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lib.rs
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//! Safe Rust wrapper for LAPACK without external dependency.
//!
//! [Lapack] trait
//! ----------------
//!
//! This crates provides LAPACK wrapper as a traits.
//! For example, LU decomposition of general matrices is provided like:
//!
//! ```ignore
//! pub trait Lapack {
//! fn lu(l: MatrixLayout, a: &mut [Self]) -> Result<Pivot>;
//! }
//! ```
//!
//! see [Lapack] for detail.
//! This trait is implemented for [f32], [f64], [c32] which is an alias to `num::Complex<f32>`,
//! and [c64] which is an alias to `num::Complex<f64>`.
//! You can use it like `f64::lu`:
//!
//! ```
//! use lax::{Lapack, layout::MatrixLayout, Transpose};
//!
//! let mut a = vec![
//! 1.0, 2.0,
//! 3.0, 4.0
//! ];
//! let mut b = vec![1.0, 2.0];
//! let layout = MatrixLayout::C { row: 2, lda: 2 };
//! let pivot = f64::lu(layout, &mut a).unwrap();
//! f64::solve(layout, Transpose::No, &a, &pivot, &mut b).unwrap();
//! ```
//!
//! When you want to write generic algorithm for real and complex matrices,
//! this trait can be used as a trait bound:
//!
//! ```
//! use lax::{Lapack, layout::MatrixLayout, Transpose};
//!
//! fn solve_at_once<T: Lapack>(layout: MatrixLayout, a: &mut [T], b: &mut [T]) -> Result<(), lax::error::Error> {
//! let pivot = T::lu(layout, a)?;
//! T::solve(layout, Transpose::No, a, &pivot, b)?;
//! Ok(())
//! }
//! ```
//!
//! There are several similar traits as described below to keep development easy.
//! They are merged into a single trait, [Lapack].
//!
//! Linear equation, Inverse matrix, Condition number
//! --------------------------------------------------
//!
//! According to the property input metrix, several types of triangular decomposition are used:
//!
//! - [solve] module provides methods for LU-decomposition for general matrix.
//! - [solveh] module provides methods for Bunch-Kaufman diagonal pivoting method for symmetric/Hermitian indefinite matrix.
//! - [cholesky] module provides methods for Cholesky decomposition for symmetric/Hermitian positive dinite matrix.
//!
//! Eigenvalue Problem
//! -------------------
//!
//! According to the property input metrix,
//! there are several types of eigenvalue problem API
//!
//! - [eig] module for eigenvalue problem for general matrix.
//! - [eigh] module for eigenvalue problem for symmetric/Hermitian matrix.
//! - [eigh_generalized] module for generalized eigenvalue problem for symmetric/Hermitian matrix.
//!
//! Singular Value Decomposition
//! -----------------------------
//!
//! - [svd] module for singular value decomposition (SVD) for general matrix
//! - [svddc] module for singular value decomposition (SVD) with divided-and-conquer algorithm for general matrix
//! - [least_squares] module for solving least square problem using SVD
//!
#![deny(rustdoc::broken_intra_doc_links, rustdoc::private_intra_doc_links)]
#[cfg(any(feature = "intel-mkl-system", feature = "intel-mkl-static"))]
extern crate intel_mkl_src as _src;
#[cfg(any(feature = "openblas-system", feature = "openblas-static"))]
extern crate openblas_src as _src;
#[cfg(any(feature = "netlib-system", feature = "netlib-static"))]
extern crate netlib_src as _src;
pub mod alloc;
pub mod cholesky;
pub mod eig;
pub mod eigh;
pub mod eigh_generalized;
pub mod error;
pub mod flags;
pub mod layout;
pub mod least_squares;
pub mod opnorm;
pub mod qr;
pub mod rcond;
pub mod solve;
pub mod solveh;
pub mod svd;
pub mod svddc;
pub mod triangular;
pub mod tridiagonal;
pub use self::flags::*;
pub use self::least_squares::LeastSquaresOwned;
pub use self::svd::{SvdOwned, SvdRef};
pub use self::tridiagonal::{LUFactorizedTridiagonal, Tridiagonal};
use self::{alloc::*, error::*, layout::*};
use cauchy::*;
use std::mem::MaybeUninit;
pub type Pivot = Vec<i32>;
#[cfg_attr(doc, katexit::katexit)]
/// Trait for primitive types which implements LAPACK subroutines
pub trait Lapack: Scalar {
/// Compute right eigenvalue and eigenvectors for a general matrix
fn eig(
calc_v: bool,
l: MatrixLayout,
a: &mut [Self],
) -> Result<(Vec<Self::Complex>, Vec<Self::Complex>)>;
/// Compute right eigenvalue and eigenvectors for a symmetric or Hermitian matrix
fn eigh(
calc_eigenvec: bool,
layout: MatrixLayout,
uplo: UPLO,
a: &mut [Self],
) -> Result<Vec<Self::Real>>;
/// Compute right eigenvalue and eigenvectors for a symmetric or Hermitian matrix
fn eigh_generalized(
calc_eigenvec: bool,
layout: MatrixLayout,
uplo: UPLO,
a: &mut [Self],
b: &mut [Self],
) -> Result<Vec<Self::Real>>;
/// Execute Householder reflection as the first step of QR-decomposition
///
/// For C-continuous array,
/// this will call LQ-decomposition of the transposed matrix $ A^T = LQ^T $
fn householder(l: MatrixLayout, a: &mut [Self]) -> Result<Vec<Self>>;
/// Reconstruct Q-matrix from Householder-reflectors
fn q(l: MatrixLayout, a: &mut [Self], tau: &[Self]) -> Result<()>;
/// Execute QR-decomposition at once
fn qr(l: MatrixLayout, a: &mut [Self]) -> Result<Vec<Self>>;
/// Compute singular-value decomposition (SVD)
fn svd(l: MatrixLayout, calc_u: bool, calc_vt: bool, a: &mut [Self]) -> Result<SvdOwned<Self>>;
/// Compute singular value decomposition (SVD) with divide-and-conquer algorithm
fn svddc(layout: MatrixLayout, jobz: JobSvd, a: &mut [Self]) -> Result<SvdOwned<Self>>;
/// Compute a vector $x$ which minimizes Euclidian norm $\| Ax - b\|$
/// for a given matrix $A$ and a vector $b$.
fn least_squares(
a_layout: MatrixLayout,
a: &mut [Self],
b: &mut [Self],
) -> Result<LeastSquaresOwned<Self>>;
/// Solve least square problems $\argmin_X \| AX - B\|$
fn least_squares_nrhs(
a_layout: MatrixLayout,
a: &mut [Self],
b_layout: MatrixLayout,
b: &mut [Self],
) -> Result<LeastSquaresOwned<Self>>;
/// Computes the LU decomposition of a general $m \times n$ matrix
/// with partial pivoting with row interchanges.
///
/// For a given matrix $A$, LU decomposition is described as $A = PLU$ where:
///
/// - $L$ is lower matrix
/// - $U$ is upper matrix
/// - $P$ is permutation matrix represented by [Pivot]
///
/// This is designed as two step computation according to LAPACK API:
///
/// 1. Factorize input matrix $A$ into $L$, $U$, and $P$.
/// 2. Solve linear equation $Ax = b$ by [Lapack::solve]
/// or compute inverse matrix $A^{-1}$ by [Lapack::inv] using the output of LU decomposition.
///
/// Output
/// -------
/// - $U$ and $L$ are stored in `a` after LU decomposition has succeeded.
/// - $P$ is returned as [Pivot]
///
/// Error
/// ------
/// - if the matrix is singular
/// - On this case, `return_code` in [Error::LapackComputationalFailure] means
/// `return_code`-th diagonal element of $U$ becomes zero.
///
fn lu(l: MatrixLayout, a: &mut [Self]) -> Result<Pivot>;
/// Compute inverse matrix $A^{-1}$ from the output of LU-decomposition
fn inv(l: MatrixLayout, a: &mut [Self], p: &Pivot) -> Result<()>;
/// Solve linear equations $Ax = b$ using the output of LU-decomposition
fn solve(l: MatrixLayout, t: Transpose, a: &[Self], p: &Pivot, b: &mut [Self]) -> Result<()>;
/// Factorize symmetric/Hermitian matrix using Bunch-Kaufman diagonal pivoting method
///
/// For a given symmetric matrix $A$,
/// this method factorizes $A = U^T D U$ or $A = L D L^T$ where
///
/// - $U$ (or $L$) are is a product of permutation and unit upper (lower) triangular matrices
/// - $D$ is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
///
/// This takes two-step approach based in LAPACK:
///
/// 1. Factorize given matrix $A$ into upper ($U$) or lower ($L$) form with diagonal matrix $D$
/// 2. Then solve linear equation $Ax = b$, and/or calculate inverse matrix $A^{-1}$
///
fn bk(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<Pivot>;
/// Compute inverse matrix $A^{-1}$ using the result of [Lapack::bk]
fn invh(l: MatrixLayout, uplo: UPLO, a: &mut [Self], ipiv: &Pivot) -> Result<()>;
/// Solve symmetric/Hermitian linear equation $Ax = b$ using the result of [Lapack::bk]
fn solveh(l: MatrixLayout, uplo: UPLO, a: &[Self], ipiv: &Pivot, b: &mut [Self]) -> Result<()>;
/// Solve symmetric/Hermitian positive-definite linear equations using Cholesky decomposition
///
/// For a given positive definite matrix $A$,
/// Cholesky decomposition is described as $A = U^T U$ or $A = LL^T$ where
///
/// - $L$ is lower matrix
/// - $U$ is upper matrix
///
/// This is designed as two step computation according to LAPACK API
///
/// 1. Factorize input matrix $A$ into $L$ or $U$
/// 2. Solve linear equation $Ax = b$ by [Lapack::solve_cholesky]
/// or compute inverse matrix $A^{-1}$ by [Lapack::inv_cholesky]
///
fn cholesky(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<()>;
/// Compute inverse matrix $A^{-1}$ using $U$ or $L$ calculated by [Lapack::cholesky]
fn inv_cholesky(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<()>;
/// Solve linear equation $Ax = b$ using $U$ or $L$ calculated by [Lapack::cholesky]
fn solve_cholesky(l: MatrixLayout, uplo: UPLO, a: &[Self], b: &mut [Self]) -> Result<()>;
/// Estimates the the reciprocal of the condition number of the matrix in 1-norm.
///
/// `anorm` should be the 1-norm of the matrix `a`.
fn rcond(l: MatrixLayout, a: &[Self], anorm: Self::Real) -> Result<Self::Real>;
/// Compute norm of matrices
///
/// For a $n \times m$ matrix
/// $$
/// A = \begin{pmatrix}
/// a_{11} & \cdots & a_{1m} \\\\
/// \vdots & \ddots & \vdots \\\\
/// a_{n1} & \cdots & a_{nm}
/// \end{pmatrix}
/// $$
/// LAPACK can compute three types of norms:
///
/// - Operator norm based on 1-norm for its domain linear space:
/// $$
/// \Vert A \Vert_1 = \sup_{\Vert x \Vert_1 = 1} \Vert Ax \Vert_1
/// = \max_{1 \le j \le m } \sum_{i=1}^n |a_{ij}|
/// $$
/// where
/// $\Vert x\Vert_1 = \sum_{j=1}^m |x_j|$
/// is 1-norm for a vector $x$.
///
/// - Operator norm based on $\infty$-norm for its domain linear space:
/// $$
/// \Vert A \Vert_\infty = \sup_{\Vert x \Vert_\infty = 1} \Vert Ax \Vert_\infty
/// = \max_{1 \le i \le n } \sum_{j=1}^m |a_{ij}|
/// $$
/// where
/// $\Vert x\Vert_\infty = \max_{j=1}^m |x_j|$
/// is $\infty$-norm for a vector $x$.
///
/// - Frobenious norm
/// $$
/// \Vert A \Vert_F = \sqrt{\mathrm{Tr} \left(AA^\dagger\right)} = \sqrt{\sum_{i=1}^n \sum_{j=1}^m |a_{ij}|^2}
/// $$
///
fn opnorm(t: NormType, l: MatrixLayout, a: &[Self]) -> Self::Real;
fn solve_triangular(
al: MatrixLayout,
bl: MatrixLayout,
uplo: UPLO,
d: Diag,
a: &[Self],
b: &mut [Self],
) -> Result<()>;
/// Computes the LU factorization of a tridiagonal `m x n` matrix `a` using
/// partial pivoting with row interchanges.
fn lu_tridiagonal(a: Tridiagonal<Self>) -> Result<LUFactorizedTridiagonal<Self>>;
fn rcond_tridiagonal(lu: &LUFactorizedTridiagonal<Self>) -> Result<Self::Real>;
fn solve_tridiagonal(
lu: &LUFactorizedTridiagonal<Self>,
bl: MatrixLayout,
t: Transpose,
b: &mut [Self],
) -> Result<()>;
}
macro_rules! impl_lapack {
($s:ty) => {
impl Lapack for $s {
fn eig(
calc_v: bool,
l: MatrixLayout,
a: &mut [Self],
) -> Result<(Vec<Self::Complex>, Vec<Self::Complex>)> {
use eig::*;
let work = EigWork::<$s>::new(calc_v, l)?;
let EigOwned { eigs, vr, vl } = work.eval(a)?;
Ok((eigs, vr.or(vl).unwrap_or_default()))
}
fn eigh(
calc_eigenvec: bool,
layout: MatrixLayout,
uplo: UPLO,
a: &mut [Self],
) -> Result<Vec<Self::Real>> {
use eigh::*;
let work = EighWork::<$s>::new(calc_eigenvec, layout)?;
work.eval(uplo, a)
}
fn eigh_generalized(
calc_eigenvec: bool,
layout: MatrixLayout,
uplo: UPLO,
a: &mut [Self],
b: &mut [Self],
) -> Result<Vec<Self::Real>> {
use eigh_generalized::*;
let work = EighGeneralizedWork::<$s>::new(calc_eigenvec, layout)?;
work.eval(uplo, a, b)
}
fn householder(l: MatrixLayout, a: &mut [Self]) -> Result<Vec<Self>> {
use qr::*;
let work = HouseholderWork::<$s>::new(l)?;
work.eval(a)
}
fn q(l: MatrixLayout, a: &mut [Self], tau: &[Self]) -> Result<()> {
use qr::*;
let mut work = QWork::<$s>::new(l)?;
work.calc(a, tau)?;
Ok(())
}
fn qr(l: MatrixLayout, a: &mut [Self]) -> Result<Vec<Self>> {
let tau = Self::householder(l, a)?;
let r = Vec::from(&*a);
Self::q(l, a, &tau)?;
Ok(r)
}
fn svd(
l: MatrixLayout,
calc_u: bool,
calc_vt: bool,
a: &mut [Self],
) -> Result<SvdOwned<Self>> {
use svd::*;
let work = SvdWork::<$s>::new(l, calc_u, calc_vt)?;
work.eval(a)
}
fn svddc(layout: MatrixLayout, jobz: JobSvd, a: &mut [Self]) -> Result<SvdOwned<Self>> {
use svddc::*;
let work = SvdDcWork::<$s>::new(layout, jobz)?;
work.eval(a)
}
fn least_squares(
l: MatrixLayout,
a: &mut [Self],
b: &mut [Self],
) -> Result<LeastSquaresOwned<Self>> {
let b_layout = l.resized(b.len() as i32, 1);
Self::least_squares_nrhs(l, a, b_layout, b)
}
fn least_squares_nrhs(
a_layout: MatrixLayout,
a: &mut [Self],
b_layout: MatrixLayout,
b: &mut [Self],
) -> Result<LeastSquaresOwned<Self>> {
use least_squares::*;
let work = LeastSquaresWork::<$s>::new(a_layout, b_layout)?;
work.eval(a, b)
}
fn lu(l: MatrixLayout, a: &mut [Self]) -> Result<Pivot> {
use solve::*;
LuImpl::lu(l, a)
}
fn inv(l: MatrixLayout, a: &mut [Self], p: &Pivot) -> Result<()> {
use solve::*;
let mut work = InvWork::<$s>::new(l)?;
work.calc(a, p)?;
Ok(())
}
fn solve(
l: MatrixLayout,
t: Transpose,
a: &[Self],
p: &Pivot,
b: &mut [Self],
) -> Result<()> {
use solve::*;
SolveImpl::solve(l, t, a, p, b)
}
fn bk(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<Pivot> {
use solveh::*;
let work = BkWork::<$s>::new(l)?;
work.eval(uplo, a)
}
fn invh(l: MatrixLayout, uplo: UPLO, a: &mut [Self], ipiv: &Pivot) -> Result<()> {
use solveh::*;
let mut work = InvhWork::<$s>::new(l)?;
work.calc(uplo, a, ipiv)
}
fn solveh(
l: MatrixLayout,
uplo: UPLO,
a: &[Self],
ipiv: &Pivot,
b: &mut [Self],
) -> Result<()> {
use solveh::*;
SolvehImpl::solveh(l, uplo, a, ipiv, b)
}
fn cholesky(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<()> {
use cholesky::*;
CholeskyImpl::cholesky(l, uplo, a)
}
fn inv_cholesky(l: MatrixLayout, uplo: UPLO, a: &mut [Self]) -> Result<()> {
use cholesky::*;
InvCholeskyImpl::inv_cholesky(l, uplo, a)
}
fn solve_cholesky(
l: MatrixLayout,
uplo: UPLO,
a: &[Self],
b: &mut [Self],
) -> Result<()> {
use cholesky::*;
SolveCholeskyImpl::solve_cholesky(l, uplo, a, b)
}
fn rcond(l: MatrixLayout, a: &[Self], anorm: Self::Real) -> Result<Self::Real> {
use rcond::*;
let mut work = RcondWork::<$s>::new(l);
work.calc(a, anorm)
}
fn opnorm(t: NormType, l: MatrixLayout, a: &[Self]) -> Self::Real {
use opnorm::*;
let mut work = OperatorNormWork::<$s>::new(t, l);
work.calc(a)
}
fn solve_triangular(
al: MatrixLayout,
bl: MatrixLayout,
uplo: UPLO,
d: Diag,
a: &[Self],
b: &mut [Self],
) -> Result<()> {
use triangular::*;
SolveTriangularImpl::solve_triangular(al, bl, uplo, d, a, b)
}
fn lu_tridiagonal(a: Tridiagonal<Self>) -> Result<LUFactorizedTridiagonal<Self>> {
use tridiagonal::*;
let work = LuTridiagonalWork::<$s>::new(a.l);
work.eval(a)
}
fn rcond_tridiagonal(lu: &LUFactorizedTridiagonal<Self>) -> Result<Self::Real> {
use tridiagonal::*;
let mut work = RcondTridiagonalWork::<$s>::new(lu.a.l);
work.calc(lu)
}
fn solve_tridiagonal(
lu: &LUFactorizedTridiagonal<Self>,
bl: MatrixLayout,
t: Transpose,
b: &mut [Self],
) -> Result<()> {
use tridiagonal::*;
SolveTridiagonalImpl::solve_tridiagonal(lu, bl, t, b)
}
}
};
}
impl_lapack!(c64);
impl_lapack!(c32);
impl_lapack!(f64);
impl_lapack!(f32);