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use crate::algebra::abstr::{Field, Scalar};
use crate::algebra::linear::matrix::Transpose;
use crate::algebra::linear::{matrix::General, vector::Vector};
use crate::elementary::Power;
impl<T> General<T>
where
T: Field + Scalar + Power,
{
/// Computes the singular value decomposition
///
/// M = U * S * V*
///
/// # Return
///
/// (u, s, v)
///
/// # Example
///
/// ```
/// use mathru::algebra::linear::matrix::General;
///
/// let a: General<f64> = General::new(4,
/// 4,
/// vec![4.0, 1.0, -2.0, 2.0, 1.0, 2.0, 0.0, -2.0, 0.0, 3.0,
/// -2.0, 2.0, 2.0, 1.0, -2.0, -1.0]);
///
/// let (u, s, v): (General<f64>, General<f64>, General<f64>) = a.dec_sv();
/// ```
pub fn dec_sv(&self) -> (Self, Self, Self) {
let (mut u, mut b, mut v): (General<T>, General<T>, General<T>) = self.householder_bidiag();
let (_m, n): (usize, usize) = b.dim();
let max_iterations: usize = 500 * n * n;
let mut u_k: General<T> = General::one(n);
for _k in 0..max_iterations {
let (u_ks, b_k, v_k): (General<T>, General<T>, General<T>) = General::msweep(u_k, b, v);
u_k = u_ks;
b = b_k;
v = v_k;
}
u *= u_k.transpose();
let (b_m, _b_n): (usize, usize) = b.dim();
// check that all singular values are positive
for l in 0..b_m {
if b[[l, l]] < T::zero() {
b[[l, l]] = -b[[l, l]];
let mut column_l: Vector<T> = u.get_column(l);
column_l = &column_l * &-T::one();
u.set_column(&column_l, l);
}
}
// null all values beneath the diagonal
for l in 0..b_m {
for k in 0..b_m {
if k != l {
b[[k, l]] = T::zero();
}
}
}
// sort singular values in descending order
(u, b, v)
}
fn msweep(
mut u: General<T>,
mut b: General<T>,
mut v: General<T>,
) -> (General<T>, General<T>, General<T>) {
let (_m, n): (usize, usize) = b.dim();
for k in 0..n - 1 {
let mut q: General<T> = General::one(n);
// Construct matrix Q and multiply on the right by Q'.
// Q annihilates both B(k-1,k+1) and B(k,k+1)
// but makes B(k+1,k) non-zero.
let (c_r, s_r, _r_r): (T, T, T) = General::rot(b[[k, k]], b[[k, k + 1]]);
q[[k, k]] = c_r;
q[[k, k + 1]] = s_r;
q[[k + 1, k]] = -s_r;
q[[k + 1, k + 1]] = c_r;
let q_t: General<T> = q.clone().transpose();
b = &b * &q_t;
v = &v * &q_t;
// Construct matrix Q and multiply on the left by Q.
// Q annihilates B(k+1,k) but makes B(k,k+1) and
// B(k,k+2) non-zero.
let (c_l, s_l, _r_l): (T, T, T) = General::rot(b[[k, k]], b[[k + 1, k]]);
q[[k, k]] = c_l;
q[[k, k + 1]] = s_l;
q[[k + 1, k]] = -s_l;
q[[k + 1, k + 1]] = c_l;
b = &q * &b;
u = &q * &u;
}
(u, b, v)
}
pub fn rot(f: T, g: T) -> (T, T, T) {
if f == T::zero() {
(T::zero(), T::one(), g)
} else {
let expo: T = T::from_f64(2.0);
let sqrt: T = T::from_f64(0.5);
if f.abs() > g.abs() {
let t: T = g / f;
let t1: T = (T::one() + t.pow(expo)).pow(sqrt);
(T::one() / t1, t / t1, f * t1)
} else {
let t: T = f / g;
let t1: T = (T::one() + t.pow(expo)).pow(sqrt);
(t / t1, T::one() / t1, g * t1)
}
}
}
///
/// self is an m times n matrix with m >= n
/// A = UBV^{T}
/// U \in T^{m \times n}
/// B \in T^{n \times n}
/// V \in T^{n \times n}
pub fn householder_bidiag(&self) -> (Self, Self, Self) {
let (m, n): (usize, usize) = self.dim();
if m < n {
panic!("Read the API");
}
let mut u: General<T> = General::one(m);
let mut v: General<T> = General::one(n);
let mut a_i: General<T> = self.clone();
for i in 0..n - 1 {
// eliminate non-zeros below the diagonal
// Keep the product U*B unchanged
let u_x: Vector<T> = a_i.clone().get_column(i);
let u_slice: Vector<T> = u_x.get_slice(i, m - 1);
let u_i: General<T> = General::householder(&u_slice, 0);
let a_i_slice = &u_i * &a_i.clone().get_slice(i, m - 1, i, n - 1);
a_i = a_i.set_slice(&a_i_slice, i, i);
let mut u_mi: General<T> = General::one(m);
u_mi = u_mi.set_slice(&u_i, i, i);
u = &u * &u_mi;
//eliminate non-zeros to the right of the
//superdiagonal by working with the transpose
// Keep the product B*V' unchanged
//B_T = B';
if i < (n - 1) {
let v_x: Vector<T> = a_i.get_row(i);
let v_x_trans: Vector<T> = v_x.transpose();
let v_x_trans_slice: Vector<T> = v_x_trans.get_slice(i + 1, n - 1);
let v_i: General<T> = General::householder(&v_x_trans_slice, 0);
let mut v_ni: General<T> = General::one(n);
v_ni = v_ni.set_slice(&v_i, i + 1, i + 1);
//let a_i_slice = &a_i.clone().get_slice(i+1, m - 1, i+1, n - 1) * &v_i;
//a_i = a_i.set_slice(&a_i_slice, i+1, i+1);
a_i = &a_i * &v_ni;
v = &v * &v_ni;
}
}
//Null all elements beneath the diagonal, and superdiagonal
for i in 0..m {
for k in 0..n {
if k != i && k != (i + 1) {
a_i[[i, k]] = T::zero();
}
}
}
(u, a_i, v)
}
}