Implemented QR algorithm with initial transformation to Hessenberg form and Wilkinson shift for symmetric matrices
This commit is contained in:
Daniel
parent
6fee70bd19
commit
c4753aaf65
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@ -151,7 +151,8 @@ pub use structs::{
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pub use linalg::{
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qr,
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householder_matrix,
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cholesky
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cholesky,
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hessenberg
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};
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mod structs;
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@ -1,5 +1,5 @@
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use traits::operations::{Transpose, ApproxEq};
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use traits::structure::{ColSlice, Eye, Indexable, Diag, SquareMat, BaseFloat};
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use traits::structure::{ColSlice, Eye, Indexable, Diag, SquareMat, BaseFloat, Cast};
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use traits::geometry::Norm;
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use std::cmp::min;
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use std::ops::{Mul, Add, Sub};
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@ -71,7 +71,7 @@ pub fn qr<N, V, M>(m: &M) -> (M, M)
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(q, r)
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}
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/// Eigendecomposition of a square matrix using the qr algorithm.
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/// Eigendecomposition of a square symmetric matrix using the qr algorithm
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pub fn eigen_qr<N, V, VS, M>(m: &M, eps: &N, niter: usize) -> (M, V)
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where N: BaseFloat,
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V: Mul<M, Output = V>,
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@ -79,39 +79,165 @@ pub fn eigen_qr<N, V, VS, M>(m: &M, eps: &N, niter: usize) -> (M, V)
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M: Indexable<(usize, usize), N> + SquareMat<N, V> + Add<M, Output = M> +
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Sub<M, Output = M> + ColSlice<VS> +
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ApproxEq<N> + Copy {
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let mut eigenvectors: M = ::one::<M>();
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let mut eigenvalues = *m;
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// let mut shifter: M = Eye::new_identity(rows);
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let (mut eigenvectors, mut eigenvalues) = hessenberg(m);
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// Allocate arrays for Givens rotation components
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let mut c = Vec::<N>::with_capacity(::dim::<M>()-1);
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let mut s = Vec::<N>::with_capacity(::dim::<M>()-1);
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if ::dim::<M>() == 1 {
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return (eigenvectors, eigenvalues.diag());
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}
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unsafe {
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c.set_len(::dim::<M>()-1);
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s.set_len(::dim::<M>()-1);
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}
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let mut iter = 0;
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for _ in 0..niter {
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let mut stop = true;
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let mut curdim = ::dim::<M>()-1;
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for _ in 0..::dim::<M>() {
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let mut stop = false;
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while !stop && iter < niter {
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let lambda;
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unsafe {
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let a = eigenvalues.unsafe_at((curdim-1, curdim-1));
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let b = eigenvalues.unsafe_at((curdim-1, curdim));
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let c = eigenvalues.unsafe_at((curdim, curdim-1));
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let d = eigenvalues.unsafe_at((curdim, curdim));
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let trace = a + d;
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let det = a * d - b * c;
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let constquarter: N = Cast::from(0.25f64);
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let consthalf: N = Cast::from(0.5f64);
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let e = (constquarter * trace * trace - det).sqrt();
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let lambda1 = consthalf * trace + e;
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let lambda2 = consthalf * trace - e;
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if (lambda1 - d).abs() < (lambda2 - d).abs() {
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lambda = lambda1;
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}
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else {
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lambda = lambda2;
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}
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}
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// Shift matrix
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for k in 0..curdim+1 {
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unsafe {
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let a = eigenvalues.unsafe_at((k,k));
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eigenvalues.unsafe_set((k,k), a - lambda);
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}
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}
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// Givens rotation from left
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for k in 0..curdim {
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let x_i = unsafe { eigenvalues.unsafe_at((k,k)) };
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let x_j = unsafe { eigenvalues.unsafe_at((k+1,k)) };
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let r = (x_i*x_i + x_j*x_j).sqrt();
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let ctmp = x_i / r;
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let stmp = -x_j / r;
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c[k] = ctmp;
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s[k] = stmp;
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for j in k..(curdim+1) {
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unsafe {
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let a = eigenvalues.unsafe_at((k,j));
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let b = eigenvalues.unsafe_at((k+1,j));
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eigenvalues.unsafe_set((k,j), ctmp * a - stmp * b);
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eigenvalues.unsafe_set((k+1,j), stmp * a + ctmp * b);
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}
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for j in 0..::dim::<M>() {
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for i in 0..j {
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if unsafe { eigenvalues.unsafe_at((i, j)) }.abs() >= *eps {
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stop = false;
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break;
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}
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}
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for i in j + 1..::dim::<M>() {
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if unsafe { eigenvalues.unsafe_at((i, j)) }.abs() >= *eps {
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// Givens rotation from right applied to eigenvalues
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for k in 0..curdim {
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for i in 0..(k+2) {
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unsafe {
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let a = eigenvalues.unsafe_at((i,k));
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let b = eigenvalues.unsafe_at((i,k+1));
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eigenvalues.unsafe_set((i,k), c[k] * a - s[k] * b);
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eigenvalues.unsafe_set((i,k+1), s[k] * a + c[k] * b);
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}
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}
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}
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// Shift back
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for k in 0..curdim+1 {
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unsafe {
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let a = eigenvalues.unsafe_at((k,k));
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eigenvalues.unsafe_set((k,k), a + lambda);
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}
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}
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// Givens rotation from right applied to eigenvectors
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for k in 0..curdim {
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for i in 0..::dim::<M>() {
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unsafe {
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let a = eigenvectors.unsafe_at((i,k));
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let b = eigenvectors.unsafe_at((i,k+1));
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eigenvectors.unsafe_set((i,k), c[k] * a - s[k] * b);
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eigenvectors.unsafe_set((i,k+1), s[k] * a + c[k] * b);
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}
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}
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}
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iter = iter + 1;
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stop = true;
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for j in 0..curdim {
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// Check last row
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if unsafe { eigenvalues.unsafe_at((curdim, j)) }.abs() >= *eps {
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stop = false;
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break;
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}
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// Check last column
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if unsafe { eigenvalues.unsafe_at((j, curdim)) }.abs() >= *eps {
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stop = false;
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break;
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}
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}
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}
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if stop {
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break;
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if curdim > 1 {
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curdim = curdim - 1;
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}
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else {
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break;
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}
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}
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iter = iter + 1;
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let (q, r) = qr(&eigenvalues);;
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eigenvalues = r * q;
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eigenvectors = eigenvectors * q;
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}
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(eigenvectors, eigenvalues.diag())
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@ -164,3 +290,48 @@ pub fn cholesky<N, V, VS, M>(m: &M) -> Result<M, &'static str>
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return Ok(out);
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}
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/// Hessenberg
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/// Returns the matrix m in Hessenberg form and the corresponding similarity transformation
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///
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/// # Arguments
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/// * `m` - matrix to transform
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///
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/// # Returns
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/// * First return value `q` - Similarity matrix p such that q * h * q^T = m
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/// * Second return value `h` - Matrix m in Hessenberg form
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pub fn hessenberg<N, V, M>(m: &M) -> (M, M)
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where N: BaseFloat,
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V: Indexable<usize, N> + Norm<N>,
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M: Copy + Eye + ColSlice<V> + Transpose + Indexable<(usize, usize), N> +
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Mul<M, Output = M> {
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let mut h = m.clone();
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let (rows, cols) = h.shape();
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let mut q : M = Eye::new_identity(cols);
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if cols <= 2 {
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return (q, h);
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}
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for ite in 0..(cols-2) {
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let mut v = h.col_slice(ite, ite+1, rows);
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let alpha = Norm::norm(&v);
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unsafe {
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let x = v.unsafe_at(0);
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v.unsafe_set(0, x - alpha);
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}
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if !::is_zero(&v.normalize_mut()) {
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let p: M = householder_matrix(rows, ite+1, v);
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q = q * p;
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h = p * h * p;
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}
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}
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return (q, h);
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}
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@ -1,4 +1,4 @@
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pub use self::decompositions::{qr, eigen_qr, householder_matrix, cholesky};
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pub use self::decompositions::{qr, eigen_qr, householder_matrix, cholesky, hessenberg};
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mod decompositions;
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156
tests/mat.rs
156
tests/mat.rs
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@ -3,7 +3,7 @@ extern crate rand;
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use rand::random;
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use na::{Vec1, Vec3, Mat1, Mat2, Mat3, Mat4, Mat5, Mat6, Rot2, Rot3, Persp3, PerspMat3, Ortho3,
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OrthoMat3, DMat, DVec, Row, Col, BaseFloat, Diag, Transpose, RowSlice, ColSlice};
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OrthoMat3, DMat, DVec, Row, Col, BaseFloat, Diag, Transpose, RowSlice, ColSlice, Shape};
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macro_rules! test_inv_mat_impl(
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($t: ty) => (
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@ -63,28 +63,49 @@ macro_rules! test_cholesky_impl(
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);
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);
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// NOTE: deactivated untile we get a better convergence rate.
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// macro_rules! test_eigen_qr_impl(
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// ($t: ty) => {
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// for _ in (0usize .. 10000) {
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// let randmat : $t = random();
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// // Make it symetric so that we can recompose the matrix to test at the end.
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// let randmat = na::transpose(&randmat) * randmat;
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//
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// let (eigenvectors, eigenvalues) = na::eigen_qr(&randmat, &Float::epsilon(), 100);
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//
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// let diag: $t = Diag::from_diag(&eigenvalues);
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//
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// let recomp = eigenvectors * diag * na::transpose(&eigenvectors);
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//
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// println!("eigenvalues: {}", eigenvalues);
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// println!(" mat: {}", randmat);
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// println!("recomp: {}", recomp);
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//
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// assert!(na::approx_eq_eps(&randmat, &recomp, &1.0e-2));
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// }
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// }
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// )
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macro_rules! test_hessenberg_impl(
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($t: ty) => (
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for _ in (0usize .. 10000) {
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let randmat : $t = random();
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let (q, h) = na::hessenberg(&randmat);
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let recomp = q * h * na::transpose(&q);
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let (rows, cols) = h.shape();
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// Check if `h` has zero entries below the first subdiagonal
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if cols > 2 {
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for j in 0..(cols-2) {
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for i in (j+2)..rows {
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assert!(na::approx_eq(&h[(i,j)], &0.0f64));
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}
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}
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}
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assert!(na::approx_eq(&randmat, &recomp));
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}
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);
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);
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macro_rules! test_eigen_qr_impl(
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($t: ty) => {
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for _ in (0usize .. 10000) {
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let randmat : $t = random();
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// Make it symetric so that we can recompose the matrix to test at the end.
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let randmat = na::transpose(&randmat) * randmat;
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let (eigenvectors, eigenvalues) = na::eigen_qr(&randmat, &1e-13, 100);
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let diag: $t = Diag::from_diag(&eigenvalues);
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let recomp = eigenvectors * diag * na::transpose(&eigenvectors);
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println!("eigenvalues: {:?}", eigenvalues);
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println!(" mat: {:?}", randmat);
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println!("recomp: {:?}", recomp);
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assert!(na::approx_eq_eps(&randmat, &recomp, &1.0e-2));
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}
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}
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);
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#[test]
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fn test_transpose_mat1() {
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@ -532,36 +553,35 @@ fn test_qr_mat6() {
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test_qr_impl!(Mat6<f64>);
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}
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// NOTE: deactivated until we get a better convergence rate.
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// #[test]
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// fn test_eigen_qr_mat1() {
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// test_eigen_qr_impl!(Mat1<f64>);
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// }
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//
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// #[test]
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// fn test_eigen_qr_mat2() {
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// test_eigen_qr_impl!(Mat2<f64>);
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// }
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//
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// #[test]
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// fn test_eigen_qr_mat3() {
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// test_eigen_qr_impl!(Mat3<f64>);
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// }
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//
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// #[test]
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// fn test_eigen_qr_mat4() {
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// test_eigen_qr_impl!(Mat4<f64>);
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// }
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//
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// #[test]
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// fn test_eigen_qr_mat5() {
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// test_eigen_qr_impl!(Mat5<f64>);
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// }
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//
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// #[test]
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// fn test_eigen_qr_mat6() {
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// test_eigen_qr_impl!(Mat6<f64>);
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// }
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#[test]
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fn test_eigen_qr_mat1() {
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test_eigen_qr_impl!(Mat1<f64>);
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}
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#[test]
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fn test_eigen_qr_mat2() {
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test_eigen_qr_impl!(Mat2<f64>);
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}
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#[test]
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fn test_eigen_qr_mat3() {
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test_eigen_qr_impl!(Mat3<f64>);
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}
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#[test]
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fn test_eigen_qr_mat4() {
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test_eigen_qr_impl!(Mat4<f64>);
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}
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#[test]
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fn test_eigen_qr_mat5() {
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test_eigen_qr_impl!(Mat5<f64>);
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}
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#[test]
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fn test_eigen_qr_mat6() {
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test_eigen_qr_impl!(Mat6<f64>);
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}
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#[test]
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fn test_from_fn() {
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@ -733,6 +753,36 @@ fn test_cholesky_mat6() {
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test_cholesky_impl!(Mat6<f64>);
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}
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#[test]
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fn test_hessenberg_mat1() {
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test_hessenberg_impl!(Mat1<f64>);
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}
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#[test]
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fn test_hessenberg_mat2() {
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test_hessenberg_impl!(Mat2<f64>);
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}
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#[test]
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fn test_hessenberg_mat3() {
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test_hessenberg_impl!(Mat3<f64>);
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}
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#[test]
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fn test_hessenberg_mat4() {
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test_hessenberg_impl!(Mat4<f64>);
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}
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#[test]
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fn test_hessenberg_mat5() {
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test_hessenberg_impl!(Mat5<f64>);
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}
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#[test]
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fn test_hessenberg_mat6() {
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test_hessenberg_impl!(Mat6<f64>);
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}
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#[test]
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fn test_transpose_square_mat() {
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let col_major_mat = &[0, 1, 2, 3,
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