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#![allow(non_snake_case)] //! Cholesky and Modified Cholesky factorisations. //! //! UdU' and LdL' factorisations of positive semi-definite matrices. Where: //! U is unit upper triangular //! d is diagonal //! L is unit lower triangular //! //! Storage: //! UD format of UdU' factor //! strict_upper_triangle(UD) = strict_upper_triangle(U), diagonal(UD) = d, strict_lower_triangle(UD) ignored or zeroed //! LD format of LdL' factor //! strict_lower_triangle(LD) = strict_lower_triangle(L), diagonal(LD) = d, strict_upper_triangle(LD) ignored or zeroed use nalgebra as na; use na::{allocator::Allocator, DefaultAllocator}; use na::{Dim, MatrixMN, RealField}; use super::rcond; pub struct UDU<N: RealField> { pub zero: N, pub one: N, pub minus_one: N, } impl<N: RealField> UDU<N> { pub fn new() -> UDU<N> { UDU { zero: N::zero(), one: N::one(), minus_one: N::one().neg(), } } /// Estimate the reciprocal condition number for inversion of the original PSD * matrix for which UD is the factor UdU' or LdL'. /// /// Additional columns are ignored. The rcond of the original matrix is simply the rcond of its d factor /// Using the d factor is fast and simple, and avoids computing any squares. pub fn UdUrcond<R: Dim, C: Dim>(UD: &MatrixMN<N, R, C>) -> N where DefaultAllocator: Allocator<N, R, C>, { rcond::rcond_symetric(&UD) } /// Estimate the reciprocal condition number for inversion of the original PSD matrix for which U is the factor UU' /// /// The rcond of the original matrix is simply the square of the rcond of diagonal(UC). pub fn UCrcond<R: Dim, C: Dim>(&self, UC: &MatrixMN<N, R, C>) -> N where DefaultAllocator: Allocator<N, R, C>, { assert_eq!(UC.nrows(), UC.nrows()); let rcond = rcond::rcond_symetric(&UC); // Square to get rcond of original matrix, take care to propogate rcond's sign! if rcond < self.zero { -(rcond * rcond) } else { rcond * rcond } } /// In place Modified upper triangular Cholesky factor of a Positive definite or semi-definite matrix M. /// /// Reference: A+G p.218 Upper Cholesky algorithm modified for UdU' /// /// Numerical stability may not be as good as M(k,i) is updated from previous results. /// Algorithm has poor locality of reference and avoided for large matrices. /// Infinity values on the diagonal can be factorised. /// /// Input: M, n=last column to be included in factorisation, Strict lower triangle of M is ignored in computation /// /// Output: M as UdU' factor /// /// strict_upper_triangle(M) = strict_upper_triangle(U), /// diagonal(M) = d, /// strict_lower_triangle(M) is unmodified /// /// Return: reciprocal condition number, -1 if negative, 0 if semi-definite (including zero) pub fn UdUfactor_variant1<R: Dim, C: Dim>(&self, M: &mut MatrixMN<N, R, C>, n: usize) -> N where DefaultAllocator: Allocator<N, R, C>, { for j in (0..n).rev() { let mut d = M[(j, j)]; // Diagonal element if d > self.zero { // Positive definite d = self.one / d; for i in 0..j { let e = M[(i, j)]; M[(i, j)] = d * e; for k in 0..=i { let mut Mk = M.row_mut(k); let t = e * Mk[j]; Mk[i] -= t; } } } else if d == self.zero { // Possibly semi-definite, check not negative for i in 0..j { if M[(i, j)] != self.zero { return self.minus_one; } } } else { // Negative return self.minus_one; } } // Estimate the reciprocal condition number UDU::UdUrcond(M) } /// In place modified upper triangular Cholesky factor of a Positive definite or semi-definite matrix M /// /// Reference: A+G p.219 right side of table /// /// Algorithm has good locality of reference and preferable for large matrices. /// Infinity values on the diagonal cannot be factorised /// /// Input: M, n=last column to be included in factorisation, Strict lower triangle of M is ignored in computation /// /// Output: M as UdU' factor /// /// strict_upper_triangle(M) = strict_upper_triangle(U), diagonal(M) = d, /// strict_lower_triangle(M) is unmodified /// /// Return: reciprocal condition number, -1 if negative, 0 if semi-definite (including zero) pub fn UdUfactor_variant2<R: Dim, C: Dim>(&self, M: &mut MatrixMN<N, R, C>, n: usize) -> N where DefaultAllocator: Allocator<N, R, C>, { for j in (0..n).rev() { let mut d = M[(j, j)]; // Diagonal element if d > self.zero { // Positive definite for i in (0..=j).rev() { let mut e = M[(i, j)]; for k in j + 1..n { e -= M[(i, k)] * M[(k, k)] * M[(j, k)]; } if i == j { d = e; M[(i, j)] = e; // Diagonal element } else { M[(i, j)] = e / d; } } } else if d == self.zero { // Possibly semi-definite, check not negative, whole row must be identically zero for k in j + 1..n { if M[(j, k)] != self.zero { return self.minus_one; } } } else { // Negative return self.minus_one; } } // Estimate the reciprocal condition number UDU::UdUrcond(M) } /// In place upper triangular Cholesky factor of a Positive definite or semi-definite matrix M. /// /// Reference: A+G p.218 /// /// Input: M, n=last std::size_t to be included in factorisation, Strict lower triangle of M is ignored in computation /// /// Output: M as UC*UC' factor, upper_triangle(M) = UC /// /// Return: reciprocal condition number, -1 if negative, 0 if semi-definite (including zero) pub fn UCfactor_n<R: Dim, C: Dim>(&self, M: &mut MatrixMN<N, R, C>, n: usize) -> N where DefaultAllocator: Allocator<N, R, C>, { for j in (0..n).rev() { let mut d = M[(j, j)]; // Diagonal element if d > self.zero { // Positive definite d = d.sqrt(); M[(j, j)] = d; d = self.one / d; for i in 0..j { let e = d * M[(i, j)]; M[(i, j)] = e; for k in 0..=i { let t = e * M[(k, j)]; M[(k, i)] -= t; } } } else if d == self.zero { // Possibly semi-definite, check not negative for i in 0..j { if M[(i, j)] != self.zero { return self.one; } } } else { // Negative return self.minus_one; } } M.fill_lower_triangle(self.zero, 1); // Estimate the reciprocal condition number self.UCrcond(M) } /// In-place (destructive) inversion of diagonal and unit upper triangular matrices in UD. /// /// BE VERY CAREFUL THIS IS NOT THE INVERSE OF UD. /// /// Inversion on d and U is separate: inv(U)*inv(d)*inv(U') = inv(U'dU) NOT EQUAL inv(UdU') /// /// Lower triangle of UD is ignored and unmodified, /// Only diagonal part d can be singular (zero elements), inverse is computed of all elements other then singular. /// /// Reference: A+G p.223 /// /// Output: UD: inv(U), inv(d) /// /// Return: singularity (of d), true iff d has a zero element pub fn UdUinverse<R: Dim, C: Dim>(&self, UD: &mut MatrixMN<N, R, C>) -> bool where DefaultAllocator: Allocator<N, R, C>, { let n = UD.nrows(); assert_eq!(n, UD.ncols()); // Invert U in place if n > 1 { for i in (0..n - 1).rev() { for j in (i + 1..n).rev() { let mut UDij = -UD[(i, j)]; for k in i + 1..j { UDij -= UD[(i, k)] * UD[(k, j)]; } UD[(i, j)] = UDij; } } } // Invert d in place let mut singular = false; for i in 0..n { // Detect singular element let UDii = UD[(i, i)]; if UDii != self.zero { UD[(i, i)] = self.one / UDii; } else { singular = true; } } singular } /// In-place (destructive) inversion of upper triangular matrix in U. /// /// Output: U: inv(U) /// /// Return: singularity (of U), true iff diagonal of U has a zero element pub fn UTinverse<R: Dim, C: Dim>(&self, U: &mut MatrixMN<N, R, C>) -> bool where DefaultAllocator: Allocator<N, R, C>, { let n = U.nrows(); assert_eq!(n, U.ncols()); let mut singular = false; // Invert U in place for i in (0..n).rev() { let mut d = U[(i, i)]; if d == self.zero { singular = true; break; } d = self.one / d; U[(i, i)] = d; for j in (i + 1..n).rev() { let mut e = self.zero; for k in i + 1..=j { e -= U[(i, k)] * U[(k, j)]; } U[(i, j)] = e * d; } } singular } /// In-place recomposition of Symmetric matrix from U'dU factor store in UD format. /// /// Generally used for recomposing result of UdUinverse. /// Note definiteness of result depends purely on diagonal(M) i.e. if d is positive definite (>0) then result is positive definite /// /// Reference: A+G p.223 /// /// In place computation uses simple structure of solution due to triangular zero elements. /// Defn: R = (U' d) row i , C = U column j -> M(i,j) = R dot C, /// However M(i,j) only dependent R(k<=i), C(k<=j) due to zeros. /// Therefore in place multiple sequences such k < i <= j /// /// Input: M - U'dU factorisation (UD format) /// /// Output: M - U'dU recomposition (symmetric) pub fn UdUrecompose_transpose<R: Dim, C: Dim>(M: &mut MatrixMN<N, R, C>) where DefaultAllocator: Allocator<N, R, C>, { let n = M.nrows(); assert_eq!(n, M.ncols()); // Recompose M = (U'dU) in place for i in (0..n).rev() { // (U' d) row i of lower triangle from upper triangle for j in 0..i { M[(i, j)] = M[(j, i)] * M[(j, j)]; } // (U' d) U in place for j in (i..n).rev() { // Compute matrix product (U'd) row i * U col j if j > i { // Optimised handling of 1 in U let mii = M[(i, i)]; M[(i, j)] *= mii; } for k in 0..i { // Inner loop k < i <=j, only strict triangular elements let t = M[(i, k)] * M[(k, j)]; M[(i, j)] += t; // M(i,k) element of U'd, M(k,j) element of U } M[(j, i)] = M[(i, j)]; } } } /// In-place recomposition of Symmetric matrix from UdU' factor store in UD format. /// /// See UdUrecompose_transpose() /// /// Input: M - UdU' factorisation (UD format) /// /// Output: M - UdU' recomposition (symmetric) pub fn UdUrecompose<R: Dim, C: Dim>(M: &mut MatrixMN<N, R, C>) where DefaultAllocator: Allocator<N, R, C>, { let n = M.nrows(); assert_eq!(n, M.ncols()); // Recompose M = (UdU') in place for i in 0..n { // (d U') col i of lower triangle from upper trinagle for j in i + 1..n { M[(j, i)] = M[(i, j)] * M[(j, j)]; } // U (d U') in place for j in 0..=i { // j<=i // Compute matrix product (U'd) row i * U col j if j > i { // Optimised handling of 1 in U let mii = M[(i, i)]; M[(i, j)] *= mii; } for k in i + 1..n { // Inner loop k > i >=j, only strict triangular elements let t = M[(i, k)] * M[(k, j)]; M[(i, j)] += t; // M(i,k) element of U'd, M(k,j) element of U } M[(j, i)] = M[(i, j)]; } } } }