mathhook-core 0.2.0

Core mathematical engine for MathHook - expressions, algebra, and solving
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
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353

//! Hessian matrix operations for second-order partial derivatives
use crate::calculus::derivatives::Derivative;
use crate::core::{Expression, Symbol};
use crate::simplify::Simplify;
/// Hessian matrix operations
pub struct HessianOperations;
impl HessianOperations {
    /// Compute Hessian matrix
    ///
    /// # Examples
    ///
    /// ```rust
    /// use mathhook_core::simplify::Simplify;
    /// use mathhook_core::calculus::derivatives::Derivative;
    /// use mathhook_core::{Expression};
    /// use mathhook_core::symbol;
    /// use mathhook_core::calculus::derivatives::HessianOperations;
    ///
    /// let x = symbol!(x);
    /// let y = symbol!(y);
    /// let expr = Expression::add(vec![
    ///     Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
    ///     Expression::pow(Expression::symbol(y.clone()), Expression::integer(2))
    /// ]);
    /// let hessian = HessianOperations::compute(&expr, &[x, y]);
    /// ```
    pub fn compute(expr: &Expression, variables: &[Symbol]) -> Vec<Vec<Expression>> {
        let n = variables.len();
        let mut hessian = Vec::with_capacity(n);
        for _ in 0..n {
            hessian.push(Vec::with_capacity(n));
        }
        for i in 0..n {
            for j in 0..n {
                if j >= i {
                    let second_partial = expr
                        .derivative(variables[i].clone())
                        .derivative(variables[j].clone())
                        .simplify();
                    hessian[i].push(second_partial);
                } else {
                    let symmetric_entry = hessian[j][i].clone();
                    hessian[i].push(symmetric_entry);
                }
            }
        }
        hessian
    }
    /// Compute Hessian determinant
    ///
    /// # Examples
    ///
    /// ```rust
    /// use mathhook_core::{Expression, Symbol};
    /// use mathhook_core::symbol;
    /// use mathhook_core::calculus::derivatives::HessianOperations;
    ///
    /// let x = symbol!(x);
    /// let y = symbol!(y);
    /// let expr = Expression::add(vec![
    ///     Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
    ///     Expression::pow(Expression::symbol(y.clone()), Expression::integer(2))
    /// ]);
    /// let det = HessianOperations::determinant(&expr, vec![x, y]);
    /// ```
    pub fn determinant(expr: &Expression, variables: Vec<Symbol>) -> Expression {
        let hessian = Self::compute(expr, &variables);
        Self::matrix_determinant(&hessian)
    }
    /// Compute matrix determinant recursively
    fn matrix_determinant(matrix: &[Vec<Expression>]) -> Expression {
        let n = matrix.len();
        match n {
            0 => Expression::integer(1),
            1 => matrix[0][0].clone(),
            2 => {
                let a = &matrix[0][0];
                let b = &matrix[0][1];
                let c = &matrix[1][0];
                let d = &matrix[1][1];
                Expression::add(vec![
                    Expression::mul(vec![a.clone(), d.clone()]),
                    Expression::mul(vec![
                        Expression::integer(-1),
                        Expression::mul(vec![b.clone(), c.clone()]),
                    ]),
                ])
                .simplify()
            }
            _ => {
                let mut det_terms = Vec::with_capacity(n);
                for j in 0..n {
                    let cofactor = Self::cofactor(matrix, 0, j);
                    let sign = if j % 2 == 0 { 1 } else { -1 };
                    det_terms.push(Expression::mul(vec![
                        Expression::integer(sign),
                        matrix[0][j].clone(),
                        cofactor,
                    ]));
                }
                Expression::add(det_terms).simplify()
            }
        }
    }
    /// Compute cofactor for matrix determinant
    fn cofactor(matrix: &[Vec<Expression>], row: usize, col: usize) -> Expression {
        let n = matrix.len();
        let minor: Vec<Vec<_>> = (0..n)
            .filter(|&i| i != row)
            .map(|i| {
                (0..n)
                    .filter(|&j| j != col)
                    .map(|j| matrix[i][j].clone())
                    .collect()
            })
            .collect();
        Self::matrix_determinant(&minor)
    }
    /// Check if Hessian is positive definite (for optimization)
    ///
    /// # Examples
    ///
    /// ```rust
    /// use mathhook_core::{Expression};
    /// use mathhook_core::symbol;
    /// use mathhook_core::calculus::derivatives::HessianOperations;
    ///
    /// let x = symbol!(x);
    /// let y = symbol!(y);
    /// let expr = Expression::add(vec![
    ///     Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
    ///     Expression::pow(Expression::symbol(y.clone()), Expression::integer(2))
    /// ]);
    /// let is_pos_def = HessianOperations::is_positive_definite(&expr, vec![x, y]);
    /// ```
    pub fn is_positive_definite(expr: &Expression, variables: Vec<Symbol>) -> bool {
        let hessian = Self::compute(expr, &variables);
        Self::check_positive_definite(&hessian)
    }
    /// Check positive definiteness using leading principal minors
    fn check_positive_definite(hessian: &[Vec<Expression>]) -> bool {
        let n = hessian.len();
        for k in 1..=n {
            let submatrix: Vec<Vec<_>> = (0..k)
                .map(|i| (0..k).map(|j| hessian[i][j].clone()).collect())
                .collect();
            let det = Self::matrix_determinant(&submatrix);
            if det.is_zero() {
                return false;
            }
        }
        true
    }
    /// Compute trace of Hessian matrix
    ///
    /// # Examples
    ///
    /// ```rust
    /// use mathhook_core::{Expression, Symbol};
    /// use mathhook_core::symbol;
    /// use mathhook_core::calculus::derivatives::HessianOperations;
    ///
    /// let x = symbol!(x);
    /// let y = symbol!(y);
    /// let expr = Expression::add(vec![
    ///     Expression::mul(vec![Expression::integer(3), Expression::pow(Expression::symbol(x.clone()), Expression::integer(2))]),
    ///     Expression::mul(vec![Expression::integer(5), Expression::pow(Expression::symbol(y.clone()), Expression::integer(2))])
    /// ]);
    /// let trace = HessianOperations::trace(&expr, vec![x, y]);
    /// ```
    pub fn trace(expr: &Expression, variables: Vec<Symbol>) -> Expression {
        let hessian = Self::compute(expr, &variables);
        let n = hessian.len();
        let mut diagonal_terms = Vec::with_capacity(n);
        diagonal_terms.extend((0..n).map(|i| hessian[i][i].clone()));
        Expression::add(diagonal_terms).simplify()
    }
}
#[cfg(test)]
mod tests {
    use std::slice::from_ref;

    use super::*;
    use crate::expr;
    use crate::symbol;
    #[test]
    fn test_quadratic_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let quadratic = Expression::add(vec![
            Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
            Expression::pow(Expression::symbol(y.clone()), Expression::integer(2)),
        ]);
        let hessian = HessianOperations::compute(&quadratic, &[x.clone(), y.clone()]);
        assert_eq!(hessian.len(), 2);
        assert_eq!(hessian[0].len(), 2);
        assert_eq!(hessian[1].len(), 2);
        assert_eq!(hessian[0][0].simplify(), Expression::integer(2));
        assert_eq!(hessian[1][1].simplify(), Expression::integer(2));
        assert_eq!(hessian[0][1].simplify(), Expression::integer(0));
        assert_eq!(hessian[1][0].simplify(), Expression::integer(0));
    }
    #[test]
    fn test_mixed_partial_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let mixed = Expression::mul(vec![
            Expression::symbol(x.clone()),
            Expression::symbol(y.clone()),
        ]);
        let hessian = HessianOperations::compute(&mixed, &[x.clone(), y.clone()]);
        assert_eq!(hessian[0][0].simplify(), Expression::integer(0));
        assert_eq!(hessian[1][1].simplify(), Expression::integer(0));
        assert_eq!(hessian[0][1].simplify(), Expression::integer(1));
        assert_eq!(hessian[1][0].simplify(), Expression::integer(1));
    }
    #[test]
    fn test_cubic_polynomial_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let cubic = Expression::add(vec![
            Expression::pow(Expression::symbol(x.clone()), Expression::integer(3)),
            Expression::mul(vec![
                Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
                Expression::symbol(y.clone()),
            ]),
            Expression::pow(Expression::symbol(y.clone()), Expression::integer(3)),
        ]);
        let hessian = HessianOperations::compute(&cubic, &[x.clone(), y.clone()]);
        assert_eq!(hessian.len(), 2);
        assert!(!hessian[0][0].is_zero());
        assert!(!hessian[1][1].is_zero());
        assert!(!hessian[0][1].is_zero());
        assert!(!hessian[1][0].is_zero());
    }
    #[test]
    fn test_single_variable_hessian() {
        let x = symbol!(x);
        let expr = Expression::pow(Expression::symbol(x.clone()), Expression::integer(4));
        let hessian = HessianOperations::compute(&expr, from_ref(&x));
        assert_eq!(hessian.len(), 1);
        assert_eq!(hessian[0].len(), 1);
        assert!(!hessian[0][0].is_zero());
    }
    #[test]
    fn test_three_variable_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let z = symbol!(z);
        let expr = Expression::add(vec![
            Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
            Expression::pow(Expression::symbol(y.clone()), Expression::integer(2)),
            Expression::pow(Expression::symbol(z.clone()), Expression::integer(2)),
        ]);
        let hessian = HessianOperations::compute(&expr, &[x.clone(), y.clone(), z.clone()]);
        assert_eq!(hessian.len(), 3);
        for (i, row) in hessian.iter().enumerate().take(3) {
            assert_eq!(row.len(), 3);
            for (j, val) in row.iter().enumerate().take(3) {
                let expected = if i == j {
                    Expression::integer(2)
                } else {
                    Expression::integer(0)
                };
                assert_eq!(val.simplify(), expected);
            }
        }
    }
    #[test]
    fn test_hessian_determinant_2x2() {
        let x = symbol!(x);
        let y = symbol!(y);
        let expr = Expression::add(vec![
            Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
            Expression::pow(Expression::symbol(y.clone()), Expression::integer(2)),
        ]);
        let det = HessianOperations::determinant(&expr, vec![x.clone(), y.clone()]);
        assert_eq!(det.simplify(), Expression::integer(4));
    }
    #[test]
    fn test_hessian_trace() {
        let x = symbol!(x);
        let y = symbol!(y);
        let expr = Expression::add(vec![
            Expression::mul(vec![
                Expression::integer(3),
                Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
            ]),
            Expression::mul(vec![
                Expression::integer(5),
                Expression::pow(Expression::symbol(y.clone()), Expression::integer(2)),
            ]),
        ]);
        let trace = HessianOperations::trace(&expr, vec![x.clone(), y.clone()]);
        assert_eq!(trace.simplify(), Expression::integer(16));
    }
    #[test]
    fn test_constant_function_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let constant = Expression::integer(42);
        let hessian = HessianOperations::compute(&constant, &[x.clone(), y.clone()]);
        for value in hessian.iter().flatten() {
            assert_eq!(value.simplify(), expr!(0));
        }
    }
    #[test]
    fn test_linear_function_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let linear = Expression::add(vec![
            Expression::mul(vec![Expression::integer(2), Expression::symbol(x.clone())]),
            Expression::mul(vec![Expression::integer(3), Expression::symbol(y.clone())]),
            Expression::integer(1),
        ]);
        let hessian = HessianOperations::compute(&linear, &[x.clone(), y.clone()]);
        for value in hessian.iter().flatten() {
            assert_eq!(value.simplify(), expr!(0));
        }
    }
    #[test]
    fn test_hessian_symmetry() {
        let x = symbol!(x);
        let y = symbol!(y);
        let expr = Expression::add(vec![
            Expression::mul(vec![
                Expression::pow(Expression::symbol(x.clone()), Expression::integer(2)),
                Expression::symbol(y.clone()),
            ]),
            Expression::mul(vec![
                Expression::symbol(x.clone()),
                Expression::pow(Expression::symbol(y.clone()), Expression::integer(2)),
            ]),
        ]);
        let hessian = HessianOperations::compute(&expr, &[x.clone(), y.clone()]);
        assert_eq!(hessian[0][1], hessian[1][0]);
    }
    #[test]
    fn test_trigonometric_hessian() {
        let x = symbol!(x);
        let y = symbol!(y);
        let trig_expr = Expression::add(vec![
            Expression::function("sin", vec![Expression::symbol(x.clone())]),
            Expression::function("cos", vec![Expression::symbol(y.clone())]),
        ]);
        let hessian = HessianOperations::compute(&trig_expr, &[x.clone(), y.clone()]);
        assert_eq!(hessian.len(), 2);
        assert!(!hessian[0][0].is_zero());
        assert!(!hessian[1][1].is_zero());
        assert_eq!(hessian[0][1].simplify(), Expression::integer(0));
        assert_eq!(hessian[1][0].simplify(), Expression::integer(0));
    }
}