func overDetProb(m, n int) (a *cmat.Mat, b, x []complex128, 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 = SolveHerm(cmat.Mul(cmat.H(a), a), cmat.MulVec(cmat.H(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 := cmat.MulVec(a, want)
got, err := lu.Solve(false, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
// Then solve conjugate-transpose system.
b = cmat.MulVec(cmat.H(a), want)
got, err = lu.Solve(true, b)
if err != nil {
t.Fatal(err)
}
testSliceEq(t, want, got)
}
func ExampleCholFact_Solve() {
// A = V' V, with V = [1, 1; 2, 1]
v := cmat.NewRows([][]complex128{
{1, 1},
{2, 1},
})
a := cmat.Mul(cmat.H(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 := []complex128{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.Println(formatSlice(x, 'f', 3))
// Output:
// [(1.000+0.000i) (2.000+0.000i)]
}
func underDetProb(m, n int) (a *cmat.Mat, b, x []complex128, 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 := SolveHerm(cmat.Mul(a, cmat.H(a)), b)
if err != nil {
return nil, nil, nil, err
}
// x <- A' y
x = cmat.MulVec(cmat.H(a), y)
return
}
func TestSolvePosDef(t *testing.T) {
n := 100
// Random symmetric positive definite matrix.
a := randMat(2*n, n)
a = cmat.Mul(cmat.H(a), a)
// Random vector.
want := randVec(n)
b := cmat.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 conjugate transpose.
qr, err := QR(cmat.H(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 = cmat.Plus(a, cmat.H(a))
// Random vector.
want := randVec(n)
b := cmat.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 = cmat.Mul(cmat.H(a), a)
// Random vector.
want := randVec(n)
b := cmat.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 TestCholFact_Inverse(t *testing.T) {
n := 100
// Random symmetric positive definite matrix.
a := randMat(2*n, n)
a = cmat.Mul(cmat.H(a), a)
// Factorize matrix.
chol, err := Chol(a)
if err != nil {
t.Fatal(err)
}
b, err := chol.Inv()
if err != nil {
t.Fatal(err)
}
right := cmat.Mul(a, b)
testMatEq(t, cmat.I(n), right)
left := cmat.Mul(a, b)
testMatEq(t, cmat.I(n), left)
}