nalgebra/linalg/
exp.rs

1//! This module provides the matrix exponent (exp) function to square matrices.
2//!
3use crate::{
4    ComplexField, OMatrix, RealField,
5    base::{
6        DefaultAllocator,
7        allocator::Allocator,
8        dimension::{Const, Dim, DimMin, DimMinimum},
9    },
10    convert, try_convert,
11};
12
13use crate::num::Zero;
14
15/// Precomputed factorials for integers in range `0..=34`.
16/// Note: `35!` does not fit into 128 bits.
17// TODO: find a better place for this array?
18const FACTORIAL: [u128; 35] = [
19    1,
20    1,
21    2,
22    6,
23    24,
24    120,
25    720,
26    5040,
27    40320,
28    362880,
29    3628800,
30    39916800,
31    479001600,
32    6227020800,
33    87178291200,
34    1307674368000,
35    20922789888000,
36    355687428096000,
37    6402373705728000,
38    121645100408832000,
39    2432902008176640000,
40    51090942171709440000,
41    1124000727777607680000,
42    25852016738884976640000,
43    620448401733239439360000,
44    15511210043330985984000000,
45    403291461126605635584000000,
46    10888869450418352160768000000,
47    304888344611713860501504000000,
48    8841761993739701954543616000000,
49    265252859812191058636308480000000,
50    8222838654177922817725562880000000,
51    263130836933693530167218012160000000,
52    8683317618811886495518194401280000000,
53    295232799039604140847618609643520000000,
54];
55
56// https://github.com/scipy/scipy/blob/c1372d8aa90a73d8a52f135529293ff4edb98fc8/scipy/sparse/linalg/matfuncs.py
57struct ExpmPadeHelper<T, D>
58where
59    T: ComplexField,
60    D: DimMin<D>,
61    DefaultAllocator: Allocator<D, D> + Allocator<DimMinimum<D, D>>,
62{
63    use_exact_norm: bool,
64    ident: OMatrix<T, D, D>,
65
66    a: OMatrix<T, D, D>,
67    a2: Option<OMatrix<T, D, D>>,
68    a4: Option<OMatrix<T, D, D>>,
69    a6: Option<OMatrix<T, D, D>>,
70    a8: Option<OMatrix<T, D, D>>,
71    a10: Option<OMatrix<T, D, D>>,
72
73    d4_exact: Option<T::RealField>,
74    d6_exact: Option<T::RealField>,
75    d8_exact: Option<T::RealField>,
76    d10_exact: Option<T::RealField>,
77
78    d4_approx: Option<T::RealField>,
79    d6_approx: Option<T::RealField>,
80    d8_approx: Option<T::RealField>,
81    d10_approx: Option<T::RealField>,
82}
83
84impl<T, D> ExpmPadeHelper<T, D>
85where
86    T: ComplexField,
87    D: DimMin<D>,
88    DefaultAllocator: Allocator<D, D> + Allocator<DimMinimum<D, D>>,
89{
90    fn new(a: OMatrix<T, D, D>, use_exact_norm: bool) -> Self {
91        let (nrows, ncols) = a.shape_generic();
92        ExpmPadeHelper {
93            use_exact_norm,
94            ident: OMatrix::<T, D, D>::identity_generic(nrows, ncols),
95            a,
96            a2: None,
97            a4: None,
98            a6: None,
99            a8: None,
100            a10: None,
101            d4_exact: None,
102            d6_exact: None,
103            d8_exact: None,
104            d10_exact: None,
105            d4_approx: None,
106            d6_approx: None,
107            d8_approx: None,
108            d10_approx: None,
109        }
110    }
111
112    fn calc_a2(&mut self) {
113        if self.a2.is_none() {
114            self.a2 = Some(&self.a * &self.a);
115        }
116    }
117
118    fn calc_a4(&mut self) {
119        if self.a4.is_none() {
120            self.calc_a2();
121            let a2 = self.a2.as_ref().unwrap();
122            self.a4 = Some(a2 * a2);
123        }
124    }
125
126    fn calc_a6(&mut self) {
127        if self.a6.is_none() {
128            self.calc_a2();
129            self.calc_a4();
130            let a2 = self.a2.as_ref().unwrap();
131            let a4 = self.a4.as_ref().unwrap();
132            self.a6 = Some(a4 * a2);
133        }
134    }
135
136    fn calc_a8(&mut self) {
137        if self.a8.is_none() {
138            self.calc_a2();
139            self.calc_a6();
140            let a2 = self.a2.as_ref().unwrap();
141            let a6 = self.a6.as_ref().unwrap();
142            self.a8 = Some(a6 * a2);
143        }
144    }
145
146    fn calc_a10(&mut self) {
147        if self.a10.is_none() {
148            self.calc_a4();
149            self.calc_a6();
150            let a4 = self.a4.as_ref().unwrap();
151            let a6 = self.a6.as_ref().unwrap();
152            self.a10 = Some(a6 * a4);
153        }
154    }
155
156    fn d4_tight(&mut self) -> T::RealField {
157        if self.d4_exact.is_none() {
158            self.calc_a4();
159            self.d4_exact = Some(self.a4.as_ref().unwrap().one_norm().powf(convert(0.25)));
160        }
161        self.d4_exact.clone().unwrap()
162    }
163
164    fn d6_tight(&mut self) -> T::RealField {
165        if self.d6_exact.is_none() {
166            self.calc_a6();
167            self.d6_exact = Some(
168                self.a6
169                    .as_ref()
170                    .unwrap()
171                    .one_norm()
172                    .powf(convert(1.0 / 6.0)),
173            );
174        }
175        self.d6_exact.clone().unwrap()
176    }
177
178    fn d8_tight(&mut self) -> T::RealField {
179        if self.d8_exact.is_none() {
180            self.calc_a8();
181            self.d8_exact = Some(
182                self.a8
183                    .as_ref()
184                    .unwrap()
185                    .one_norm()
186                    .powf(convert(1.0 / 8.0)),
187            );
188        }
189        self.d8_exact.clone().unwrap()
190    }
191
192    fn d10_tight(&mut self) -> T::RealField {
193        if self.d10_exact.is_none() {
194            self.calc_a10();
195            self.d10_exact = Some(
196                self.a10
197                    .as_ref()
198                    .unwrap()
199                    .one_norm()
200                    .powf(convert(1.0 / 10.0)),
201            );
202        }
203        self.d10_exact.clone().unwrap()
204    }
205
206    fn d4_loose(&mut self) -> T::RealField {
207        if self.use_exact_norm {
208            return self.d4_tight();
209        }
210
211        if self.d4_exact.is_some() {
212            return self.d4_exact.clone().unwrap();
213        }
214
215        if self.d4_approx.is_none() {
216            self.calc_a4();
217            self.d4_approx = Some(self.a4.as_ref().unwrap().one_norm().powf(convert(0.25)));
218        }
219
220        self.d4_approx.clone().unwrap()
221    }
222
223    fn d6_loose(&mut self) -> T::RealField {
224        if self.use_exact_norm {
225            return self.d6_tight();
226        }
227
228        if self.d6_exact.is_some() {
229            return self.d6_exact.clone().unwrap();
230        }
231
232        if self.d6_approx.is_none() {
233            self.calc_a6();
234            self.d6_approx = Some(
235                self.a6
236                    .as_ref()
237                    .unwrap()
238                    .one_norm()
239                    .powf(convert(1.0 / 6.0)),
240            );
241        }
242
243        self.d6_approx.clone().unwrap()
244    }
245
246    fn d8_loose(&mut self) -> T::RealField {
247        if self.use_exact_norm {
248            return self.d8_tight();
249        }
250
251        if self.d8_exact.is_some() {
252            return self.d8_exact.clone().unwrap();
253        }
254
255        if self.d8_approx.is_none() {
256            self.calc_a8();
257            self.d8_approx = Some(
258                self.a8
259                    .as_ref()
260                    .unwrap()
261                    .one_norm()
262                    .powf(convert(1.0 / 8.0)),
263            );
264        }
265
266        self.d8_approx.clone().unwrap()
267    }
268
269    fn d10_loose(&mut self) -> T::RealField {
270        if self.use_exact_norm {
271            return self.d10_tight();
272        }
273
274        if self.d10_exact.is_some() {
275            return self.d10_exact.clone().unwrap();
276        }
277
278        if self.d10_approx.is_none() {
279            self.calc_a10();
280            self.d10_approx = Some(
281                self.a10
282                    .as_ref()
283                    .unwrap()
284                    .one_norm()
285                    .powf(convert(1.0 / 10.0)),
286            );
287        }
288
289        self.d10_approx.clone().unwrap()
290    }
291
292    fn pade3(&mut self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
293        let b: [T; 4] = [convert(120.0), convert(60.0), convert(12.0), convert(1.0)];
294        self.calc_a2();
295        let a2 = self.a2.as_ref().unwrap();
296        let u = &self.a * (a2 * b[3].clone() + &self.ident * b[1].clone());
297        let v = a2 * b[2].clone() + &self.ident * b[0].clone();
298        (u, v)
299    }
300
301    fn pade5(&mut self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
302        let b: [T; 6] = [
303            convert(30240.0),
304            convert(15120.0),
305            convert(3360.0),
306            convert(420.0),
307            convert(30.0),
308            convert(1.0),
309        ];
310        self.calc_a2();
311        self.calc_a6();
312        let u = &self.a
313            * (self.a4.as_ref().unwrap() * b[5].clone()
314                + self.a2.as_ref().unwrap() * b[3].clone()
315                + &self.ident * b[1].clone());
316        let v = self.a4.as_ref().unwrap() * b[4].clone()
317            + self.a2.as_ref().unwrap() * b[2].clone()
318            + &self.ident * b[0].clone();
319        (u, v)
320    }
321
322    fn pade7(&mut self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
323        let b: [T; 8] = [
324            convert(17_297_280.0),
325            convert(8_648_640.0),
326            convert(1_995_840.0),
327            convert(277_200.0),
328            convert(25_200.0),
329            convert(1_512.0),
330            convert(56.0),
331            convert(1.0),
332        ];
333        self.calc_a2();
334        self.calc_a4();
335        self.calc_a6();
336        let u = &self.a
337            * (self.a6.as_ref().unwrap() * b[7].clone()
338                + self.a4.as_ref().unwrap() * b[5].clone()
339                + self.a2.as_ref().unwrap() * b[3].clone()
340                + &self.ident * b[1].clone());
341        let v = self.a6.as_ref().unwrap() * b[6].clone()
342            + self.a4.as_ref().unwrap() * b[4].clone()
343            + self.a2.as_ref().unwrap() * b[2].clone()
344            + &self.ident * b[0].clone();
345        (u, v)
346    }
347
348    fn pade9(&mut self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
349        let b: [T; 10] = [
350            convert(17_643_225_600.0),
351            convert(8_821_612_800.0),
352            convert(2_075_673_600.0),
353            convert(302_702_400.0),
354            convert(30_270_240.0),
355            convert(2_162_160.0),
356            convert(110_880.0),
357            convert(3_960.0),
358            convert(90.0),
359            convert(1.0),
360        ];
361        self.calc_a2();
362        self.calc_a4();
363        self.calc_a6();
364        self.calc_a8();
365        let u = &self.a
366            * (self.a8.as_ref().unwrap() * b[9].clone()
367                + self.a6.as_ref().unwrap() * b[7].clone()
368                + self.a4.as_ref().unwrap() * b[5].clone()
369                + self.a2.as_ref().unwrap() * b[3].clone()
370                + &self.ident * b[1].clone());
371        let v = self.a8.as_ref().unwrap() * b[8].clone()
372            + self.a6.as_ref().unwrap() * b[6].clone()
373            + self.a4.as_ref().unwrap() * b[4].clone()
374            + self.a2.as_ref().unwrap() * b[2].clone()
375            + &self.ident * b[0].clone();
376        (u, v)
377    }
378
379    fn pade13_scaled(&mut self, s: u64) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
380        let b: [T; 14] = [
381            convert(64_764_752_532_480_000.0),
382            convert(32_382_376_266_240_000.0),
383            convert(7_771_770_303_897_600.0),
384            convert(1_187_353_796_428_800.0),
385            convert(129_060_195_264_000.0),
386            convert(10_559_470_521_600.0),
387            convert(670_442_572_800.0),
388            convert(33_522_128_640.0),
389            convert(1_323_241_920.0),
390            convert(40_840_800.0),
391            convert(960_960.0),
392            convert(16_380.0),
393            convert(182.0),
394            convert(1.0),
395        ];
396        let s = s as f64;
397
398        let mb = &self.a * convert::<f64, T>(2.0_f64.powf(-s));
399        self.calc_a2();
400        self.calc_a4();
401        self.calc_a6();
402        let mb2 = self.a2.as_ref().unwrap() * convert::<f64, T>(2.0_f64.powf(-2.0 * s));
403        let mb4 = self.a4.as_ref().unwrap() * convert::<f64, T>(2.0.powf(-4.0 * s));
404        let mb6 = self.a6.as_ref().unwrap() * convert::<f64, T>(2.0.powf(-6.0 * s));
405
406        let u2 = &mb6 * (&mb6 * b[13].clone() + &mb4 * b[11].clone() + &mb2 * b[9].clone());
407        let u = &mb
408            * (&u2
409                + &mb6 * b[7].clone()
410                + &mb4 * b[5].clone()
411                + &mb2 * b[3].clone()
412                + &self.ident * b[1].clone());
413        let v2 = &mb6 * (&mb6 * b[12].clone() + &mb4 * b[10].clone() + &mb2 * b[8].clone());
414        let v = v2
415            + &mb6 * b[6].clone()
416            + &mb4 * b[4].clone()
417            + &mb2 * b[2].clone()
418            + &self.ident * b[0].clone();
419        (u, v)
420    }
421}
422
423/// Compute `n!`
424#[inline(always)]
425fn factorial(n: usize) -> u128 {
426    match FACTORIAL.get(n) {
427        Some(f) => *f,
428        None => panic!("{n}! is greater than u128::MAX"),
429    }
430}
431
432/// Compute the 1-norm of a non-negative integer power of a non-negative matrix.
433fn onenorm_matrix_power_nonm<T, D>(a: &OMatrix<T, D, D>, p: usize) -> T
434where
435    T: RealField,
436    D: Dim,
437    DefaultAllocator: Allocator<D, D> + Allocator<D>,
438{
439    let nrows = a.shape_generic().0;
440    let mut v = crate::OVector::<T, D>::repeat_generic(nrows, Const::<1>, convert(1.0));
441    let m = a.transpose();
442
443    for _ in 0..p {
444        v = &m * v;
445    }
446
447    v.max()
448}
449
450fn ell<T, D>(a: &OMatrix<T, D, D>, m: usize) -> u64
451where
452    T: ComplexField,
453    D: Dim,
454    DefaultAllocator: Allocator<D, D> + Allocator<D> + Allocator<D> + Allocator<D, D>,
455{
456    let a_abs = a.map(|x| x.abs());
457
458    let a_abs_onenorm = onenorm_matrix_power_nonm(&a_abs, 2 * m + 1);
459
460    if a_abs_onenorm == <T as ComplexField>::RealField::zero() {
461        return 0;
462    }
463
464    // 2m choose m = (2m)!/(m! * (2m-m)!) = (2m)!/((m!)^2)
465    let m_factorial = factorial(m);
466    let choose_2m_m = factorial(2 * m) / (m_factorial * m_factorial);
467
468    let abs_c_recip = choose_2m_m * factorial(2 * m + 1);
469    let alpha = a_abs_onenorm / a.one_norm();
470    let alpha: f64 = try_convert::<_, f64>(alpha).unwrap() / abs_c_recip as f64;
471
472    let u = 2_f64.powf(-53.0);
473    let log2_alpha_div_u = (alpha / u).log2();
474    let value = (log2_alpha_div_u / (2.0 * m as f64)).ceil();
475    if value > 0.0 { value as u64 } else { 0 }
476}
477
478fn solve_p_q<T, D>(u: OMatrix<T, D, D>, v: OMatrix<T, D, D>) -> OMatrix<T, D, D>
479where
480    T: ComplexField,
481    D: DimMin<D, Output = D>,
482    DefaultAllocator: Allocator<D, D> + Allocator<DimMinimum<D, D>>,
483{
484    let p = &u + &v;
485    let q = &v - &u;
486
487    q.lu().solve(&p).unwrap()
488}
489
490impl<T: ComplexField, D> OMatrix<T, D, D>
491where
492    D: DimMin<D, Output = D>,
493    DefaultAllocator: Allocator<D, D>
494        + Allocator<DimMinimum<D, D>>
495        + Allocator<D>
496        + Allocator<D>
497        + Allocator<D, D>,
498{
499    /// Computes exponential of this matrix
500    #[must_use]
501    pub fn exp(&self) -> Self {
502        // Simple case
503        if self.nrows() == 1 {
504            return self.map(|v| v.exp());
505        }
506
507        let mut helper = ExpmPadeHelper::new(self.clone(), true);
508
509        let eta_1 = T::RealField::max(helper.d4_loose(), helper.d6_loose());
510        if eta_1 < convert(1.495_585_217_958_292e-2) && ell(&helper.a, 3) == 0 {
511            let (u, v) = helper.pade3();
512            return solve_p_q(u, v);
513        }
514
515        let eta_2 = T::RealField::max(helper.d4_tight(), helper.d6_loose());
516        if eta_2 < convert(2.539_398_330_063_23e-1) && ell(&helper.a, 5) == 0 {
517            let (u, v) = helper.pade5();
518            return solve_p_q(u, v);
519        }
520
521        let eta_3 = T::RealField::max(helper.d6_tight(), helper.d8_loose());
522        if eta_3 < convert(9.504_178_996_162_932e-1) && ell(&helper.a, 7) == 0 {
523            let (u, v) = helper.pade7();
524            return solve_p_q(u, v);
525        }
526        if eta_3 < convert(2.097_847_961_257_068e0) && ell(&helper.a, 9) == 0 {
527            let (u, v) = helper.pade9();
528            return solve_p_q(u, v);
529        }
530
531        let eta_4 = T::RealField::max(helper.d8_loose(), helper.d10_loose());
532        let eta_5 = T::RealField::min(eta_3, eta_4);
533        let theta_13 = convert(4.25);
534
535        let mut s = if eta_5 == T::RealField::zero() {
536            0
537        } else {
538            let l2 = try_convert::<_, f64>((eta_5 / theta_13).log2().ceil()).unwrap();
539
540            if l2 < 0.0 { 0 } else { l2 as u64 }
541        };
542
543        s += ell(
544            &(&helper.a * convert::<f64, T>(2.0_f64.powf(-(s as f64)))),
545            13,
546        );
547
548        let (u, v) = helper.pade13_scaled(s);
549        let mut x = solve_p_q(u, v);
550
551        for _ in 0..s {
552            x = &x * &x;
553        }
554        x
555    }
556}
nalgebra/linalg/exp.rs

nalgebra/linalg/
exp.rs
