From ad8250c3613868816364bca5bf63a6ba0813e4c9 Mon Sep 17 00:00:00 2001
From: Max Verevkin <me@maxverevkin.tk>
Date: Mon, 20 Sep 2021 22:32:48 +0300
Subject: [PATCH] exp.rs: factorial(): use precomputed factorial array

---
 src/linalg/exp.rs | 69 ++++++++++++++++++++++++++++++++++++++---------
 1 file changed, 56 insertions(+), 13 deletions(-)

diff --git a/src/linalg/exp.rs b/src/linalg/exp.rs
index 074e801c..d6f61956 100644
--- a/src/linalg/exp.rs
+++ b/src/linalg/exp.rs
@@ -11,6 +11,47 @@ use crate::{
 
 use crate::num::Zero;
 
+/// Precomputed factorials for integers in range `0..=34`.
+/// Note: `35!` does not fit into 128 bits.
+// TODO: find a better place for this array?
+const FACTORIAL: [u128; 35] = [
+    1,
+    1,
+    2,
+    6,
+    24,
+    120,
+    720,
+    5040,
+    40320,
+    362880,
+    3628800,
+    39916800,
+    479001600,
+    6227020800,
+    87178291200,
+    1307674368000,
+    20922789888000,
+    355687428096000,
+    6402373705728000,
+    121645100408832000,
+    2432902008176640000,
+    51090942171709440000,
+    1124000727777607680000,
+    25852016738884976640000,
+    620448401733239439360000,
+    15511210043330985984000000,
+    403291461126605635584000000,
+    10888869450418352160768000000,
+    304888344611713860501504000000,
+    8841761993739701954543616000000,
+    265252859812191058636308480000000,
+    8222838654177922817725562880000000,
+    263130836933693530167218012160000000,
+    8683317618811886495518194401280000000,
+    295232799039604140847618609643520000000,
+];
+
 // https://github.com/scipy/scipy/blob/c1372d8aa90a73d8a52f135529293ff4edb98fc8/scipy/sparse/linalg/matfuncs.py
 struct ExpmPadeHelper<T, D>
 where
@@ -321,8 +362,8 @@ where
         self.calc_a2();
         self.calc_a4();
         self.calc_a6();
-        let mb2 = self.a2.as_ref().unwrap() * convert::<f64, T>(2.0_f64.powf(-2.0 * s.clone()));
-        let mb4 = self.a4.as_ref().unwrap() * convert::<f64, T>(2.0.powf(-4.0 * s.clone()));
+        let mb2 = self.a2.as_ref().unwrap() * convert::<f64, T>(2.0_f64.powf(-2.0 * s));
+        let mb4 = self.a4.as_ref().unwrap() * convert::<f64, T>(2.0.powf(-4.0 * s));
         let mb6 = self.a6.as_ref().unwrap() * convert::<f64, T>(2.0.powf(-6.0 * s));
 
         let u2 = &mb6 * (&mb6 * b[13].clone() + &mb4 * b[11].clone() + &mb2 * b[9].clone());
@@ -342,15 +383,17 @@ where
     }
 }
 
-fn factorial(n: u128) -> u128 {
-    if n == 1 {
-        return 1;
+/// Compute `n!`
+#[inline(always)]
+fn factorial(n: usize) -> u128 {
+    match FACTORIAL.get(n) {
+        Some(f) => *f,
+        None => panic!("{}! is greater than u128::MAX", n),
     }
-    n * factorial(n - 1)
 }
 
 /// Compute the 1-norm of a non-negative integer power of a non-negative matrix.
-fn onenorm_matrix_power_nonm<T, D>(a: &OMatrix<T, D, D>, p: u64) -> T
+fn onenorm_matrix_power_nonm<T, D>(a: &OMatrix<T, D, D>, p: usize) -> T
 where
     T: RealField,
     D: Dim,
@@ -367,7 +410,7 @@ where
     v.max()
 }
 
-fn ell<T, D>(a: &OMatrix<T, D, D>, m: u64) -> u64
+fn ell<T, D>(a: &OMatrix<T, D, D>, m: usize) -> u64
 where
     T: ComplexField,
     D: Dim,
@@ -376,8 +419,6 @@ where
         + Allocator<T::RealField, D>
         + Allocator<T::RealField, D, D>,
 {
-    // 2m choose m = (2m)!/(m! * (2m-m)!)
-
     let a_abs = a.map(|x| x.abs());
 
     let a_abs_onenorm = onenorm_matrix_power_nonm(&a_abs, 2 * m + 1);
@@ -386,9 +427,11 @@ where
         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);
+    // 2m choose m = (2m)!/(m! * (2m-m)!) = (2m)!/((m!)^2)
+    let m_factorial = factorial(m);
+    let choose_2m_m = factorial(2 * m) / (m_factorial * m_factorial);
+
+    let abs_c_recip = choose_2m_m * factorial(2 * m + 1);
     let alpha = a_abs_onenorm / one_norm(a);
     let alpha: f64 = try_convert(alpha).unwrap() / abs_c_recip as f64;