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//! This crate implements the matrix 1-norm estimator by [Higham and Tisseur]. //! //! [Higham and Tisseur]: http://eprints.ma.man.ac.uk/321/1/covered/MIMS_ep2006_145.pdf use ndarray::{ prelude::*, ArrayBase, Data, DataMut, Dimension, Ix1, Ix2, s, }; use ordered_float::NotNan; use rand::{ Rng, SeedableRng, thread_rng, }; use rand_xoshiro::Xoshiro256StarStar; use std::collections::BTreeSet; use std::cmp; use std::slice; pub struct Normest1 { n: usize, t: usize, rng: Xoshiro256StarStar, x_matrix: Array2<f64>, y_matrix: Array2<f64>, z_matrix: Array2<f64>, w_vector: Array1<f64>, sign_matrix: Array2<f64>, sign_matrix_old: Array2<f64>, column_is_parallel: Vec<bool>, indices: Vec<usize>, indices_history: BTreeSet<usize>, h: Vec<NotNan<f64>>, } /// A trait to generalize over 1-norm estimates of a matrix `A`, matrix powers `A^m`, /// or matrix products `A1 * A2 * ... * An`. /// /// In the 1-norm estimator, one repeatedly constructs a matrix-matrix product between some n×n /// matrix X and some other n×t matrix Y. If one wanted to estimate the 1-norm of a matrix m times /// itself, X^m, it might thus be computationally less expensive to repeatedly apply /// X * ( * ( X ... ( X * Y ) rather than to calculate Z = X^m = X * X * ... * X and then apply Z * /// Y. In the first case, one has several matrix-matrix multiplications with complexity O(m*n*n*t), /// while in the latter case one has O(m*n*n*n) (plus one more O(n*n*t)). /// /// So in case of t << n, it is cheaper to repeatedly apply matrix multiplication to the smaller /// matrix on the RHS, rather than to construct one definite matrix on the LHS. Of course, this is /// modified by the number of iterations needed when performing the norm estimate, sustained /// performance of the matrix multiplication method used, etc. /// /// It is at the designation of the user to check what is more efficient: to pass in one definite /// matrix or choose the alternative route described here. trait LinearOperator { fn multiply_matrix<S>(&self, b: &mut ArrayBase<S, Ix2>, c: &mut ArrayBase<S, Ix2>, transpose: bool) where S: DataMut<Elem=f64>; } impl<S1> LinearOperator for ArrayBase<S1, Ix2> where S1: Data<Elem=f64>, { fn multiply_matrix<S2>(&self, b: &mut ArrayBase<S2, Ix2>, c: &mut ArrayBase<S2, Ix2>, transpose: bool) where S2: DataMut<Elem=f64> { let (n_rows, n_cols) = self.dim(); assert_eq!(n_rows, n_cols, "Number of rows and columns does not match: `self` has to be a square matrix"); let n = n_rows; let (b_n, b_t) = b.dim(); let (c_n, c_t) = b.dim(); assert_eq!(n, b_n, "Number of rows of b not equal to number of rows of `self`."); assert_eq!(n, c_n, "Number of rows of c not equal to number of rows of `self`."); assert_eq!(b_t, c_t, "Number of columns of b not equal to number of columns of c."); let t = b_t; let (a_slice, a_layout) = as_slice_with_layout(self).expect("Matrix `self` not contiguous."); let (b_slice, b_layout) = as_slice_with_layout(b).expect("Matrix `b` not contiguous."); let (c_slice, c_layout) = as_slice_with_layout_mut(c).expect("Matrix `c` not contiguous."); assert_eq!(a_layout, b_layout); assert_eq!(a_layout, c_layout); let layout = a_layout; let a_transpose = if transpose { cblas::Transpose::Ordinary } else { cblas::Transpose::None }; unsafe { cblas::dgemm( layout, a_transpose, cblas::Transpose::None, n as i32, t as i32, n as i32, 1.0, a_slice, n as i32, b_slice, t as i32, 0.0, c_slice, t as i32, ) } } } impl<S1> LinearOperator for [&ArrayBase<S1, Ix2>] where S1: Data<Elem=f64> { fn multiply_matrix<S2>(&self, b: &mut ArrayBase<S2, Ix2>, c: &mut ArrayBase<S2, Ix2>, transpose: bool) where S2: DataMut<Elem=f64> { if self.len() > 0 { let mut reversed; let mut forward; // TODO: Investigate, if an enum instead of a trait object might be more performant. // This probably doesn't matter for large matrices, but could have a measurable impact // on small ones. let a_iter: &mut dyn DoubleEndedIterator<Item=_> = if transpose { reversed = self.iter().rev(); &mut reversed } else { forward = self.iter(); &mut forward }; let a = a_iter.next().unwrap(); // Ok because of if condition a.multiply_matrix(b, c, transpose); // NOTE: The swap in the loop body makes use of the fact that in all instances where // `multiply_matrix` is used, the values potentially stored in `b` are not required // anymore. for a in a_iter { std::mem::swap(b, c); a.multiply_matrix(b, c, transpose); } } } } impl<S1> LinearOperator for (&ArrayBase<S1, Ix2>, usize) where S1: Data<Elem=f64> { fn multiply_matrix<S2>(&self, b: &mut ArrayBase<S2, Ix2>, c: &mut ArrayBase< S2, Ix2>, transpose: bool) where S2: DataMut<Elem=f64> { let a = self.0; let m = self.1; if m > 0 { a.multiply_matrix(b, c, transpose); for _ in 1..m { std::mem::swap(b, c); self.0.multiply_matrix(b, c, transpose); } } } } impl Normest1 { pub fn new(n: usize, t: usize) -> Self { assert!(t <= n, "Cannot have more iteration columns t than columns in the matrix."); let rng = Xoshiro256StarStar::from_rng(&mut thread_rng()).expect("Rng initialization failed."); let x_matrix = unsafe { Array2::<f64>::uninitialized((n, t)) }; let y_matrix = unsafe { Array2::<f64>::uninitialized((n, t)) }; let z_matrix = unsafe { Array2::<f64>::uninitialized((n, t)) }; let w_vector = unsafe { Array1::uninitialized(n) }; let sign_matrix = unsafe { Array2::<f64>::uninitialized((n, t)) }; let sign_matrix_old = unsafe { Array2::<f64>::uninitialized((n, t)) }; let column_is_parallel = vec![false; t]; let indices = (0..n).collect(); let indices_history = BTreeSet::new(); let h = vec![unsafe { NotNan::unchecked_new(0.0) }; n]; Normest1 { n, t, rng, x_matrix, y_matrix, z_matrix, w_vector, sign_matrix, sign_matrix_old, column_is_parallel, indices, indices_history, h, } } fn calculate<L>(&mut self, a_linear_operator: &L, itmax: usize) -> f64 where L: LinearOperator + ?Sized { assert!(itmax > 1, "normest1 is undefined for iterations itmax < 2"); // Explicitly empty the index history; all other quantities will be overwritten at some // point. self.indices_history.clear(); let n = self.n; let t = self.t; let sample = [-1., 1.0]; // “We now explain our choice of starting matrix. We take the first column of X to be the // vector of 1s, which is the starting vector used in Algorithm 2.1. This has the advantage // that for a matrix with nonnegative elements the algorithm converges with an exact estimate // on the second iteration, and such matrices arise in applications, for example as a // stochastic matrix or as the inverse of an M -matrix.” // // “The remaining columns are chosen as rand {− 1 , 1 } , with a check for and correction of // parallel columns, exactly as for S in the body of the algorithm. We choose random vectors // because it is difficult to argue for any particular fixed vectors and because randomness // lessens the importance of counterexamples (see the comments in the next section).” { let rng_mut = &mut self.rng; self.x_matrix.mapv_inplace(|_| sample[rng_mut.gen_range(0, sample.len())]); self.x_matrix.column_mut(0).fill(1.); } // Resample the x_matrix to make sure no columns are parallel find_parallel_columns_in(&self.x_matrix, &mut self.y_matrix, &mut self.column_is_parallel); for (i, is_parallel) in self.column_is_parallel.iter().enumerate() { if *is_parallel { resample_column(&mut self.x_matrix, i, &mut self.rng, &sample); } } // Set all columns to unit vectors self.x_matrix.mapv_inplace(|x| x / n as f64); let mut estimate = 0.0; let mut best_index = 0; 'optimization_loop: for k in 0..itmax { // Y = A X a_linear_operator.multiply_matrix(&mut self.x_matrix, &mut self.y_matrix, false); // est = max{‖Y(:,j)‖₁ : j = 1:t} let (max_norm_index, max_norm) = matrix_onenorm_with_index(&self.y_matrix); // if est > est_old or k=2 if max_norm > estimate || k == 1 { // ind_best = indⱼ where est = ‖Y(:,j)‖₁, w = Y(:, ind_best) estimate = max_norm; best_index = self.indices[max_norm_index]; self.w_vector.assign(&self.y_matrix.column(max_norm_index)); } else if k > 1 && max_norm <= estimate { break 'optimization_loop } if k >= itmax { break 'optimization_loop } // S = sign(Y) assign_signum_of_array( &self.y_matrix, &mut self.sign_matrix ); // TODO: Combine the test checking for parallelity between _all_ columns between S // and S_old with the “if t > 1” test below. // // > If every column of S is parallel to a column of Sold, goto (6), end // // NOTE: We are reusing `y_matrix` here as a temporary value. if are_all_columns_parallel_between(&self.sign_matrix_old, &self.sign_matrix, &mut self.y_matrix) { break 'optimization_loop; } // FIXME: Is an explicit if condition here necessary? if t > 1 { // > Ensure that no column of S is parallel to another column of S // > or to a column of Sold by replacing columns of S by rand{-1,+1} // // NOTE: We are reusing `y_matrix` here as a temporary value. resample_parallel_columns( &mut self.sign_matrix, &self.sign_matrix_old, &mut self.y_matrix, &mut self.column_is_parallel, &mut self.rng, &sample, ); } // > est_old = est, Sold = S // NOTE: Other than in the original algorithm, we store the sign matrix at this point // already. This way, we can reuse the sign matrix as additional workspace which is // useful when performing matrix multiplication with A^m or A1 A2 ... An (see the // description of the LinearOperator trait for explanation). // // NOTE: We don't “save” the old estimate, because we are using max_norm as another name // for the new estimate instead of overwriting/reusing est. self.sign_matrix_old.assign(&self.sign_matrix); // Z = A^T S a_linear_operator.multiply_matrix(&mut self.sign_matrix, &mut self.z_matrix, true); // hᵢ= ‖Z(i,:)‖_∞ let mut max_h = 0.0; for (row, h_element) in self.z_matrix.genrows().into_iter().zip(self.h.iter_mut()) { let h = vector_maxnorm(&row); max_h = if h > max_h { h } else { max_h }; // Convert f64 to NotNan for using sort_unstable_by below *h_element = h.into(); } // TODO: This test for equality needs an approximate equality test instead. if k > 0 && max_h == self.h[best_index].into() { break 'optimization_loop } // > Sort h so that h_1 >= ... >= h_n and re-order correspondingly. // NOTE: h itself doesn't need to be reordered. Only the order of // the indices is relevant. { let h_ref = &self.h; self.indices.sort_unstable_by(|i, j| h_ref[*j].cmp(&h_ref[*i])); } self.x_matrix.fill(0.0); if t > 1 { // > Replace ind(1:t) by the first t indices in ind(1:n) that are not in ind_hist. // // > X(:, j) = e_ind_j, j = 1:t // // > ind_hist = [ind_hist ind(1:t)] // // NOTE: It's not actually needed to operate on the `indices` vector. What's important // is that the history of indices, `indices_history`, gets updated with visited indices, // and that each column of `x_matrix` is assigned that unit vector that is defined by the // respective index. // // If so many indices have already been used that `n_cols - indices_history.len() < t` // (which means that we have less than `t` unused indices remaining), we have to use a few // historical indices when filling up the columns in `x_matrix`. For that, we put the // historical indices after the fresh indices, but otherwise keep the order induced by `h` // above. let fresh_indices = cmp::min(t, n - self.indices_history.len()); if fresh_indices == 0 { break 'optimization_loop; } let mut current_column_fresh = 0; let mut current_column_historical = fresh_indices; let mut index_iterator = self.indices.iter(); let mut all_first_t_in_history = true; // First, iterate over the first t sorted indices. for i in (&mut index_iterator).take(t) { if !self.indices_history.contains(i) { all_first_t_in_history = false; self.x_matrix[(*i, current_column_fresh)] = 1.0; current_column_fresh += 1; self.indices_history.insert(*i); } else if current_column_historical < t { self.x_matrix[(*i, current_column_historical)] = 1.0; current_column_historical += 1; } } // > if ind(1:t) is contained in ind_hist, goto (6), end if all_first_t_in_history { break 'optimization_loop; } // Iterate over the remaining indices 'fill_x: for i in index_iterator { if current_column_fresh >= t { break 'fill_x; } if !self.indices_history.contains(i) { self.x_matrix[(*i, current_column_fresh)] = 1.0; current_column_fresh += 1; self.indices_history.insert(*i); } else if current_column_historical < t { self.x_matrix[(*i, current_column_historical)] = 1.0; current_column_historical += 1; } } } } estimate } /// Estimate the 1-norm of matrix `a` using up to `itmax` iterations. pub fn normest1<S>(&mut self, a: &ArrayBase<S, Ix2>, itmax: usize) -> f64 where S: Data<Elem=f64>, { self.calculate(a, itmax) } /// Estimate the 1-norm of a marix `a` to the power `m` up to `itmax` iterations. pub fn normest1_pow<S>(&mut self, a: &ArrayBase<S, Ix2>, m: usize, itmax: usize) -> f64 where S: Data<Elem=f64>, { self.calculate(&(a, m), itmax) } /// Estimate the 1-norm of a product of matrices `a1 a2 ... an` up to `itmax` iterations. pub fn normest1_prod<S>(&mut self, aprod: &[&ArrayBase<S, Ix2>], itmax: usize) -> f64 where S: Data<Elem=f64>, { self.calculate(aprod, itmax) } } /// Estimates the 1-norm of matrix `a`. /// /// The parameter `t` is the number of vectors that have to fulfill some bound. See [Higham, /// Tisseur] for more information. `itmax` is the maximum number of sweeps permitted. /// /// **NOTE:** This function allocates on every call. If you want to repeatedly estimate the /// 1-norm on matrices of the same size, construct a [`Normest1`] first, and call its methods. /// /// [Higham, Tisseur]: http://eprints.ma.man.ac.uk/321/1/covered/MIMS_ep2006_145.pdf /// [`Normest1`]: struct.Normest1.html pub fn normest1(a_matrix: &Array2<f64>, t: usize, itmax: usize) -> f64 { // Assume the matrix is square and take the columns as n. If it's not square, the assertion in // normest.calculate will fail. let n = a_matrix.dim().1; let mut normest1 = Normest1::new(n, t); normest1.normest1(a_matrix, itmax) } /// Estimates the 1-norm of a matrix `a` to the power `m`, `a^m`. /// /// The parameter `t` is the number of vectors that have to fulfill some bound. See [Higham, /// Tisseur] for more information. `itmax` is the maximum number of sweeps permitted. /// /// **NOTE:** This function allocates on every call. If you want to repeatedly estimate the /// 1-norm on matrices of the same size, construct a [`Normest1`] first, and call its methods. /// /// [Higham, Tisseur]: http://eprints.ma.man.ac.uk/321/1/covered/MIMS_ep2006_145.pdf pub fn normest1_pow(a_matrix: &Array2<f64>, m: usize, t: usize, itmax: usize) -> f64 { // Assume the matrix is square and take the columns as n. If it's not square, the assertion in // normest.calculate will fail. let n = a_matrix.dim().1; let mut normest1 = Normest1::new(n, t); normest1.normest1_pow(a_matrix, m, itmax) } /// Estimates the 1-norm of a product of matrices `a1`, `a2`, ..., `an` passed in as a slice of /// references. /// /// The parameter `t` is the number of vectors that have to fulfill some bound. See [Higham, /// Tisseur] for more information. `itmax` is the maximum number of sweeps permitted. /// /// **NOTE:** This function allocates on every call. If you want to repeatedly estimate the /// 1-norm on matrices of the same size, construct a [`Normest1`] first, and call its methods. /// /// [Higham, Tisseur]: http://eprints.ma.man.ac.uk/321/1/covered/MIMS_ep2006_145.pdf pub fn normest1_prod(a_matrices: &[&Array2<f64>], t: usize, itmax: usize) -> f64 { assert!(a_matrices.len() > 0); let n = a_matrices[0].dim().1; let mut normest1 = Normest1::new(n, t); normest1.normest1_prod(a_matrices, itmax) } /// Assigns the sign of matrix `a` to matrix `b`. /// /// Panics if matrices `a` and `b` have different shape and strides, or if either underlying array is /// non-contiguous. This is to make sure that the iteration order over the matrices is the same. fn assign_signum_of_array<S1, S2, D>(a: &ArrayBase<S1, D>, b: &mut ArrayBase<S2, D>) where S1: Data<Elem=f64>, S2: DataMut<Elem=f64>, D: Dimension { assert_eq!(a.strides(), b.strides()); let (a_slice, a_layout) = as_slice_with_layout(a).expect("Matrix `a` is not contiguous."); let (b_slice, b_layout) = as_slice_with_layout_mut(b).expect("Matrix `b` is not contiguous."); assert_eq!(a_layout, b_layout); signum_of_slice(a_slice, b_slice); } fn signum_of_slice(source: &[f64], destination: &mut [f64]) { for (s, d) in source.iter().zip(destination) { *d = s.signum(); } } /// Calculate the onenorm of a vector (an `ArrayBase` with dimension `Ix1`). fn vector_onenorm<S>(a: &ArrayBase<S, Ix1>) -> f64 where S: Data<Elem=f64>, { let stride = a.strides()[0]; assert!(stride >= 0); let stride = stride as usize; let n_elements = a.len(); let a_slice = { let a = a.as_ptr(); let total_len = n_elements * stride; unsafe { slice::from_raw_parts(a, total_len) } }; unsafe { cblas::dasum(n_elements as i32, a_slice, stride as i32) } } /// Calculate the maximum norm of a vector (an `ArrayBase` with dimension `Ix1`). fn vector_maxnorm<S>(a: &ArrayBase<S, Ix1>) -> f64 where S: Data<Elem=f64> { let stride = a.strides()[0]; assert!(stride >= 0); let stride = stride as usize; let n_elements = a.len(); let a_slice = { let a = a.as_ptr(); let total_len = n_elements * stride; unsafe { slice::from_raw_parts(a, total_len) } }; let idx = unsafe { cblas::idamax( n_elements as i32, a_slice, stride as i32, ) as usize }; f64::abs(a[idx]) } // /// Calculate the onenorm of a matrix (an `ArrayBase` with dimension `Ix2`). // fn matrix_onenorm<S>(a: &ArrayBase<S, Ix2>) -> f64 // where S: Data<Elem=f64>, // { // let (n_rows, n_cols) = a.dim(); // if let Some((a_slice, layout)) = as_slice_with_layout(a) { // let layout = match layout { // cblas::Layout::RowMajor => lapacke::Layout::RowMajor, // cblas::Layout::ColumnMajor => lapacke::Layout::ColumnMajor, // }; // unsafe { // lapacke::dlange( // layout, // b'1', // n_rows as i32, // n_cols as i32, // a_slice, // n_rows as i32, // ) // } // // Fall through case for non-contiguous arrays. // } else { // a.gencolumns().into_iter() // .fold(0.0, |max, column| { // let onenorm = column.fold(0.0, |acc, element| { acc + f64::abs(*element) }); // if onenorm > max { onenorm } else { max } // }) // } // } /// Returns the one-norm of a matrix `a` together with the index of that column for /// which the norm is maximal. fn matrix_onenorm_with_index<S>(a: &ArrayBase<S, Ix2>) -> (usize, f64) where S: Data<Elem=f64>, { let mut max_norm = 0.0; let mut max_norm_index = 0; for (i, column) in a.gencolumns().into_iter().enumerate() { let norm = vector_onenorm(&column); if norm > max_norm { max_norm = norm; max_norm_index = i; } } (max_norm_index, max_norm) } /// Finds columns in the matrix `a` that are parallel to to some other column in `a`. /// /// Assumes that all entries of `a` are either +1 or -1. /// /// If column `j` of matrix `a` is parallel to some column `i`, `column_is_parallel[i]` is set to /// `true`. The matrix `c` is used as an intermediate value for the matrix product `a^t * a`. /// /// This function does not reset `column_is_parallel` to `false`. Entries that are `true` will be /// assumed to be parallel and not checked. /// /// Panics if arrays `a` and `c` don't have the same dimensions, or if the length of the slice /// `column_is_parallel` is not equal to the number of columns in `a`. fn find_parallel_columns_in<S1, S2> ( a: &ArrayBase<S1, Ix2>, c: &mut ArrayBase<S2, Ix2>, column_is_parallel: &mut [bool] ) where S1: Data<Elem=f64>, S2: DataMut<Elem=f64> { let a_dim = a.dim(); let c_dim = c.dim(); assert_eq!(a_dim, c_dim); let (n_rows, n_cols) = a_dim; assert_eq!(column_is_parallel.len(), n_cols); { let (a_slice, a_layout) = as_slice_with_layout(a).expect("Matrix `a` is not contiguous."); let (c_slice, c_layout) = as_slice_with_layout_mut(c).expect("Matrix `c` is not contiguous."); assert_eq!(a_layout, c_layout); let layout = a_layout; // NOTE: When calling the wrapped Fortran dsyrk subroutine with row major layout, // cblas::*syrk changes `'U'` to `'L'` (`Upper` to `Lower`), and `'O'` to `'N'` (`Ordinary` // to `None`). Different from `cblas::*gemm`, however, it does not automatically make sure // that the other arguments are changed to make sense in a routine expecting column major // order (in `cblas::*gemm`, this happens by flipping the matrices `a` and `b` as // arguments). // // So while `cblas::dsyrk` changes transposition and the position of where the results are // written to, it passes the other arguments on to the Fortran routine as is. // // For example, in case matrix `a` is a 4x2 matrix in column major order, and we want to // perform the operation `a^T a` on it (resulting in a symmetric 2x2 matrix), we would pass // TRANS='T', N=2 (order of c), K=4 (number of rows because of 'T'), LDA=4 (max(1,k) // because of 'T'), LDC=2. // // But if `a` is in row major order and we want to perform the same operation, we pass // TRANS='T' (gets translated to 'N'), N=2, K=2 (number of columns, because we 'T' -> 'N'), // LDA=2 (max(1,n) because of 'N'), LDC=2. // // In other words, because of row major order, the Fortran routine actually sees our 4x2 // matrix as a 2x4 matrix, and if we want to calculate `a^T a`, `cblas::dsyrk` makes sure // `'N'` is passed. let (k, lda) = match layout { cblas::Layout::ColumnMajor => (n_cols, n_rows), cblas::Layout::RowMajor => (n_rows, n_cols), }; unsafe { cblas::dsyrk( layout, cblas::Part::Upper, cblas::Transpose::Ordinary, n_cols as i32, k as i32, 1.0, a_slice, lda as i32, 0.0, c_slice, n_cols as i32, ); } } // c is upper triangular and contains all pair-wise vector products: // // x x x x x // . x x x x // . . x x x // . . . x x // . . . . x // Don't check more rows than we have columns 'rows: for (i, row) in c.genrows().into_iter().enumerate().take(n_cols) { // Skip if the column is already found to be parallel or if we are checking // the last column if column_is_parallel[i] || i >= n_cols - 1 { continue 'rows; } for (j, element) in row.slice(s![i+1..]).iter().enumerate() { // Check if the vectors are parallel or anti-parallel if f64::abs(*element) == n_rows as f64 { column_is_parallel[i+j+1] = true; } } } } /// Checks whether any columns of the matrix `a` are parallel to any columns of `b`. /// /// Assumes that we have parallelity only if all entries of two columns `a` and `b` are either +1 /// or -1. /// /// `The matrix `c` is used as an intermediate value for the matrix product `a^t * b`. /// /// `column_is_parallel[j]` is set to `true` if column `j` of matrix `a` is parallel to some column /// `i` of the matrix `b`, /// /// This function does not reset `column_is_parallel` to `false`. Entries that are `true` will be /// assumed to be parallel and not checked. /// /// Panics if arrays `a`, `b`, and `c` don't have the same dimensions, or if the length of the slice /// `column_is_parallel` is not equal to the number of columns in `a`. fn find_parallel_columns_between<S1, S2, S3> ( a: &ArrayBase<S1, Ix2>, b: &ArrayBase<S2, Ix2>, c: &mut ArrayBase<S3, Ix2>, column_is_parallel: &mut [bool], ) where S1: Data<Elem=f64>, S2: Data<Elem=f64>, S3: DataMut<Elem=f64> { let a_dim = a.dim(); let b_dim = b.dim(); let c_dim = c.dim(); assert_eq!(a_dim, b_dim); assert_eq!(a_dim, c_dim); let (n_rows, n_cols) = a_dim; assert_eq!(column_is_parallel.len(), n_cols); // Extra scope, because c_slice needs to be dropped after the dgemm { let (a_slice, a_layout) = as_slice_with_layout(&a).expect("Matrix `a` not contiguous."); let (b_slice, b_layout) = as_slice_with_layout(&b).expect("Matrix `b` not contiguous."); let (c_slice, c_layout) = as_slice_with_layout_mut(c).expect("Matrix `c` not contiguous."); assert_eq!(a_layout, b_layout); assert_eq!(a_layout, c_layout); let layout = a_layout; unsafe { cblas::dgemm( layout, cblas::Transpose::Ordinary, cblas::Transpose::None, n_cols as i32, n_cols as i32, n_rows as i32, 1.0, a_slice, n_cols as i32, b_slice, n_cols as i32, 0.0, c_slice, n_cols as i32, ); } } // We are iterating over the rows because it's more memory efficient (for row-major array). In // terms of logic there is no difference: we simply check if the current column of a (that's // the outer loop) is parallel to any column of b (inner loop). By iterating via columns we would check if // any column of a is parallel to the, in that case, current column of b. // TODO: Implement for column major arrays. 'rows: for (i, row) in c.genrows().into_iter().enumerate().take(n_cols) { // Skip if the column is already found to be parallel the last column. if column_is_parallel[i] { continue 'rows; } for element in row { if f64::abs(*element) == n_rows as f64 { column_is_parallel[i] = true; continue 'rows; } } } } /// Check if every column in `a` is parallel to some column in `b`. /// /// Assumes that we have parallelity only if all entries of two columns `a` and `b` are either +1 /// or -1. fn are_all_columns_parallel_between<S1, S2> ( a: &ArrayBase<S1, Ix2>, b: &ArrayBase<S1, Ix2>, c: &mut ArrayBase<S2, Ix2>, ) -> bool where S1: Data<Elem=f64>, S2: DataMut<Elem=f64> { let a_dim = a.dim(); let b_dim = b.dim(); let c_dim = c.dim(); assert_eq!(a_dim, b_dim); assert_eq!(a_dim, c_dim); let (n_rows, n_cols) = a_dim; // Extra scope, because c_slice needs to be dropped after the dgemm { let (a_slice, a_layout) = as_slice_with_layout(&a).expect("Matrix `a` not contiguous."); let (b_slice, b_layout) = as_slice_with_layout(&b).expect("Matrix `b` not contiguous."); let (c_slice, c_layout) = as_slice_with_layout_mut(c).expect("Matrix `c` not contiguous."); assert_eq!(a_layout, b_layout); assert_eq!(a_layout, c_layout); let layout = a_layout; unsafe { cblas::dgemm( layout, cblas::Transpose::Ordinary, cblas::Transpose::None, n_cols as i32, n_cols as i32, n_rows as i32, 1.0, a_slice, n_cols as i32, b_slice, n_cols as i32, 0.0, c_slice, n_rows as i32, ); } } // We are iterating over the rows because it's more memory efficient (for row-major array). In // terms of logic there is no difference: we simply check if a specific column of a is parallel // to any column of b. By iterating via columns we would check if any column of a is parallel // to a specific column of b. 'rows: for row in c.genrows() { for element in row { // If a parallel column was found, cut to the next one. if f64::abs(*element) == n_rows as f64 { continue 'rows; } } // This return statement should only be reached if not a single column parallel to the // current one was found. return false; } true } /// Find parallel columns in matrix `a` and columns in `a` that are parallel to any columns in /// matrix `b`, and replace those with random vectors. Returns `true` if resampling has taken place. fn resample_parallel_columns<S1, S2, S3, R>( a: &mut ArrayBase<S1, Ix2>, b: &ArrayBase<S2, Ix2>, c: &mut ArrayBase<S3, Ix2>, column_is_parallel: &mut [bool], rng: &mut R, sample: &[f64], ) -> bool where S1: DataMut<Elem=f64>, S2: Data<Elem=f64>, S3: DataMut<Elem=f64>, R: Rng { column_is_parallel.iter_mut().for_each(|x| {*x = false;}); find_parallel_columns_in(a, c, column_is_parallel); find_parallel_columns_between(a, b, c, column_is_parallel); let mut has_resampled = false; for (i, is_parallel) in column_is_parallel.into_iter().enumerate() { if *is_parallel { resample_column(a, i, rng, sample); has_resampled = true; } } has_resampled } /// Resamples column `i` of matrix `a` with elements drawn from `sample` using `rng`. /// /// Panics if `i` exceeds the number of columns in `a`. fn resample_column<R, S>(a: &mut ArrayBase<S, Ix2>, i: usize, rng: &mut R, sample: &[f64]) where S: DataMut<Elem=f64>, R: Rng { assert!(i < a.dim().1, "Trying to resample column with index exceeding matrix dimensions"); assert!(sample.len() > 0); a.column_mut(i).mapv_inplace(|_| sample[rng.gen_range(0, sample.len())]); } /// Returns slice and layout underlying an array `a`. fn as_slice_with_layout<S, T, D>(a: &ArrayBase<S, D>) -> Option<(&[T], cblas::Layout)> where S: Data<Elem=T>, D: Dimension { if let Some(a_slice) = a.as_slice() { Some((a_slice, cblas::Layout::RowMajor)) } else if let Some(a_slice) = a.as_slice_memory_order() { Some((a_slice, cblas::Layout::ColumnMajor)) } else { None } } /// Returns mutable slice and layout underlying an array `a`. fn as_slice_with_layout_mut<S, T, D>(a: &mut ArrayBase<S, D>) -> Option<(&mut [T], cblas::Layout)> where S: DataMut<Elem=T>, D: Dimension { if a.as_slice_mut().is_some() { Some((a.as_slice_mut().unwrap(), cblas::Layout::RowMajor)) } else if a.as_slice_memory_order_mut().is_some() { Some((a.as_slice_memory_order_mut().unwrap(), cblas::Layout::ColumnMajor)) } else { None } // XXX: The above is a workaround for Rust not having non-lexical lifetimes yet. // More information here: // http://smallcultfollowing.com/babysteps/blog/2016/04/27/non-lexical-lifetimes-introduction/#problem-case-3-conditional-control-flow-across-functions // // if let Some(slice) = a.as_slice_mut() { // Some((slice, cblas::Layout::RowMajor)) // } else if let Some(slice) = a.as_slice_memory_order_mut() { // Some((slice, cblas::Layout::ColumnMajor)) // } else { // None // } } #[cfg(test)] mod tests { extern crate openblas_src; use ndarray::{ prelude::*, Zip, }; use ndarray_rand::RandomExt; use rand::{ SeedableRng, }; use rand::distributions::StandardNormal; use rand_xoshiro::Xoshiro256Plus; #[test] fn equality_between_methods() { let t = 2; let n = 100; let itmax = 5; let mut rng = Xoshiro256Plus::seed_from_u64(1234); let distribution = StandardNormal; let mut a_matrix = Array::random_using((n, n), distribution, &mut rng); a_matrix.mapv_inplace(|x| 1.0/x); let unity = Array::eye(n); let estimate_onlymatrix = crate::normest1(&a_matrix, t, itmax); let estimate_matrixpow = crate::normest1_pow(&a_matrix, 1, t, itmax); let estimate_matrixprod = crate::normest1_prod(&[&a_matrix, &unity], t, itmax); assert_eq!(estimate_onlymatrix, estimate_matrixpow); assert_eq!(estimate_onlymatrix, estimate_matrixprod); } #[test] fn pow_2_is_prod_2() { let t = 2; let n = 100; let itmax = 5; let mut rng = Xoshiro256Plus::seed_from_u64(1234); let distribution = StandardNormal; let mut a_matrix = Array::random_using((n, n), distribution, &mut rng); a_matrix.mapv_inplace(|x| 1.0/x); let estimate_matrixpow = crate::normest1_pow(&a_matrix, 2, t, itmax); let estimate_matrixprod = crate::normest1_prod(&[&a_matrix, &a_matrix], t, itmax); assert_eq!(estimate_matrixpow, estimate_matrixprod); } #[test] /// This performs tests inspired by Table 3 of [Higham and Tisseur]. /// /// NOTE: Due to (most likely) floating point precision), the ratio `calculated/expected` (that /// is, the ratio of the estimated condition number to the explicitly calculated one) can /// exceed 1.0. However, when running the tests I have observed at most a ratio exceeding 1.0 /// by 3 bits in the significand/mantissa. In other words, the estimated condition number appears to be /// within 4 ULPS of the calculated/expected one. /// /// One can probably explain this with different ordering of summation/addition/multiplication. /// /// During tests run performed by the author(s) of this library, running the tets below with /// `nsamples = 5000` happened to always let the test pass. fn table_3_t_2() { let t = 2; let n = 100; let itmax = 5; let nsamples = 5000; let mut calculated = Vec::with_capacity(nsamples); let mut expected = Vec::with_capacity(nsamples); let mut rng = Xoshiro256Plus::seed_from_u64(1234); let distribution = StandardNormal; for _ in 0..nsamples { let mut a_matrix = Array::random_using((n, n), distribution, &mut rng); a_matrix.mapv_inplace(|x| 1.0/x); let estimate = crate::normest1(&a_matrix, t, itmax); calculated.push(estimate); expected.push({ let (a_slice, a_layout) = crate::as_slice_with_layout(&a_matrix).expect("a matrix not contiguous"); let a_layout = match a_layout { cblas::Layout::ColumnMajor => lapacke::Layout::ColumnMajor, cblas::Layout::RowMajor => lapacke::Layout::RowMajor, }; unsafe { lapacke::dlange( a_layout, b'1', n as i32, n as i32, a_slice, n as i32, )} }); } let calculated = Array1::from_vec(calculated); let expected = Array1::from_vec(expected); let mut underestimation_ratio = unsafe { Array1::<f64>::uninitialized(nsamples) }; Zip::from(&calculated) .and(&expected) .and(&mut underestimation_ratio) .apply(|c, e, u| { *u = *c / *e; }); let underestimation_mean = underestimation_ratio.mean_axis(Axis(0)).into_scalar(); assert!(0.99 < underestimation_mean); assert!(underestimation_mean < 1.0); } }