Merge potential next release v0.4 (#187) Breaking Changes
* First draft of the new n-dimensional arrays + NB use case * Improves default implementation of multiple Array methods * Refactors tree methods * Adds matrix decomposition routines * Adds matrix decomposition methods to ndarray and nalgebra bindings * Refactoring + linear regression now uses array2 * Ridge & Linear regression * LBFGS optimizer & logistic regression * LBFGS optimizer & logistic regression * Changes linear methods, metrics and model selection methods to new n-dimensional arrays * Switches KNN and clustering algorithms to new n-d array layer * Refactors distance metrics * Optimizes knn and clustering methods * Refactors metrics module * Switches decomposition methods to n-dimensional arrays * Linalg refactoring - cleanup rng merge (#172) * Remove legacy DenseMatrix and BaseMatrix implementation. Port the new Number, FloatNumber and Array implementation into module structure. * Exclude AUC metrics. Needs reimplementation * Improve developers walkthrough New traits system in place at `src/numbers` and `src/linalg` Co-authored-by: Lorenzo <tunedconsulting@gmail.com> * Provide SupervisedEstimator with a constructor to avoid explicit dynamical box allocation in 'cross_validate' and 'cross_validate_predict' as required by the use of 'dyn' as per Rust 2021 * Implement getters to use as_ref() in src/neighbors * Implement getters to use as_ref() in src/naive_bayes * Implement getters to use as_ref() in src/linear * Add Clone to src/naive_bayes * Change signature for cross_validate and other model_selection functions to abide to use of dyn in Rust 2021 * Implement ndarray-bindings. Remove FloatNumber from implementations * Drop nalgebra-bindings support (as decided in conf-call to go for ndarray) * Remove benches. Benches will have their own repo at smartcore-benches * Implement SVC * Implement SVC serialization. Move search parameters in dedicated module * Implement SVR. Definitely too slow * Fix compilation issues for wasm (#202) Co-authored-by: Luis Moreno <morenol@users.noreply.github.com> * Fix tests (#203) * Port linalg/traits/stats.rs * Improve methods naming * Improve Display for DenseMatrix Co-authored-by: Montana Low <montanalow@users.noreply.github.com> Co-authored-by: VolodymyrOrlov <volodymyr.orlov@gmail.com>
This commit is contained in:
Lorenzo
@@ -0,0 +1,909 @@
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//! # Eigen Decomposition
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//!
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//! Eigendecomposition is one of the most useful matrix factorization methods in machine learning that decomposes a matrix into eigenvectors and eigenvalues.
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//! This decomposition plays an important role in the the [Principal Component Analysis (PCA)](../../decomposition/pca/index.html).
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//!
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//! Eigendecomposition decomposes a square matrix into a set of eigenvectors and eigenvalues.
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//!
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//! \\[A = Q \Lambda Q^{-1}\\]
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//!
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//! where \\(Q\\) is a matrix comprised of the eigenvectors, \\(\Lambda\\) is a diagonal matrix comprised of the eigenvalues along the diagonal,
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//! and \\(Q{-1}\\) is the inverse of the matrix comprised of the eigenvectors.
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//!
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//! Example:
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//! ```
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//! use smartcore::linalg::basic::matrix::DenseMatrix;
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//! use smartcore::linalg::traits::evd::*;
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//!
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//! let A = DenseMatrix::from_2d_array(&[
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//! &[0.9000, 0.4000, 0.7000],
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//! &[0.4000, 0.5000, 0.3000],
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//! &[0.7000, 0.3000, 0.8000],
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//! ]);
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//!
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//! let evd = A.evd(true).unwrap();
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//! let eigenvectors: DenseMatrix<f64> = evd.V;
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//! let eigenvalues: Vec<f64> = evd.d;
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//! ```
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//!
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//! ## References:
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//! * ["Numerical Recipes: The Art of Scientific Computing", Press W.H., Teukolsky S.A., Vetterling W.T, Flannery B.P, 3rd ed., Section 11 Eigensystems](http://numerical.recipes/)
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//! * ["Introduction to Linear Algebra", Gilbert Strang, 5rd ed., ch. 6 Eigenvalues and Eigenvectors](https://math.mit.edu/~gs/linearalgebra/)
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//!
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//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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#![allow(non_snake_case)]
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use crate::error::Failed;
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use crate::linalg::basic::arrays::Array2;
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use crate::numbers::basenum::Number;
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use crate::numbers::realnum::RealNumber;
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use num::complex::Complex;
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use std::fmt::Debug;
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#[derive(Debug, Clone)]
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/// Results of eigen decomposition
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pub struct EVD<T: Number + RealNumber, M: Array2<T>> {
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/// Real part of eigenvalues.
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pub d: Vec<T>,
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/// Imaginary part of eigenvalues.
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pub e: Vec<T>,
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/// Eigenvectors
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pub V: M,
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}
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/// Trait that implements EVD decomposition routine for any matrix.
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pub trait EVDDecomposable<T: Number + RealNumber>: Array2<T> {
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/// Compute the eigen decomposition of a square matrix.
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/// * `symmetric` - whether the matrix is symmetric
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fn evd(&self, symmetric: bool) -> Result<EVD<T, Self>, Failed> {
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self.clone().evd_mut(symmetric)
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}
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/// Compute the eigen decomposition of a square matrix. The input matrix
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/// will be used for factorization.
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/// * `symmetric` - whether the matrix is symmetric
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fn evd_mut(mut self, symmetric: bool) -> Result<EVD<T, Self>, Failed> {
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let (nrows, ncols) = self.shape();
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if ncols != nrows {
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panic!("Matrix is not square: {} x {}", nrows, ncols);
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}
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let n = nrows;
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let mut d = vec![T::zero(); n];
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let mut e = vec![T::zero(); n];
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let mut V;
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if symmetric {
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V = self;
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// Tridiagonalize.
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tred2(&mut V, &mut d, &mut e);
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// Diagonalize.
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tql2(&mut V, &mut d, &mut e);
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} else {
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let scale = balance(&mut self);
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let perm = elmhes(&mut self);
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V = Self::eye(n);
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eltran(&self, &mut V, &perm);
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hqr2(&mut self, &mut V, &mut d, &mut e);
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balbak(&mut V, &scale);
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sort(&mut d, &mut e, &mut V);
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}
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Ok(EVD { V, d, e })
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}
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}
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fn tred2<T: Number + RealNumber, M: Array2<T>>(V: &mut M, d: &mut [T], e: &mut [T]) {
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let (n, _) = V.shape();
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for (i, d_i) in d.iter_mut().enumerate().take(n) {
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*d_i = *V.get((n - 1, i));
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}
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for i in (1..n).rev() {
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let mut scale = T::zero();
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let mut h = T::zero();
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for d_k in d.iter().take(i) {
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scale += d_k.abs();
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}
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if scale == T::zero() {
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e[i] = d[i - 1];
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for (j, d_j) in d.iter_mut().enumerate().take(i) {
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*d_j = *V.get((i - 1, j));
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V.set((i, j), T::zero());
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V.set((j, i), T::zero());
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}
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} else {
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for d_k in d.iter_mut().take(i) {
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*d_k /= scale;
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h += (*d_k) * (*d_k);
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}
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let mut f = d[i - 1];
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let mut g = h.sqrt();
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if f > T::zero() {
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g = -g;
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}
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e[i] = scale * g;
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h -= f * g;
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d[i - 1] = f - g;
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for e_j in e.iter_mut().take(i) {
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*e_j = T::zero();
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}
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for j in 0..i {
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f = d[j];
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V.set((j, i), f);
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g = e[j] + *V.get((j, j)) * f;
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for k in j + 1..=i - 1 {
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g += *V.get((k, j)) * d[k];
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e[k] += *V.get((k, j)) * f;
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}
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e[j] = g;
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}
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f = T::zero();
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for j in 0..i {
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e[j] /= h;
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f += e[j] * d[j];
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}
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let hh = f / (h + h);
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for j in 0..i {
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e[j] -= hh * d[j];
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}
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for j in 0..i {
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f = d[j];
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g = e[j];
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for k in j..=i - 1 {
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V.sub_element_mut((k, j), f * e[k] + g * d[k]);
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}
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d[j] = *V.get((i - 1, j));
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V.set((i, j), T::zero());
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}
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}
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d[i] = h;
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}
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for i in 0..n - 1 {
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V.set((n - 1, i), *V.get((i, i)));
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V.set((i, i), T::one());
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let h = d[i + 1];
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if h != T::zero() {
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for (k, d_k) in d.iter_mut().enumerate().take(i + 1) {
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*d_k = *V.get((k, i + 1)) / h;
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}
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for j in 0..=i {
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let mut g = T::zero();
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for k in 0..=i {
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g += *V.get((k, i + 1)) * *V.get((k, j));
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}
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for (k, d_k) in d.iter().enumerate().take(i + 1) {
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V.sub_element_mut((k, j), g * (*d_k));
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}
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}
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}
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for k in 0..=i {
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V.set((k, i + 1), T::zero());
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}
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}
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for (j, d_j) in d.iter_mut().enumerate().take(n) {
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*d_j = *V.get((n - 1, j));
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V.set((n - 1, j), T::zero());
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}
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V.set((n - 1, n - 1), T::one());
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e[0] = T::zero();
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}
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fn tql2<T: Number + RealNumber, M: Array2<T>>(V: &mut M, d: &mut [T], e: &mut [T]) {
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let (n, _) = V.shape();
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for i in 1..n {
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e[i - 1] = e[i];
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}
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e[n - 1] = T::zero();
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let mut f = T::zero();
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let mut tst1 = T::zero();
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for l in 0..n {
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tst1 = T::max(tst1, d[l].abs() + e[l].abs());
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let mut m = l;
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loop {
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if m < n {
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if e[m].abs() <= tst1 * T::epsilon() {
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break;
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}
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m += 1;
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} else {
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break;
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}
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}
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if m > l {
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let mut iter = 0;
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loop {
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iter += 1;
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if iter >= 30 {
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panic!("Too many iterations");
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}
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let mut g = d[l];
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let mut p = (d[l + 1] - g) / (T::two() * e[l]);
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let mut r = p.hypot(T::one());
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if p < T::zero() {
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r = -r;
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}
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d[l] = e[l] / (p + r);
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d[l + 1] = e[l] * (p + r);
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let dl1 = d[l + 1];
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let mut h = g - d[l];
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for d_i in d.iter_mut().take(n).skip(l + 2) {
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*d_i -= h;
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}
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f += h;
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p = d[m];
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let mut c = T::one();
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let mut c2 = c;
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let mut c3 = c;
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let el1 = e[l + 1];
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let mut s = T::zero();
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let mut s2 = T::zero();
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for i in (l..m).rev() {
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c3 = c2;
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c2 = c;
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s2 = s;
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g = c * e[i];
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h = c * p;
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r = p.hypot(e[i]);
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e[i + 1] = s * r;
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s = e[i] / r;
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c = p / r;
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p = c * d[i] - s * g;
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d[i + 1] = h + s * (c * g + s * d[i]);
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for k in 0..n {
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h = *V.get((k, i + 1));
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V.set((k, i + 1), s * *V.get((k, i)) + c * h);
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V.set((k, i), c * *V.get((k, i)) - s * h);
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}
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}
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p = -s * s2 * c3 * el1 * e[l] / dl1;
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e[l] = s * p;
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d[l] = c * p;
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if e[l].abs() <= tst1 * T::epsilon() {
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break;
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}
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}
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}
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d[l] += f;
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e[l] = T::zero();
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}
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for i in 0..n - 1 {
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let mut k = i;
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let mut p = d[i];
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for (j, d_j) in d.iter().enumerate().take(n).skip(i + 1) {
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if *d_j > p {
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k = j;
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p = *d_j;
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}
|
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}
|
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if k != i {
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d[k] = d[i];
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d[i] = p;
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for j in 0..n {
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p = *V.get((j, i));
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V.set((j, i), *V.get((j, k)));
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V.set((j, k), p);
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}
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}
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}
|
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}
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fn balance<T: Number + RealNumber, M: Array2<T>>(A: &mut M) -> Vec<T> {
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let radix = T::two();
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let sqrdx = radix * radix;
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|
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let (n, _) = A.shape();
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let mut scale = vec![T::one(); n];
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let t = T::from(0.95).unwrap();
|
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|
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let mut done = false;
|
||||
while !done {
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done = true;
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for (i, scale_i) in scale.iter_mut().enumerate().take(n) {
|
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let mut r = T::zero();
|
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let mut c = T::zero();
|
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for j in 0..n {
|
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if j != i {
|
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c += A.get((j, i)).abs();
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r += A.get((i, j)).abs();
|
||||
}
|
||||
}
|
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if c != T::zero() && r != T::zero() {
|
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let mut g = r / radix;
|
||||
let mut f = T::one();
|
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let s = c + r;
|
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while c < g {
|
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f *= radix;
|
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c *= sqrdx;
|
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}
|
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g = r * radix;
|
||||
while c > g {
|
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f /= radix;
|
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c /= sqrdx;
|
||||
}
|
||||
if (c + r) / f < t * s {
|
||||
done = false;
|
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g = T::one() / f;
|
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*scale_i *= f;
|
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for j in 0..n {
|
||||
A.mul_element_mut((i, j), g);
|
||||
}
|
||||
for j in 0..n {
|
||||
A.mul_element_mut((j, i), f);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
scale
|
||||
}
|
||||
|
||||
fn elmhes<T: Number + RealNumber, M: Array2<T>>(A: &mut M) -> Vec<usize> {
|
||||
let (n, _) = A.shape();
|
||||
let mut perm = vec![0; n];
|
||||
|
||||
for (m, perm_m) in perm.iter_mut().enumerate().take(n - 1).skip(1) {
|
||||
let mut x = T::zero();
|
||||
let mut i = m;
|
||||
for j in m..n {
|
||||
if A.get((j, m - 1)).abs() > x.abs() {
|
||||
x = *A.get((j, m - 1));
|
||||
i = j;
|
||||
}
|
||||
}
|
||||
*perm_m = i;
|
||||
if i != m {
|
||||
for j in (m - 1)..n {
|
||||
A.swap((i, j), (m, j));
|
||||
}
|
||||
for j in 0..n {
|
||||
A.swap((j, i), (j, m));
|
||||
}
|
||||
}
|
||||
if x != T::zero() {
|
||||
for i in (m + 1)..n {
|
||||
let mut y = *A.get((i, m - 1));
|
||||
if y != T::zero() {
|
||||
y /= x;
|
||||
A.set((i, m - 1), y);
|
||||
for j in m..n {
|
||||
A.sub_element_mut((i, j), y * *A.get((m, j)));
|
||||
}
|
||||
for j in 0..n {
|
||||
A.add_element_mut((j, m), y * *A.get((j, i)));
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
perm
|
||||
}
|
||||
|
||||
fn eltran<T: Number + RealNumber, M: Array2<T>>(A: &M, V: &mut M, perm: &[usize]) {
|
||||
let (n, _) = A.shape();
|
||||
for mp in (1..n - 1).rev() {
|
||||
for k in mp + 1..n {
|
||||
V.set((k, mp), *A.get((k, mp - 1)));
|
||||
}
|
||||
let i = perm[mp];
|
||||
if i != mp {
|
||||
for j in mp..n {
|
||||
V.set((mp, j), *V.get((i, j)));
|
||||
V.set((i, j), T::zero());
|
||||
}
|
||||
V.set((i, mp), T::one());
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn hqr2<T: Number + RealNumber, M: Array2<T>>(A: &mut M, V: &mut M, d: &mut [T], e: &mut [T]) {
|
||||
let (n, _) = A.shape();
|
||||
let mut z = T::zero();
|
||||
let mut s = T::zero();
|
||||
let mut r = T::zero();
|
||||
let mut q = T::zero();
|
||||
let mut p = T::zero();
|
||||
let mut anorm = T::zero();
|
||||
|
||||
for i in 0..n {
|
||||
for j in i32::max(i as i32 - 1, 0)..n as i32 {
|
||||
anorm += A.get((i, j as usize)).abs();
|
||||
}
|
||||
}
|
||||
|
||||
let mut nn = n - 1;
|
||||
let mut t = T::zero();
|
||||
'outer: loop {
|
||||
let mut its = 0;
|
||||
loop {
|
||||
let mut l = nn;
|
||||
while l > 0 {
|
||||
s = A.get((l - 1, l - 1)).abs() + A.get((l, l)).abs();
|
||||
if s == T::zero() {
|
||||
s = anorm;
|
||||
}
|
||||
if A.get((l, l - 1)).abs() <= T::epsilon() * s {
|
||||
A.set((l, l - 1), T::zero());
|
||||
break;
|
||||
}
|
||||
l -= 1;
|
||||
}
|
||||
let mut x = *A.get((nn, nn));
|
||||
if l == nn {
|
||||
d[nn] = x + t;
|
||||
A.set((nn, nn), x + t);
|
||||
if nn == 0 {
|
||||
break 'outer;
|
||||
} else {
|
||||
nn -= 1;
|
||||
}
|
||||
} else {
|
||||
let mut y = *A.get((nn - 1, nn - 1));
|
||||
let mut w = *A.get((nn, nn - 1)) * *A.get((nn - 1, nn));
|
||||
if l == nn - 1 {
|
||||
p = T::half() * (y - x);
|
||||
q = p * p + w;
|
||||
z = q.abs().sqrt();
|
||||
x += t;
|
||||
A.set((nn, nn), x);
|
||||
A.set((nn - 1, nn - 1), y + t);
|
||||
if q >= T::zero() {
|
||||
z = p + <T as RealNumber>::copysign(z, p);
|
||||
d[nn - 1] = x + z;
|
||||
d[nn] = x + z;
|
||||
if z != T::zero() {
|
||||
d[nn] = x - w / z;
|
||||
}
|
||||
x = *A.get((nn, nn - 1));
|
||||
s = x.abs() + z.abs();
|
||||
p = x / s;
|
||||
q = z / s;
|
||||
r = (p * p + q * q).sqrt();
|
||||
p /= r;
|
||||
q /= r;
|
||||
for j in nn - 1..n {
|
||||
z = *A.get((nn - 1, j));
|
||||
A.set((nn - 1, j), q * z + p * *A.get((nn, j)));
|
||||
A.set((nn, j), q * *A.get((nn, j)) - p * z);
|
||||
}
|
||||
for i in 0..=nn {
|
||||
z = *A.get((i, nn - 1));
|
||||
A.set((i, nn - 1), q * z + p * *A.get((i, nn)));
|
||||
A.set((i, nn), q * *A.get((i, nn)) - p * z);
|
||||
}
|
||||
for i in 0..n {
|
||||
z = *V.get((i, nn - 1));
|
||||
V.set((i, nn - 1), q * z + p * *V.get((i, nn)));
|
||||
V.set((i, nn), q * *V.get((i, nn)) - p * z);
|
||||
}
|
||||
} else {
|
||||
d[nn] = x + p;
|
||||
e[nn] = -z;
|
||||
d[nn - 1] = d[nn];
|
||||
e[nn - 1] = -e[nn];
|
||||
}
|
||||
|
||||
if nn <= 1 {
|
||||
break 'outer;
|
||||
} else {
|
||||
nn -= 2;
|
||||
}
|
||||
} else {
|
||||
if its == 30 {
|
||||
panic!("Too many iterations in hqr");
|
||||
}
|
||||
if its == 10 || its == 20 {
|
||||
t += x;
|
||||
for i in 0..nn + 1 {
|
||||
A.sub_element_mut((i, i), x);
|
||||
}
|
||||
s = A.get((nn, nn - 1)).abs() + A.get((nn - 1, nn - 2)).abs();
|
||||
y = T::from_f64(0.75).unwrap() * s;
|
||||
x = T::from_f64(0.75).unwrap() * s;
|
||||
w = T::from_f64(-0.4375).unwrap() * s * s;
|
||||
}
|
||||
its += 1;
|
||||
let mut m = nn - 2;
|
||||
while m >= l {
|
||||
z = *A.get((m, m));
|
||||
r = x - z;
|
||||
s = y - z;
|
||||
p = (r * s - w) / *A.get((m + 1, m)) + *A.get((m, m + 1));
|
||||
q = *A.get((m + 1, m + 1)) - z - r - s;
|
||||
r = *A.get((m + 2, m + 1));
|
||||
s = p.abs() + q.abs() + r.abs();
|
||||
p /= s;
|
||||
q /= s;
|
||||
r /= s;
|
||||
if m == l {
|
||||
break;
|
||||
}
|
||||
let u = A.get((m, m - 1)).abs() * (q.abs() + r.abs());
|
||||
let v = p.abs()
|
||||
* (A.get((m - 1, m - 1)).abs() + z.abs() + A.get((m + 1, m + 1)).abs());
|
||||
if u <= T::epsilon() * v {
|
||||
break;
|
||||
}
|
||||
m -= 1;
|
||||
}
|
||||
for i in m..nn - 1 {
|
||||
A.set((i + 2, i), T::zero());
|
||||
if i != m {
|
||||
A.set((i + 2, i - 1), T::zero());
|
||||
}
|
||||
}
|
||||
for k in m..nn {
|
||||
if k != m {
|
||||
p = *A.get((k, k - 1));
|
||||
q = *A.get((k + 1, k - 1));
|
||||
r = T::zero();
|
||||
if k + 1 != nn {
|
||||
r = *A.get((k + 2, k - 1));
|
||||
}
|
||||
x = p.abs() + q.abs() + r.abs();
|
||||
if x != T::zero() {
|
||||
p /= x;
|
||||
q /= x;
|
||||
r /= x;
|
||||
}
|
||||
}
|
||||
let s = <T as RealNumber>::copysign((p * p + q * q + r * r).sqrt(), p);
|
||||
if s != T::zero() {
|
||||
if k == m {
|
||||
if l != m {
|
||||
A.set((k, k - 1), -*A.get((k, k - 1)));
|
||||
}
|
||||
} else {
|
||||
A.set((k, k - 1), -s * x);
|
||||
}
|
||||
p += s;
|
||||
x = p / s;
|
||||
y = q / s;
|
||||
z = r / s;
|
||||
q /= p;
|
||||
r /= p;
|
||||
for j in k..n {
|
||||
p = *A.get((k, j)) + q * *A.get((k + 1, j));
|
||||
if k + 1 != nn {
|
||||
p += r * *A.get((k + 2, j));
|
||||
A.sub_element_mut((k + 2, j), p * z);
|
||||
}
|
||||
A.sub_element_mut((k + 1, j), p * y);
|
||||
A.sub_element_mut((k, j), p * x);
|
||||
}
|
||||
|
||||
let mmin = if nn < k + 3 { nn } else { k + 3 };
|
||||
for i in 0..(mmin + 1) {
|
||||
p = x * *A.get((i, k)) + y * *A.get((i, k + 1));
|
||||
if k + 1 != nn {
|
||||
p += z * *A.get((i, k + 2));
|
||||
A.sub_element_mut((i, k + 2), p * r);
|
||||
}
|
||||
A.sub_element_mut((i, k + 1), p * q);
|
||||
A.sub_element_mut((i, k), p);
|
||||
}
|
||||
for i in 0..n {
|
||||
p = x * *V.get((i, k)) + y * *V.get((i, k + 1));
|
||||
if k + 1 != nn {
|
||||
p += z * *V.get((i, k + 2));
|
||||
V.sub_element_mut((i, k + 2), p * r);
|
||||
}
|
||||
V.sub_element_mut((i, k + 1), p * q);
|
||||
V.sub_element_mut((i, k), p);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
if l + 1 >= nn {
|
||||
break;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
if anorm != T::zero() {
|
||||
for nn in (0..n).rev() {
|
||||
p = d[nn];
|
||||
q = e[nn];
|
||||
let na = nn.wrapping_sub(1);
|
||||
if q == T::zero() {
|
||||
let mut m = nn;
|
||||
A.set((nn, nn), T::one());
|
||||
if nn > 0 {
|
||||
let mut i = nn - 1;
|
||||
loop {
|
||||
let w = *A.get((i, i)) - p;
|
||||
r = T::zero();
|
||||
for j in m..=nn {
|
||||
r += *A.get((i, j)) * *A.get((j, nn));
|
||||
}
|
||||
if e[i] < T::zero() {
|
||||
z = w;
|
||||
s = r;
|
||||
} else {
|
||||
m = i;
|
||||
|
||||
if e[i] == T::zero() {
|
||||
t = w;
|
||||
if t == T::zero() {
|
||||
t = T::epsilon() * anorm;
|
||||
}
|
||||
A.set((i, nn), -r / t);
|
||||
} else {
|
||||
let x = *A.get((i, i + 1));
|
||||
let y = *A.get((i + 1, i));
|
||||
q = (d[i] - p).powf(T::two()) + e[i].powf(T::two());
|
||||
t = (x * s - z * r) / q;
|
||||
A.set((i, nn), t);
|
||||
if x.abs() > z.abs() {
|
||||
A.set((i + 1, nn), (-r - w * t) / x);
|
||||
} else {
|
||||
A.set((i + 1, nn), (-s - y * t) / z);
|
||||
}
|
||||
}
|
||||
t = A.get((i, nn)).abs();
|
||||
if T::epsilon() * t * t > T::one() {
|
||||
for j in i..=nn {
|
||||
A.div_element_mut((j, nn), t);
|
||||
}
|
||||
}
|
||||
}
|
||||
if i == 0 {
|
||||
break;
|
||||
} else {
|
||||
i -= 1;
|
||||
}
|
||||
}
|
||||
}
|
||||
} else if q < T::zero() {
|
||||
let mut m = na;
|
||||
if A.get((nn, na)).abs() > A.get((na, nn)).abs() {
|
||||
A.set((na, na), q / *A.get((nn, na)));
|
||||
A.set((na, nn), -(*A.get((nn, nn)) - p) / *A.get((nn, na)));
|
||||
} else {
|
||||
let temp = Complex::new(T::zero(), -*A.get((na, nn)))
|
||||
/ Complex::new(*A.get((na, na)) - p, q);
|
||||
A.set((na, na), temp.re);
|
||||
A.set((na, nn), temp.im);
|
||||
}
|
||||
A.set((nn, na), T::zero());
|
||||
A.set((nn, nn), T::one());
|
||||
if nn >= 2 {
|
||||
for i in (0..nn - 1).rev() {
|
||||
let w = *A.get((i, i)) - p;
|
||||
let mut ra = T::zero();
|
||||
let mut sa = T::zero();
|
||||
for j in m..=nn {
|
||||
ra += *A.get((i, j)) * *A.get((j, na));
|
||||
sa += *A.get((i, j)) * *A.get((j, nn));
|
||||
}
|
||||
if e[i] < T::zero() {
|
||||
z = w;
|
||||
r = ra;
|
||||
s = sa;
|
||||
} else {
|
||||
m = i;
|
||||
if e[i] == T::zero() {
|
||||
let temp = Complex::new(-ra, -sa) / Complex::new(w, q);
|
||||
A.set((i, na), temp.re);
|
||||
A.set((i, nn), temp.im);
|
||||
} else {
|
||||
let x = *A.get((i, i + 1));
|
||||
let y = *A.get((i + 1, i));
|
||||
let mut vr =
|
||||
(d[i] - p).powf(T::two()) + (e[i]).powf(T::two()) - q * q;
|
||||
let vi = T::two() * q * (d[i] - p);
|
||||
if vr == T::zero() && vi == T::zero() {
|
||||
vr = T::epsilon()
|
||||
* anorm
|
||||
* (w.abs() + q.abs() + x.abs() + y.abs() + z.abs());
|
||||
}
|
||||
let temp =
|
||||
Complex::new(x * r - z * ra + q * sa, x * s - z * sa - q * ra)
|
||||
/ Complex::new(vr, vi);
|
||||
A.set((i, na), temp.re);
|
||||
A.set((i, nn), temp.im);
|
||||
if x.abs() > z.abs() + q.abs() {
|
||||
A.set(
|
||||
(i + 1, na),
|
||||
(-ra - w * *A.get((i, na)) + q * *A.get((i, nn))) / x,
|
||||
);
|
||||
A.set(
|
||||
(i + 1, nn),
|
||||
(-sa - w * *A.get((i, nn)) - q * *A.get((i, na))) / x,
|
||||
);
|
||||
} else {
|
||||
let temp = Complex::new(
|
||||
-r - y * *A.get((i, na)),
|
||||
-s - y * *A.get((i, nn)),
|
||||
) / Complex::new(z, q);
|
||||
A.set((i + 1, na), temp.re);
|
||||
A.set((i + 1, nn), temp.im);
|
||||
}
|
||||
}
|
||||
}
|
||||
t = T::max(A.get((i, na)).abs(), A.get((i, nn)).abs());
|
||||
if T::epsilon() * t * t > T::one() {
|
||||
for j in i..=nn {
|
||||
A.div_element_mut((j, na), t);
|
||||
A.div_element_mut((j, nn), t);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
for j in (0..n).rev() {
|
||||
for i in 0..n {
|
||||
z = T::zero();
|
||||
for k in 0..=j {
|
||||
z += *V.get((i, k)) * *A.get((k, j));
|
||||
}
|
||||
V.set((i, j), z);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn balbak<T: Number + RealNumber, M: Array2<T>>(V: &mut M, scale: &[T]) {
|
||||
let (n, _) = V.shape();
|
||||
for (i, scale_i) in scale.iter().enumerate().take(n) {
|
||||
for j in 0..n {
|
||||
V.mul_element_mut((i, j), *scale_i);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn sort<T: Number + RealNumber, M: Array2<T>>(d: &mut [T], e: &mut [T], V: &mut M) {
|
||||
let n = d.len();
|
||||
let mut temp = vec![T::zero(); n];
|
||||
for j in 1..n {
|
||||
let real = d[j];
|
||||
let img = e[j];
|
||||
for (k, temp_k) in temp.iter_mut().enumerate().take(n) {
|
||||
*temp_k = *V.get((k, j));
|
||||
}
|
||||
let mut i = j as i32 - 1;
|
||||
while i >= 0 {
|
||||
if d[i as usize] >= d[j] {
|
||||
break;
|
||||
}
|
||||
d[i as usize + 1] = d[i as usize];
|
||||
e[i as usize + 1] = e[i as usize];
|
||||
for k in 0..n {
|
||||
V.set((k, i as usize + 1), *V.get((k, i as usize)));
|
||||
}
|
||||
i -= 1;
|
||||
}
|
||||
d[i as usize + 1] = real;
|
||||
e[i as usize + 1] = img;
|
||||
for (k, temp_k) in temp.iter().enumerate().take(n) {
|
||||
V.set((k, i as usize + 1), *temp_k);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
use crate::linalg::basic::matrix::DenseMatrix;
|
||||
use approx::relative_eq;
|
||||
|
||||
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
||||
#[test]
|
||||
fn decompose_symmetric() {
|
||||
let A = DenseMatrix::from_2d_array(&[
|
||||
&[0.9000, 0.4000, 0.7000],
|
||||
&[0.4000, 0.5000, 0.3000],
|
||||
&[0.7000, 0.3000, 0.8000],
|
||||
]);
|
||||
|
||||
let eigen_values: Vec<f64> = vec![1.7498382, 0.3165784, 0.1335834];
|
||||
|
||||
let eigen_vectors = DenseMatrix::from_2d_array(&[
|
||||
&[0.6881997, -0.07121225, 0.7220180],
|
||||
&[0.3700456, 0.89044952, -0.2648886],
|
||||
&[0.6240573, -0.44947578, -0.6391588],
|
||||
]);
|
||||
|
||||
let evd = A.evd(true).unwrap();
|
||||
|
||||
assert!(relative_eq!(
|
||||
eigen_vectors.abs(),
|
||||
evd.V.abs(),
|
||||
epsilon = 1e-4
|
||||
));
|
||||
for i in 0..eigen_values.len() {
|
||||
assert!((eigen_values[i] - evd.d[i]).abs() < 1e-4);
|
||||
}
|
||||
for i in 0..eigen_values.len() {
|
||||
assert!((0f64 - evd.e[i]).abs() < std::f64::EPSILON);
|
||||
}
|
||||
}
|
||||
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
||||
#[test]
|
||||
fn decompose_asymmetric() {
|
||||
let A = DenseMatrix::from_2d_array(&[
|
||||
&[0.9000, 0.4000, 0.7000],
|
||||
&[0.4000, 0.5000, 0.3000],
|
||||
&[0.8000, 0.3000, 0.8000],
|
||||
]);
|
||||
|
||||
let eigen_values: Vec<f64> = vec![1.79171122, 0.31908143, 0.08920735];
|
||||
|
||||
let eigen_vectors = DenseMatrix::from_2d_array(&[
|
||||
&[0.7178958, 0.05322098, 0.6812010],
|
||||
&[0.3837711, -0.84702111, -0.1494582],
|
||||
&[0.6952105, 0.43984484, -0.7036135],
|
||||
]);
|
||||
|
||||
let evd = A.evd(false).unwrap();
|
||||
|
||||
assert!(relative_eq!(
|
||||
eigen_vectors.abs(),
|
||||
evd.V.abs(),
|
||||
epsilon = 1e-4
|
||||
));
|
||||
for i in 0..eigen_values.len() {
|
||||
assert!((eigen_values[i] - evd.d[i]).abs() < 1e-4);
|
||||
}
|
||||
for i in 0..eigen_values.len() {
|
||||
assert!((0f64 - evd.e[i]).abs() < std::f64::EPSILON);
|
||||
}
|
||||
}
|
||||
#[cfg_attr(target_arch = "wasm32", wasm_bindgen_test::wasm_bindgen_test)]
|
||||
#[test]
|
||||
fn decompose_complex() {
|
||||
let A = DenseMatrix::from_2d_array(&[
|
||||
&[3.0, -2.0, 1.0, 1.0],
|
||||
&[4.0, -1.0, 1.0, 1.0],
|
||||
&[1.0, 1.0, 3.0, -2.0],
|
||||
&[1.0, 1.0, 4.0, -1.0],
|
||||
]);
|
||||
|
||||
let eigen_values_d: Vec<f64> = vec![0.0, 2.0, 2.0, 0.0];
|
||||
let eigen_values_e: Vec<f64> = vec![2.2361, 0.9999, -0.9999, -2.2361];
|
||||
|
||||
let eigen_vectors = DenseMatrix::from_2d_array(&[
|
||||
&[-0.9159, -0.1378, 0.3816, -0.0806],
|
||||
&[-0.6707, 0.1059, 0.901, 0.6289],
|
||||
&[0.9159, -0.1378, 0.3816, 0.0806],
|
||||
&[0.6707, 0.1059, 0.901, -0.6289],
|
||||
]);
|
||||
|
||||
let evd = A.evd(false).unwrap();
|
||||
|
||||
assert!(relative_eq!(
|
||||
eigen_vectors.abs(),
|
||||
evd.V.abs(),
|
||||
epsilon = 1e-4
|
||||
));
|
||||
for i in 0..eigen_values_d.len() {
|
||||
assert!((eigen_values_d[i] - evd.d[i]).abs() < 1e-4);
|
||||
}
|
||||
for i in 0..eigen_values_e.len() {
|
||||
assert!((eigen_values_e[i] - evd.e[i]).abs() < 1e-4);
|
||||
}
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user