la-stack 0.4.6

Fast, stack-allocated linear algebra for fixed dimensions
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213

#![forbid(unsafe_code)]

//! Independent integer matrix-product oracles for Gram construction.

use core::array::from_fn;

use approx::assert_abs_diff_eq;
use pastey::paste;
use proptest::prelude::*;

use la_stack::prelude::*;

#[path = "common/proptest_config.rs"]
mod proptest_config;
use proptest_config::with_default_cases;

fn check_product<const M: usize, const N: usize>(entries: &[i16]) {
    let integers: [[i16; N]; M] = from_fn(|i| from_fn(|k| entries[i * N + k]));
    let vectors = integers.map(|row| Vector::try_new(row.map(f64::from)).unwrap());
    let result = gram_matrix(&vectors).unwrap();
    for i in 0..M {
        for j in 0..M {
            // Separate integer multiplication and summation, not the FMA kernel.
            let expected: i32 = (0..N)
                .map(|k| i32::from(integers[i][k]) * i32::from(integers[j][k]))
                .sum();
            assert_abs_diff_eq!(result.as_rows()[i][j], f64::from(expected), epsilon = 0.0);
            assert_eq!(
                result.as_rows()[i][j].to_bits(),
                result.as_rows()[j][i].to_bits()
            );
        }
    }
}

fn check_orthogonal_and_dependent<const D: usize>() {
    let vectors: [Vector<D>; D] =
        from_fn(|i| Vector::try_new(from_fn(|j| if i == j { 2.0 } else { 0.0 })).unwrap());
    let gram = gram_matrix(&vectors).unwrap();
    for i in 0..D {
        for j in 0..D {
            assert_abs_diff_eq!(
                gram.as_rows()[i][j],
                if i == j { 4.0 } else { 0.0 },
                epsilon = 0.0
            );
        }
    }
    let expected = (0..D).fold(1.0, |product, _| product * 4.0);
    assert_abs_diff_eq!(
        gram.ldlt(Tolerance::try_new(0.0).unwrap())
            .unwrap()
            .det()
            .unwrap(),
        expected,
        epsilon = 0.0
    );
    let dependent = [vectors[0]; D];
    let dependent_gram = gram_matrix(&dependent).unwrap();
    assert_eq!(dependent_gram.as_rows(), &[[4.0; D]; D]);
    assert_eq!(
        dependent_gram
            .ldlt(Tolerance::try_new(0.0).unwrap())
            .unwrap_err(),
        LaError::singular_numerical(1, FactorizationKind::Ldlt, 0.0, 0.0)
    );
}

// A separately rounded product or reversed reduction loses the exact residual.
fn check_fused_reduction<const D: usize>() {
    let delta = 2.0_f64.powi(-27);
    let left: Vector<D> = Vector::try_new(from_fn(|k| match k {
        0 => 1.0,
        1 => 1.0 + delta,
        _ => 0.0,
    }))
    .unwrap();
    let right = Vector::try_new(from_fn(|k| match k {
        0 => -1.0,
        1 => 1.0 - delta,
        _ => 0.0,
    }))
    .unwrap();
    let gram = gram_matrix(&[left, right]).unwrap();
    // -1 + (1 + 2^-27)(1 - 2^-27) = -2^-54, exactly representable.
    let expected = (-2.0_f64.powi(-54)).to_bits();
    assert_eq!(gram.as_rows()[0][1].to_bits(), expected);
    assert_eq!(gram.as_rows()[1][0].to_bits(), expected);
}

macro_rules! gram_cases {
    ($d:literal) => {
        paste! {
            proptest! {
                #![proptest_config(with_default_cases(64))]
                #[test]
                fn [<gram_product_ $d d>](entries in prop::collection::vec(-100i16..=100, 64)) {
                    check_product::<$d, $d>(&entries);
                    if $d != 8 {
                        check_product::<$d, 8>(&entries);
                        check_product::<8, $d>(&entries);
                    }
                }
            }

            #[test]
            fn [<gram_orthogonal_and_dependent_ $d d>]() {
                check_orthogonal_and_dependent::<$d>();
            }

            #[test]
            fn [<gram_fused_reduction_ $d d>]() {
                check_fused_reduction::<$d>();
            }
        }
    };
}

gram_cases!(2);
gram_cases!(3);
gram_cases!(4);
gram_cases!(5);
gram_cases!(8);

#[test]
fn empty_dimensions_and_const_evaluation() {
    const EMPTY: Result<Matrix<0>, LaError> = gram_matrix::<0, 8>(&[]);
    const ZERO: Result<Matrix<3>, LaError> = gram_matrix(&[Vector::<0>::zero(); 3]);
    assert_eq!(EMPTY.unwrap().as_rows(), &[] as &[[f64; 0]; 0]);
    assert_eq!(ZERO.unwrap().as_rows(), &[[0.0; 3]; 3]);
    assert_eq!(
        gram_matrix::<0, 0>(&[]).unwrap().as_rows(),
        &[] as &[[f64; 0]; 0]
    );
}

#[test]
fn nonempty_const_evaluation() {
    const VECTOR: Vector<2> = match Vector::try_new([3.0, 4.0]) {
        Ok(vector) => vector,
        Err(_) => panic!("finite literal input"),
    };
    const GRAM: Result<Matrix<1>, LaError> = gram_matrix(&[VECTOR]);
    assert_eq!(GRAM.unwrap().as_rows(), &[[25.0]]);
}

#[test]
fn underflow_preserves_mirrored_negative_zero() {
    let vectors = [
        Vector::try_new([2.0_f64.powi(-537)]).unwrap(),
        Vector::try_new([-2.0_f64.powi(-538)]).unwrap(),
    ];
    let gram = gram_matrix(&vectors).unwrap();
    assert_eq!(gram.as_rows()[0][0].to_bits(), 1); // Smallest positive subnormal.
    assert_eq!(gram.as_rows()[0][1].to_bits(), (-0.0_f64).to_bits());
    assert_eq!(gram.as_rows()[1][0].to_bits(), (-0.0_f64).to_bits());
    assert_eq!(gram.as_rows()[1][1].to_bits(), 0);
}

#[test]
fn overflow_checks_off_diagonal_before_later_diagonal() {
    let vectors = [
        Vector::try_new([0.0, 2.0]).unwrap(),
        Vector::try_new([f64::MAX, f64::MAX]).unwrap(),
    ];
    // (0,0) succeeds; (0,1) fails at coordinate 1. Evaluating (1,1) first
    // would instead report coordinate 0, violating upper-triangle row order.
    assert_eq!(
        gram_matrix(&vectors).unwrap_err(),
        LaError::non_finite_computation_step(ArithmeticOperation::VectorDotProduct, 1)
    );

    let later_diagonal = [
        Vector::try_new([1.0, 0.0]).unwrap(),
        Vector::try_new([0.0, f64::MAX]).unwrap(),
    ];
    assert_eq!(
        gram_matrix(&later_diagonal).unwrap_err(),
        LaError::non_finite_computation_step(ArithmeticOperation::VectorDotProduct, 1)
    );
}

#[test]
fn mixed_scale_and_signed_zero() {
    let vectors = [
        Vector::try_new([2.0_f64.powi(500), 2.0_f64.powi(-500), -0.0]).unwrap(),
        Vector::try_new([2.0_f64.powi(-500), -2.0_f64.powi(500), 0.0]).unwrap(),
    ];
    let gram = gram_matrix(&vectors).unwrap();
    assert_eq!(
        gram.as_rows(),
        &[[2.0_f64.powi(1000), 0.0], [0.0, 2.0_f64.powi(1000)]]
    );
    assert_eq!(
        gram.as_rows()[0][1].to_bits(),
        gram.as_rows()[1][0].to_bits()
    );
}

#[test]
fn overflow_preserves_dot_diagnostics() {
    for (data, index) in [([f64::MAX, 0.0], 0), ([1.0e154, 1.0e154], 1)] {
        let vectors = [Vector::try_new(data).unwrap()];
        assert!(matches!(
            gram_matrix(&vectors),
            Err(LaError::NonFinite {
                origin: NonFiniteOrigin::Computation { operation: ArithmeticOperation::VectorDotProduct, .. },
                location: NonFiniteLocation::Step { index: actual, .. },
                ..
            }) if actual == index
        ));
    }
}