a)
if err != nil {
t.Fatal(err)
}
// Sum eigenvalues.
var sum float64
for _, eig := range d {
sum += eig
}
if !epsEq(tr, sum, eps) {
t.Errorf("want %.4g, got %.4g", tr, sum)
}
}