f9f7169558
Added matrix exponential for complex matrices.
506 lines
14 KiB
Rust
506 lines
14 KiB
Rust
//! This module provides the matrix exponent (exp) function to square matrices.
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//!
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use crate::{
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base::{
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allocator::Allocator,
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dimension::{Dim, DimMin, DimMinimum, U1},
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storage::Storage,
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DefaultAllocator,
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},
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convert, try_convert, ComplexField, MatrixN, RealField,
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};
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use crate::num::Zero;
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// https://github.com/scipy/scipy/blob/c1372d8aa90a73d8a52f135529293ff4edb98fc8/scipy/sparse/linalg/matfuncs.py
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struct ExpmPadeHelper<N, D>
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where
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N: ComplexField,
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D: DimMin<D>,
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DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
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{
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use_exact_norm: bool,
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ident: MatrixN<N, D>,
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a: MatrixN<N, D>,
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a2: Option<MatrixN<N, D>>,
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a4: Option<MatrixN<N, D>>,
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a6: Option<MatrixN<N, D>>,
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a8: Option<MatrixN<N, D>>,
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a10: Option<MatrixN<N, D>>,
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d4_exact: Option<N::RealField>,
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d6_exact: Option<N::RealField>,
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d8_exact: Option<N::RealField>,
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d10_exact: Option<N::RealField>,
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d4_approx: Option<N::RealField>,
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d6_approx: Option<N::RealField>,
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d8_approx: Option<N::RealField>,
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d10_approx: Option<N::RealField>,
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}
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impl<N, D> ExpmPadeHelper<N, D>
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where
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N: ComplexField,
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D: DimMin<D>,
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DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
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{
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fn new(a: MatrixN<N, D>, use_exact_norm: bool) -> Self {
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let (nrows, ncols) = a.data.shape();
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ExpmPadeHelper {
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use_exact_norm,
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ident: MatrixN::<N, D>::identity_generic(nrows, ncols),
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a,
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a2: None,
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a4: None,
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a6: None,
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a8: None,
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a10: None,
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d4_exact: None,
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d6_exact: None,
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d8_exact: None,
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d10_exact: None,
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d4_approx: None,
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d6_approx: None,
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d8_approx: None,
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d10_approx: None,
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}
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}
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fn calc_a2(&mut self) {
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if self.a2.is_none() {
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self.a2 = Some(&self.a * &self.a);
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}
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}
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fn calc_a4(&mut self) {
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if self.a4.is_none() {
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self.calc_a2();
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let a2 = self.a2.as_ref().unwrap();
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self.a4 = Some(a2 * a2);
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}
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}
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fn calc_a6(&mut self) {
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if self.a6.is_none() {
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self.calc_a2();
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self.calc_a4();
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let a2 = self.a2.as_ref().unwrap();
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let a4 = self.a4.as_ref().unwrap();
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self.a6 = Some(a4 * a2);
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}
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}
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fn calc_a8(&mut self) {
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if self.a8.is_none() {
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self.calc_a2();
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self.calc_a6();
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let a2 = self.a2.as_ref().unwrap();
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let a6 = self.a6.as_ref().unwrap();
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self.a8 = Some(a6 * a2);
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}
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}
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fn calc_a10(&mut self) {
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if self.a10.is_none() {
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self.calc_a4();
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self.calc_a6();
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let a4 = self.a4.as_ref().unwrap();
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let a6 = self.a6.as_ref().unwrap();
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self.a10 = Some(a6 * a4);
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}
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}
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fn d4_tight(&mut self) -> N::RealField {
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if self.d4_exact.is_none() {
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self.calc_a4();
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self.d4_exact = Some(one_norm(self.a4.as_ref().unwrap()).powf(convert(0.25)));
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}
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self.d4_exact.unwrap()
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}
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fn d6_tight(&mut self) -> N::RealField {
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if self.d6_exact.is_none() {
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self.calc_a6();
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self.d6_exact = Some(one_norm(self.a6.as_ref().unwrap()).powf(convert(1.0 / 6.0)));
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}
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self.d6_exact.unwrap()
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}
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fn d8_tight(&mut self) -> N::RealField {
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if self.d8_exact.is_none() {
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self.calc_a8();
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self.d8_exact = Some(one_norm(self.a8.as_ref().unwrap()).powf(convert(1.0 / 8.0)));
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}
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self.d8_exact.unwrap()
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}
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fn d10_tight(&mut self) -> N::RealField {
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if self.d10_exact.is_none() {
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self.calc_a10();
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self.d10_exact = Some(one_norm(self.a10.as_ref().unwrap()).powf(convert(1.0 / 10.0)));
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}
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self.d10_exact.unwrap()
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}
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fn d4_loose(&mut self) -> N::RealField {
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if self.use_exact_norm {
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return self.d4_tight();
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}
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if self.d4_exact.is_some() {
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return self.d4_exact.unwrap();
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}
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if self.d4_approx.is_none() {
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self.calc_a4();
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self.d4_approx = Some(one_norm(self.a4.as_ref().unwrap()).powf(convert(0.25)));
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}
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self.d4_approx.unwrap()
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}
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fn d6_loose(&mut self) -> N::RealField {
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if self.use_exact_norm {
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return self.d6_tight();
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}
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if self.d6_exact.is_some() {
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return self.d6_exact.unwrap();
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}
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if self.d6_approx.is_none() {
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self.calc_a6();
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self.d6_approx = Some(one_norm(self.a6.as_ref().unwrap()).powf(convert(1.0 / 6.0)));
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}
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self.d6_approx.unwrap()
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}
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fn d8_loose(&mut self) -> N::RealField {
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if self.use_exact_norm {
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return self.d8_tight();
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}
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if self.d8_exact.is_some() {
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return self.d8_exact.unwrap();
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}
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if self.d8_approx.is_none() {
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self.calc_a8();
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self.d8_approx = Some(one_norm(self.a8.as_ref().unwrap()).powf(convert(1.0 / 8.0)));
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}
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self.d8_approx.unwrap()
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}
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fn d10_loose(&mut self) -> N::RealField {
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if self.use_exact_norm {
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return self.d10_tight();
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}
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if self.d10_exact.is_some() {
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return self.d10_exact.unwrap();
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}
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if self.d10_approx.is_none() {
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self.calc_a10();
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self.d10_approx = Some(one_norm(self.a10.as_ref().unwrap()).powf(convert(1.0 / 10.0)));
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}
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self.d10_approx.unwrap()
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}
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fn pade3(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
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let b: [N; 4] = [convert(120.0), convert(60.0), convert(12.0), convert(1.0)];
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self.calc_a2();
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let a2 = self.a2.as_ref().unwrap();
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let u = &self.a * (a2 * b[3] + &self.ident * b[1]);
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let v = a2 * b[2] + &self.ident * b[0];
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(u, v)
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}
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fn pade5(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
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let b: [N; 6] = [
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convert(30240.0),
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convert(15120.0),
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convert(3360.0),
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convert(420.0),
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convert(30.0),
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convert(1.0),
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];
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self.calc_a2();
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self.calc_a6();
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let u = &self.a
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* (self.a4.as_ref().unwrap() * b[5]
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+ self.a2.as_ref().unwrap() * b[3]
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+ &self.ident * b[1]);
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let v = self.a4.as_ref().unwrap() * b[4]
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+ self.a2.as_ref().unwrap() * b[2]
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+ &self.ident * b[0];
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(u, v)
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}
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fn pade7(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
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let b: [N; 8] = [
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convert(17297280.0),
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convert(8648640.0),
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convert(1995840.0),
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convert(277200.0),
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convert(25200.0),
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convert(1512.0),
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convert(56.0),
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convert(1.0),
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];
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self.calc_a2();
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self.calc_a4();
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self.calc_a6();
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let u = &self.a
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* (self.a6.as_ref().unwrap() * b[7]
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+ self.a4.as_ref().unwrap() * b[5]
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+ self.a2.as_ref().unwrap() * b[3]
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+ &self.ident * b[1]);
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let v = self.a6.as_ref().unwrap() * b[6]
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+ self.a4.as_ref().unwrap() * b[4]
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+ self.a2.as_ref().unwrap() * b[2]
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+ &self.ident * b[0];
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(u, v)
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}
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fn pade9(&mut self) -> (MatrixN<N, D>, MatrixN<N, D>) {
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let b: [N; 10] = [
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convert(17643225600.0),
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convert(8821612800.0),
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convert(2075673600.0),
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convert(302702400.0),
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convert(30270240.0),
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convert(2162160.0),
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convert(110880.0),
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convert(3960.0),
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convert(90.0),
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convert(1.0),
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];
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self.calc_a2();
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self.calc_a4();
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self.calc_a6();
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self.calc_a8();
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let u = &self.a
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* (self.a8.as_ref().unwrap() * b[9]
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+ self.a6.as_ref().unwrap() * b[7]
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+ self.a4.as_ref().unwrap() * b[5]
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+ self.a2.as_ref().unwrap() * b[3]
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+ &self.ident * b[1]);
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let v = self.a8.as_ref().unwrap() * b[8]
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+ self.a6.as_ref().unwrap() * b[6]
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+ self.a4.as_ref().unwrap() * b[4]
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+ self.a2.as_ref().unwrap() * b[2]
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+ &self.ident * b[0];
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(u, v)
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}
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fn pade13_scaled(&mut self, s: u64) -> (MatrixN<N, D>, MatrixN<N, D>) {
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let b: [N; 14] = [
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convert(64764752532480000.0),
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convert(32382376266240000.0),
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convert(7771770303897600.0),
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convert(1187353796428800.0),
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convert(129060195264000.0),
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convert(10559470521600.0),
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convert(670442572800.0),
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convert(33522128640.0),
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convert(1323241920.0),
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convert(40840800.0),
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convert(960960.0),
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convert(16380.0),
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convert(182.0),
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convert(1.0),
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];
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let s = s as f64;
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let mb = &self.a * convert::<f64, N>(2.0_f64.powf(-s));
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self.calc_a2();
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self.calc_a4();
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self.calc_a6();
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let mb2 = self.a2.as_ref().unwrap() * convert::<f64, N>(2.0_f64.powf(-2.0 * s));
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let mb4 = self.a4.as_ref().unwrap() * convert::<f64, N>(2.0.powf(-4.0 * s));
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let mb6 = self.a6.as_ref().unwrap() * convert::<f64, N>(2.0.powf(-6.0 * s));
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let u2 = &mb6 * (&mb6 * b[13] + &mb4 * b[11] + &mb2 * b[9]);
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let u = &mb * (&u2 + &mb6 * b[7] + &mb4 * b[5] + &mb2 * b[3] + &self.ident * b[1]);
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let v2 = &mb6 * (&mb6 * b[12] + &mb4 * b[10] + &mb2 * b[8]);
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let v = v2 + &mb6 * b[6] + &mb4 * b[4] + &mb2 * b[2] + &self.ident * b[0];
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(u, v)
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}
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}
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fn factorial(n: u128) -> u128 {
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if n == 1 {
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return 1;
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}
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n * factorial(n - 1)
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}
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/// Compute the 1-norm of a non-negative integer power of a non-negative matrix.
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fn onenorm_matrix_power_nonm<N, D>(a: &MatrixN<N, D>, p: u64) -> N
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where
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N: RealField,
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D: Dim,
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DefaultAllocator: Allocator<N, D, D> + Allocator<N, D>,
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{
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let nrows = a.data.shape().0;
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let mut v = crate::VectorN::<N, D>::repeat_generic(nrows, U1, convert(1.0));
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let m = a.transpose();
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for _ in 0..p {
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v = &m * v;
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}
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v.max()
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}
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fn ell<N, D>(a: &MatrixN<N, D>, m: u64) -> u64
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where
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N: ComplexField,
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D: Dim,
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DefaultAllocator: Allocator<N, D, D>
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+ Allocator<N, D>
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+ Allocator<N::RealField, D>
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+ Allocator<N::RealField, D, D>,
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{
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// 2m choose m = (2m)!/(m! * (2m-m)!)
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let a_abs = a.map(|x| x.abs());
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let a_abs_onenorm = onenorm_matrix_power_nonm(&a_abs, 2 * m + 1);
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if a_abs_onenorm == <N as ComplexField>::RealField::zero() {
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return 0;
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}
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let choose_2m_m =
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factorial(2 * m as u128) / (factorial(m as u128) * factorial(2 * m as u128 - m as u128));
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let abs_c_recip = choose_2m_m * factorial(2 * m as u128 + 1);
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let alpha = a_abs_onenorm / one_norm(a);
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let alpha: f64 = try_convert(alpha).unwrap() / abs_c_recip as f64;
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let u = 2_f64.powf(-53.0);
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let log2_alpha_div_u = (alpha / u).log2();
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let value = (log2_alpha_div_u / (2.0 * m as f64)).ceil();
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if value > 0.0 {
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value as u64
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} else {
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0
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}
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}
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fn solve_p_q<N, D>(u: MatrixN<N, D>, v: MatrixN<N, D>) -> MatrixN<N, D>
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where
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N: ComplexField,
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D: DimMin<D, Output = D>,
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DefaultAllocator: Allocator<N, D, D> + Allocator<(usize, usize), DimMinimum<D, D>>,
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{
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let p = &u + &v;
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let q = &v - &u;
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q.lu().solve(&p).unwrap()
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}
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fn one_norm<N, D>(m: &MatrixN<N, D>) -> N::RealField
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where
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N: ComplexField,
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D: Dim,
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DefaultAllocator: Allocator<N, D, D>,
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{
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let mut max = <N as ComplexField>::RealField::zero();
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for i in 0..m.ncols() {
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let col = m.column(i);
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max = max.max(
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col.iter()
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.fold(<N as ComplexField>::RealField::zero(), |a, b| a + b.abs()),
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);
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}
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max
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}
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impl<N: ComplexField, D> MatrixN<N, D>
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where
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D: DimMin<D, Output = D>,
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DefaultAllocator: Allocator<N, D, D>
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+ Allocator<(usize, usize), DimMinimum<D, D>>
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+ Allocator<N, D>
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+ Allocator<N::RealField, D>
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+ Allocator<N::RealField, D, D>,
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{
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/// Computes exponential of this matrix
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pub fn exp(&self) -> Self {
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// Simple case
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if self.nrows() == 1 {
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return self.map(|v| v.exp());
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}
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let mut h = ExpmPadeHelper::new(self.clone(), true);
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let eta_1 = N::RealField::max(h.d4_loose(), h.d6_loose());
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if eta_1 < convert(1.495585217958292e-002) && ell(&h.a, 3) == 0 {
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let (u, v) = h.pade3();
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return solve_p_q(u, v);
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}
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let eta_2 = N::RealField::max(h.d4_tight(), h.d6_loose());
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if eta_2 < convert(2.539398330063230e-001) && ell(&h.a, 5) == 0 {
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let (u, v) = h.pade5();
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return solve_p_q(u, v);
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}
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let eta_3 = N::RealField::max(h.d6_tight(), h.d8_loose());
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if eta_3 < convert(9.504178996162932e-001) && ell(&h.a, 7) == 0 {
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let (u, v) = h.pade7();
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return solve_p_q(u, v);
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}
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if eta_3 < convert(2.097847961257068e+000) && ell(&h.a, 9) == 0 {
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let (u, v) = h.pade9();
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return solve_p_q(u, v);
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}
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let eta_4 = N::RealField::max(h.d8_loose(), h.d10_loose());
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let eta_5 = N::RealField::min(eta_3, eta_4);
|
|
let theta_13 = convert(4.25);
|
|
|
|
let mut s = if eta_5 == N::RealField::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);
|
|
}
|
|
}
|