Merge pull request #708 from fredrik-jansson-se/expm

Addition of matrix exponent for static size matrices.
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Sébastien Crozet 2020-04-12 14:28:46 +02:00 committed by GitHub
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4 changed files with 627 additions and 1 deletions

494
src/linalg/exp.rs Normal file
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@ -0,0 +1,494 @@
//! This module provides the matrix exponent (exp) function to square matrices.
//!
use crate::{
base::{
allocator::Allocator,
dimension::{Dim, DimMin, DimMinimum, U1},
storage::Storage,
DefaultAllocator,
},
convert, try_convert, ComplexField, MatrixN, RealField,
};
// https://github.com/scipy/scipy/blob/c1372d8aa90a73d8a52f135529293ff4edb98fc8/scipy/sparse/linalg/matfuncs.py
struct ExpmPadeHelper<N, D>
where
N: RealField,
D: DimMin<D>,
DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
{
use_exact_norm: bool,
ident: MatrixN<N, D>,
a: MatrixN<N, D>,
a2: Option<MatrixN<N, D>>,
a4: Option<MatrixN<N, D>>,
a6: Option<MatrixN<N, D>>,
a8: Option<MatrixN<N, D>>,
a10: Option<MatrixN<N, D>>,
d4_exact: Option<N>,
d6_exact: Option<N>,
d8_exact: Option<N>,
d10_exact: Option<N>,
d4_approx: Option<N>,
d6_approx: Option<N>,
d8_approx: Option<N>,
d10_approx: Option<N>,
}
impl<N, D> ExpmPadeHelper<N, D>
where
N: RealField,
D: DimMin<D>,
DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
{
fn new(a: MatrixN<N, D>, use_exact_norm: bool) -> Self {
let (nrows, ncols) = a.data.shape();
ExpmPadeHelper {
use_exact_norm,
ident: MatrixN::<N, D>::identity_generic(nrows, ncols),
a,
a2: None,
a4: None,
a6: None,
a8: None,
a10: None,
d4_exact: None,
d6_exact: None,
d8_exact: None,
d10_exact: None,
d4_approx: None,
d6_approx: None,
d8_approx: None,
d10_approx: None,
}
}
fn calc_a2(&mut self) {
if self.a2.is_none() {
self.a2 = Some(&self.a * &self.a);
}
}
fn calc_a4(&mut self) {
if self.a4.is_none() {
self.calc_a2();
let a2 = self.a2.as_ref().unwrap();
self.a4 = Some(a2 * a2);
}
}
fn calc_a6(&mut self) {
if self.a6.is_none() {
self.calc_a2();
self.calc_a4();
let a2 = self.a2.as_ref().unwrap();
let a4 = self.a4.as_ref().unwrap();
self.a6 = Some(a4 * a2);
}
}
fn calc_a8(&mut self) {
if self.a8.is_none() {
self.calc_a2();
self.calc_a6();
let a2 = self.a2.as_ref().unwrap();
let a6 = self.a6.as_ref().unwrap();
self.a8 = Some(a6 * a2);
}
}
fn calc_a10(&mut self) {
if self.a10.is_none() {
self.calc_a4();
self.calc_a6();
let a4 = self.a4.as_ref().unwrap();
let a6 = self.a6.as_ref().unwrap();
self.a10 = Some(a6 * a4);
}
}
fn d4_tight(&mut self) -> N {
if self.d4_exact.is_none() {
self.calc_a4();
self.d4_exact = Some(one_norm(self.a4.as_ref().unwrap()).powf(convert(0.25)));
}
self.d4_exact.unwrap()
}
fn d6_tight(&mut self) -> N {
if self.d6_exact.is_none() {
self.calc_a6();
self.d6_exact = Some(one_norm(self.a6.as_ref().unwrap()).powf(convert(1.0 / 6.0)));
}
self.d6_exact.unwrap()
}
fn d8_tight(&mut self) -> N {
if self.d8_exact.is_none() {
self.calc_a8();
self.d8_exact = Some(one_norm(self.a8.as_ref().unwrap()).powf(convert(1.0 / 8.0)));
}
self.d8_exact.unwrap()
}
fn d10_tight(&mut self) -> N {
if self.d10_exact.is_none() {
self.calc_a10();
self.d10_exact = Some(one_norm(self.a10.as_ref().unwrap()).powf(convert(1.0 / 10.0)));
}
self.d10_exact.unwrap()
}
fn d4_loose(&mut self) -> N {
if self.use_exact_norm {
return self.d4_tight();
}
if self.d4_exact.is_some() {
return self.d4_exact.unwrap();
}
if self.d4_approx.is_none() {
self.calc_a4();
self.d4_approx = Some(one_norm(self.a4.as_ref().unwrap()).powf(convert(0.25)));
}
self.d4_approx.unwrap()
}
fn d6_loose(&mut self) -> N {
if self.use_exact_norm {
return self.d6_tight();
}
if self.d6_exact.is_some() {
return self.d6_exact.unwrap();
}
if self.d6_approx.is_none() {
self.calc_a6();
self.d6_approx = Some(one_norm(self.a6.as_ref().unwrap()).powf(convert(1.0 / 6.0)));
}
self.d6_approx.unwrap()
}
fn d8_loose(&mut self) -> N {
if self.use_exact_norm {
return self.d8_tight();
}
if self.d8_exact.is_some() {
return self.d8_exact.unwrap();
}
if self.d8_approx.is_none() {
self.calc_a8();
self.d8_approx = Some(one_norm(self.a8.as_ref().unwrap()).powf(convert(1.0 / 8.0)));
}
self.d8_approx.unwrap()
}
fn d10_loose(&mut self) -> N {
if self.use_exact_norm {
return self.d10_tight();
}
if self.d10_exact.is_some() {
return self.d10_exact.unwrap();
}
if self.d10_approx.is_none() {
self.calc_a10();
self.d10_approx = Some(one_norm(self.a10.as_ref().unwrap()).powf(convert(1.0 / 10.0)));
}
self.d10_approx.unwrap()
}
fn pade3(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
let b: [N; 4] = [convert(120.0), convert(60.0), convert(12.0), convert(1.0)];
self.calc_a2();
let a2 = self.a2.as_ref().unwrap();
let u = &self.a * (a2 * b[3] + &self.ident * b[1]);
let v = a2 * b[2] + &self.ident * b[0];
(u, v)
}
fn pade5(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
let b: [N; 6] = [
convert(30240.0),
convert(15120.0),
convert(3360.0),
convert(420.0),
convert(30.0),
convert(1.0),
];
self.calc_a2();
self.calc_a6();
let u = &self.a
* (self.a4.as_ref().unwrap() * b[5]
+ self.a2.as_ref().unwrap() * b[3]
+ &self.ident * b[1]);
let v = self.a4.as_ref().unwrap() * b[4]
+ self.a2.as_ref().unwrap() * b[2]
+ &self.ident * b[0];
(u, v)
}
fn pade7(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
let b: [N; 8] = [
convert(17297280.0),
convert(8648640.0),
convert(1995840.0),
convert(277200.0),
convert(25200.0),
convert(1512.0),
convert(56.0),
convert(1.0),
];
self.calc_a2();
self.calc_a4();
self.calc_a6();
let u = &self.a
* (self.a6.as_ref().unwrap() * b[7]
+ self.a4.as_ref().unwrap() * b[5]
+ self.a2.as_ref().unwrap() * b[3]
+ &self.ident * b[1]);
let v = self.a6.as_ref().unwrap() * b[6]
+ self.a4.as_ref().unwrap() * b[4]
+ self.a2.as_ref().unwrap() * b[2]
+ &self.ident * b[0];
(u, v)
}
fn pade9(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
let b: [N; 10] = [
convert(17643225600.0),
convert(8821612800.0),
convert(2075673600.0),
convert(302702400.0),
convert(30270240.0),
convert(2162160.0),
convert(110880.0),
convert(3960.0),
convert(90.0),
convert(1.0),
];
self.calc_a2();
self.calc_a4();
self.calc_a6();
self.calc_a8();
let u = &self.a
* (self.a8.as_ref().unwrap() * b[9]
+ self.a6.as_ref().unwrap() * b[7]
+ self.a4.as_ref().unwrap() * b[5]
+ self.a2.as_ref().unwrap() * b[3]
+ &self.ident * b[1]);
let v = self.a8.as_ref().unwrap() * b[8]
+ self.a6.as_ref().unwrap() * b[6]
+ self.a4.as_ref().unwrap() * b[4]
+ self.a2.as_ref().unwrap() * b[2]
+ &self.ident * b[0];
(u, v)
}
fn pade13_scaled(&mut self, s: u64) -> (MatrixN<N, D>, MatrixN<N, D>) {
let b: [N; 14] = [
convert(64764752532480000.0),
convert(32382376266240000.0),
convert(7771770303897600.0),
convert(1187353796428800.0),
convert(129060195264000.0),
convert(10559470521600.0),
convert(670442572800.0),
convert(33522128640.0),
convert(1323241920.0),
convert(40840800.0),
convert(960960.0),
convert(16380.0),
convert(182.0),
convert(1.0),
];
let s = s as f64;
let mb = &self.a * convert::<f64, N>(2.0_f64.powf(-s));
self.calc_a2();
self.calc_a4();
self.calc_a6();
let mb2 = self.a2.as_ref().unwrap() * convert::<f64, N>(2.0_f64.powf(-2.0 * s));
let mb4 = self.a4.as_ref().unwrap() * convert::<f64, N>(2.0.powf(-4.0 * s));
let mb6 = self.a6.as_ref().unwrap() * convert::<f64, N>(2.0.powf(-6.0 * s));
let u2 = &mb6 * (&mb6 * b[13] + &mb4 * b[11] + &mb2 * b[9]);
let u = &mb * (&u2 + &mb6 * b[7] + &mb4 * b[5] + &mb2 * b[3] + &self.ident * b[1]);
let v2 = &mb6 * (&mb6 * b[12] + &mb4 * b[10] + &mb2 * b[8]);
let v = v2 + &mb6 * b[6] + &mb4 * b[4] + &mb2 * b[2] + &self.ident * b[0];
(u, v)
}
}
fn factorial(n: u128) -> u128 {
if n == 1 {
return 1;
}
n * factorial(n - 1)
}
/// Compute the 1-norm of a non-negative integer power of a non-negative matrix.
fn onenorm_matrix_power_nonm<N, D>(a: &MatrixN<N, D>, p: u64) -> N
where
N: RealField,
D: Dim,
DefaultAllocator: Allocator<N, D, D> + Allocator<N, D>,
{
let nrows = a.data.shape().0;
let mut v = crate::VectorN::<N, D>::repeat_generic(nrows, U1, convert(1.0));
let m = a.transpose();
for _ in 0..p {
v = &m * v;
}
v.max()
}
fn ell<N, D>(a: &MatrixN<N, D>, m: u64) -> u64
where
N: RealField,
D: Dim,
DefaultAllocator: Allocator<N, D, D> + Allocator<N, D>,
{
// 2m choose m = (2m)!/(m! * (2m-m)!)
let a_abs_onenorm = onenorm_matrix_power_nonm(&a.abs(), 2 * m + 1);
if a_abs_onenorm == N::zero() {
return 0;
}
let choose_2m_m =
factorial(2 * m as u128) / (factorial(m as u128) * factorial(2 * m as u128 - m as u128));
let abs_c_recip = choose_2m_m * factorial(2 * m as u128 + 1);
let alpha = a_abs_onenorm / one_norm(a);
let alpha: f64 = try_convert(alpha).unwrap() / abs_c_recip as f64;
let u = 2_f64.powf(-53.0);
let log2_alpha_div_u = (alpha / u).log2();
let value = (log2_alpha_div_u / (2.0 * m as f64)).ceil();
if value > 0.0 {
value as u64
} else {
0
}
}
fn solve_p_q<N, D>(u: MatrixN<N, D>, v: MatrixN<N, D>) -> MatrixN<N, D>
where
N: ComplexField,
D: DimMin<D, Output = D>,
DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
{
let p = &u + &v;
let q = &v - &u;
q.lu().solve(&p).unwrap()
}
fn one_norm<N, D>(m: &MatrixN<N, D>) -> N
where
N: RealField,
D: Dim,
DefaultAllocator: Allocator<N, D, D>,
{
let mut col_sums = vec![N::zero(); m.ncols()];
for i in 0..m.ncols() {
let col = m.column(i);
col.iter().for_each(|v| col_sums[i] += v.abs());
}
let mut max = col_sums[0];
for i in 1..col_sums.len() {
max = N::max(max, col_sums[i]);
}
max
}
impl<N: RealField, D> MatrixN<N, D>
where
D: DimMin<D, Output = D>,
DefaultAllocator:
Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>> + Allocator<N, D>,
{
/// Computes exponential of this matrix
pub fn exp(&self) -> Self {
// Simple case
if self.nrows() == 1 {
return self.clone().map(|v| v.exp());
}
let mut h = ExpmPadeHelper::new(self.clone(), true);
let eta_1 = N::max(h.d4_loose(), h.d6_loose());
if eta_1 < convert(1.495585217958292e-002) && ell(&h.a, 3) == 0 {
let (u, v) = h.pade3();
return solve_p_q(u, v);
}
let eta_2 = N::max(h.d4_tight(), h.d6_loose());
if eta_2 < convert(2.539398330063230e-001) && ell(&h.a, 5) == 0 {
let (u, v) = h.pade5();
return solve_p_q(u, v);
}
let eta_3 = N::max(h.d6_tight(), h.d8_loose());
if eta_3 < convert(9.504178996162932e-001) && ell(&h.a, 7) == 0 {
let (u, v) = h.pade7();
return solve_p_q(u, v);
}
if eta_3 < convert(2.097847961257068e+000) && ell(&h.a, 9) == 0 {
let (u, v) = h.pade9();
return solve_p_q(u, v);
}
let eta_4 = N::max(h.d8_loose(), h.d10_loose());
let eta_5 = N::min(eta_3, eta_4);
let theta_13 = convert(4.25);
let mut s = if eta_5 == N::zero() {
0
} else {
let l2 = try_convert((eta_5 / theta_13).log2().ceil()).unwrap();
if l2 < 0.0 {
0
} else {
l2 as u64
}
};
s += ell(&(&h.a * convert::<f64, N>(2.0_f64.powf(-(s as f64)))), 13);
let (u, v) = h.pade13_scaled(s);
let mut x = solve_p_q(u, v);
for _ in 0..s {
x = &x * &x;
}
x
}
}
#[cfg(test)]
mod tests {
#[test]
fn one_norm() {
use crate::Matrix3;
let m = Matrix3::new(-3.0, 5.0, 7.0, 2.0, 6.0, 4.0, 0.0, 2.0, 8.0);
assert_eq!(super::one_norm(&m), 19.0);
}
}

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@ -5,6 +5,7 @@ mod bidiagonal;
mod cholesky;
mod convolution;
mod determinant;
mod exp;
mod full_piv_lu;
pub mod givens;
mod hessenberg;
@ -26,6 +27,7 @@ mod symmetric_tridiagonal;
pub use self::bidiagonal::*;
pub use self::cholesky::*;
pub use self::convolution::*;
pub use self::exp::*;
pub use self::full_piv_lu::*;
pub use self::hessenberg::*;
pub use self::lu::*;

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tests/linalg/exp.rs Normal file
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@ -0,0 +1,129 @@
#[cfg(test)]
mod tests {
//https://github.com/scipy/scipy/blob/c1372d8aa90a73d8a52f135529293ff4edb98fc8/scipy/sparse/linalg/tests/test_matfuncs.py
#[test]
fn exp_static() {
use nalgebra::{Matrix1, Matrix2, Matrix3};
{
let m = Matrix1::new(1.0);
let f = m.exp();
assert!(relative_eq!(f, Matrix1::new(1_f64.exp()), epsilon = 1.0e-7));
}
{
let m = Matrix2::new(0.0, 1.0, 0.0, 0.0);
assert!(relative_eq!(
m.exp(),
Matrix2::new(1.0, 1.0, 0.0, 1.0),
epsilon = 1.0e-7
));
}
{
let a: f64 = 1.0;
let b: f64 = 2.0;
let c: f64 = 3.0;
let d: f64 = 4.0;
let m = Matrix2::new(a, b, c, d);
let delta = ((a - d).powf(2.0) + 4.0 * b * c).sqrt();
let delta_2 = delta / 2.0;
let ad_2 = (a + d) / 2.0;
let m11 = ad_2.exp() * (delta * delta_2.cosh() + (a - d) * delta_2.sinh());
let m12 = 2.0 * b * ad_2.exp() * delta_2.sinh();
let m21 = 2.0 * c * ad_2.exp() * delta_2.sinh();
let m22 = ad_2.exp() * (delta * delta_2.cosh() + (d - a) * delta_2.sinh());
let f = Matrix2::new(m11, m12, m21, m22) / delta;
assert!(relative_eq!(f, m.exp(), epsilon = 1.0e-7));
}
{
// https://mathworld.wolfram.com/MatrixExponential.html
use rand::{
distributions::{Distribution, Uniform},
thread_rng,
};
let mut rng = thread_rng();
let dist = Uniform::new(-10.0, 10.0);
loop {
let a: f64 = dist.sample(&mut rng);
let b: f64 = dist.sample(&mut rng);
let c: f64 = dist.sample(&mut rng);
let d: f64 = dist.sample(&mut rng);
let m = Matrix2::new(a, b, c, d);
let delta_sq = (a - d).powf(2.0) + 4.0 * b * c;
if delta_sq < 0.0 {
continue;
}
let delta = delta_sq.sqrt();
let delta_2 = delta / 2.0;
let ad_2 = (a + d) / 2.0;
let m11 = ad_2.exp() * (delta * delta_2.cosh() + (a - d) * delta_2.sinh());
let m12 = 2.0 * b * ad_2.exp() * delta_2.sinh();
let m21 = 2.0 * c * ad_2.exp() * delta_2.sinh();
let m22 = ad_2.exp() * (delta * delta_2.cosh() + (d - a) * delta_2.sinh());
let f = Matrix2::new(m11, m12, m21, m22) / delta;
println!("a: {}", m);
assert!(relative_eq!(f, m.exp(), epsilon = 1.0e-7));
break;
}
}
{
let m = Matrix3::new(1.0, 3.0, 0.0, 0.0, 1.0, 5.0, 0.0, 0.0, 2.0);
let e1 = 1.0_f64.exp();
let e2 = 2.0_f64.exp();
let f = Matrix3::new(
e1,
3.0 * e1,
15.0 * (e2 - 2.0 * e1),
0.0,
e1,
5.0 * (e2 - e1),
0.0,
0.0,
e2,
);
assert!(relative_eq!(f, m.exp(), epsilon = 1.0e-7));
}
}
#[test]
fn exp_dynamic() {
use nalgebra::DMatrix;
let m = DMatrix::from_row_slice(3, 3, &[1.0, 3.0, 0.0, 0.0, 1.0, 5.0, 0.0, 0.0, 2.0]);
let e1 = 1.0_f64.exp();
let e2 = 2.0_f64.exp();
let f = DMatrix::from_row_slice(
3,
3,
&[
e1,
3.0 * e1,
15.0 * (e2 - 2.0 * e1),
0.0,
e1,
5.0 * (e2 - e1),
0.0,
0.0,
e2,
],
);
assert!(relative_eq!(f, m.exp(), epsilon = 1.0e-7));
}
}

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@ -1,7 +1,9 @@
mod balancing;
mod bidiagonal;
mod cholesky;
mod convolution;
mod eigen;
mod exp;
mod full_piv_lu;
mod hessenberg;
mod inverse;
@ -10,5 +12,4 @@ mod qr;
mod schur;
mod solve;
mod svd;
mod convolution;
mod tridiagonal;