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// Copyright ©2015 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 mat_test
import (
"fmt"
"gonum.org/v1/gonum/mat"
)
func ExampleCholesky() {
// Construct a symmetric positive definite matrix.
tmp := mat.NewDense(4, 4, []float64{
2, 6, 8, -4,
1, 8, 7, -2,
2, 2, 1, 7,
8, -2, -2, 1,
})
var a mat.SymDense
a.SymOuterK(1, tmp)
fmt.Printf("a = %0.4v\n", mat.Formatted(&a, mat.Prefix(" ")))
// Compute the cholesky factorization.
var chol mat.Cholesky
if ok := chol.Factorize(&a); !ok {
fmt.Println("a matrix is not positive semi-definite.")
}
// Find the determinant.
fmt.Printf("\nThe determinant of a is %0.4g\n\n", chol.Det())
// Use the factorization to solve the system of equations a * x = b.
b := mat.NewVecDense(4, []float64{1, 2, 3, 4})
var x mat.VecDense
if err := chol.SolveVecTo(&x, b); err != nil {
fmt.Println("Matrix is near singular: ", err)
}
fmt.Println("Solve a * x = b")
fmt.Printf("x = %0.4v\n", mat.Formatted(&x, mat.Prefix(" ")))
// Extract the factorization and check that it equals the original matrix.
var t mat.TriDense
chol.LTo(&t)
var test mat.Dense
test.Mul(&t, t.T())
fmt.Println()
fmt.Printf("L * Lᵀ = %0.4v\n", mat.Formatted(&a, mat.Prefix(" ")))
// Output:
// a = ⎡120 114 -4 -16⎤
// ⎢114 118 11 -24⎥
// ⎢ -4 11 58 17⎥
// ⎣-16 -24 17 73⎦
//
// The determinant of a is 1.543e+06
//
// Solve a * x = b
// x = ⎡ -0.239⎤
// ⎢ 0.2732⎥
// ⎢-0.04681⎥
// ⎣ 0.1031⎦
//
// L * Lᵀ = ⎡120 114 -4 -16⎤
// ⎢114 118 11 -24⎥
// ⎢ -4 11 58 17⎥
// ⎣-16 -24 17 73⎦
}
func ExampleCholesky_SymRankOne() {
a := mat.NewSymDense(4, []float64{
1, 1, 1, 1,
0, 2, 3, 4,
0, 0, 6, 10,
0, 0, 0, 20,
})
fmt.Printf("A = %0.4v\n", mat.Formatted(a, mat.Prefix(" ")))
// Compute the Cholesky factorization.
var chol mat.Cholesky
if ok := chol.Factorize(a); !ok {
fmt.Println("matrix a is not positive definite.")
}
x := mat.NewVecDense(4, []float64{0, 0, 0, 1})
fmt.Printf("\nx = %0.4v\n", mat.Formatted(x, mat.Prefix(" ")))
// Rank-1 update the factorization.
chol.SymRankOne(&chol, 1, x)
// Rank-1 update the matrix a.
a.SymRankOne(a, 1, x)
var au mat.SymDense
chol.ToSym(&au)
// Print the matrix that was updated directly.
fmt.Printf("\nA' = %0.4v\n", mat.Formatted(a, mat.Prefix(" ")))
// Print the matrix recovered from the factorization.
fmt.Printf("\nU'ᵀ * U' = %0.4v\n", mat.Formatted(&au, mat.Prefix(" ")))
// Output:
// A = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 20⎦
//
// x = ⎡0⎤
// ⎢0⎥
// ⎢0⎥
// ⎣1⎦
//
// A' = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 21⎦
//
// U'ᵀ * U' = ⎡ 1 1 1 1⎤
// ⎢ 1 2 3 4⎥
// ⎢ 1 3 6 10⎥
// ⎣ 1 4 10 21⎦
}
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