hologram 0.1.2

Interpolation library with multipurpose Radial Basis Function (RBF).
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

use crate::numeric::Numeric;

#[cfg(any(feature = "openblas", feature = "intel-mkl"))]
use ndarray::{Array1, Array2};
#[cfg(any(feature = "openblas", feature = "intel-mkl"))]
use ndarray_linalg::Solve;

pub fn lu_linear_solver<T>(mat: &[Vec<f64>], rhs: &[T]) -> Result<Vec<T>, String>
where
    T: Numeric,
{
    let mat_rows = mat.len();

    if mat_rows != rhs.len() {
        return Err(String::from(
            "Incompatible design matrix and right-hand side sizes!",
        ));
    }

    let (lu, p) = lu_decomposition(mat)?;

    // Forward substitution: Solve Ly = Pb
    let mut y = T::zeros(mat_rows, &rhs[p[0]]);
    y[0] = rhs[p[0]].subtract(&T::zero(&rhs[p[0]]));

    for i in 1..mat_rows {
        let sum = T::sum((0..i).map(|j| y[j].multiply_scalar(lu[i][j])));
        y[i] = rhs[p[i]].subtract(&sum);
    }

    // Backward substitution: Solve Ux = y
    let mut x = T::zeros(mat_rows, &y[mat_rows - 1]);
    x[mat_rows - 1] = y[mat_rows - 1].divide_scalar(lu[mat_rows - 1][mat_rows - 1]);

    for i in (0..mat_rows - 1).rev() {
        let sum = T::sum((i + 1..mat_rows).map(|j| x[j].multiply_scalar(lu[i][j])));
        x[i] = y[i].subtract(&sum).divide_scalar(lu[i][i]);

        if x[i].is_instance_nan() {
            return Err(format!(
                "NaN detected in backward substitution at index {}",
                i
            ));
        }
    }

    Ok(x)
}

/// Performs LU decomposition with partial pivoting on a given square matrix.
/// The function decomposes a square matrix `mat` into its lower triangular
/// (L) and upper triangular (U) components, along with a permutation vector `p`
/// to handle row swaps due to partial pivoting.
///
/// # Arguments
/// * `mat` - A reference to a square matrix.
///
/// # Returns
/// A tuple of `lu` and `p` as `Ok((lu, p))`
/// - `lu` is a matrix where the diagonal and above contain elements of U,
///                       and below the diagonal contains elements of L.
/// - `p` is a permutation vector, representing the row swaps applied to `mat`.
///
/// Or an `Err(String)` if the decomposition fails, such as encountering a zero pivot.
pub fn lu_decomposition(mat: &[Vec<f64>]) -> Result<(Vec<Vec<f64>>, Vec<usize>), String> {
    let mut lu = mat.to_vec();
    let mut p = (0..mat.len()).collect::<Vec<usize>>();

    for k in 0..mat.len() - 1 {
        let mut max_row = k;
        for i in k + 1..mat.len() {
            if lu[i][k].abs() > lu[max_row][k].abs() {
                max_row = i;
            }
        }
        if lu[max_row][k] == 0.0 {
            return Err(String::from("Zero obtained in LU[max_row][k]!"));
        }
        if max_row != k {
            lu.swap(k, max_row);
            p.swap(k, max_row);
        }
        for i in k + 1..mat.len() {
            let factor = lu[i][k] / lu[k][k];
            lu[i][k] = factor;
            for j in k + 1..mat.len() {
                lu[i][j] -= factor * lu[k][j];
            }
        }
    }

    Ok((lu, p))
}

/// Builds a design matrix based on two sets of input vectors and a kernel function.
///
/// # Arguments
/// * `x0`: A slice of vectors representing the first set of input data.
/// * `x1`: A slice of vectors representing the second set of input data.
/// * `kernel`: A function that takes two f64 values and returns a f64 value.
/// * `epsilon`: The kernel function's parameter.
///
/// # Returns
/// A vector of vectors representing the design matrix.
pub fn build_design_matrix<X>(
    x0: &[X],
    x1: &[X],
    kernel: &fn(f64, f64) -> f64,
    epsilon: f64,
) -> Vec<Vec<f64>>
where
    X: Numeric,
{
    (0..x0.len())
        .map(|i| {
            (0..x1.len())
                .map(|j| {
                    let dist = x0[i].squared_distance(&x1[j]).max(f64::EPSILON);
                    kernel(dist, epsilon)
                })
                .collect::<Vec<f64>>()
        })
        .collect::<Vec<Vec<f64>>>()
}

pub fn compute_determinant(mat: &[Vec<f64>]) -> f64 {
    let n = mat.len();

    if mat.iter().any(|row| row.len() != n) {
        panic!("Matrix must be square to compute determinant!");
    }

    // Perform LU decomposition
    let (lu, p) = match lu_decomposition(mat) {
        Ok((lu, p)) => (lu, p),
        Err(_) => return 0.0, // If decomposition fails, assume singular matrix
    };

    // Determinant is product of diagonal elements of U
    let mut det = 1.0;
    for i in 0..n {
        det *= lu[i][i]; // Product of diagonal elements
    }

    // Adjust sign based on number of row swaps in permutation vector `p`
    let num_swaps = permutation_parity(&p);
    if num_swaps % 2 != 0 {
        det = -det;
    }

    det
}

// Helper function to determine permutation parity (odd or even swaps)
fn permutation_parity(p: &[usize]) -> usize {
    let mut visited = vec![false; p.len()];
    let mut swaps = 0;

    for i in 0..p.len() {
        if visited[i] {
            continue;
        }

        let mut j = i;
        while !visited[j] {
            visited[j] = true;
            j = p[j];
            if j != i {
                swaps += 1;
            }
        }
    }

    swaps
}

#[cfg(any(feature = "openblas", feature = "intel-mkl"))]
pub fn ndarray_linear_solver<T: Numeric>(
    design_matrix: &[Vec<f64>],
    rhs: &[T],
) -> Result<Vec<T>, String> {
    let n = design_matrix.len();

    if rhs.len() != n {
        return Err(format!(
            "Design matrix and rhs have different lengths: {} vs {}",
            n,
            rhs.len()
        ));
    }

    // Convert design matrix to Array2
    let design_array =
        Array2::from_shape_vec((n, n), design_matrix.iter().flatten().copied().collect())
            .map_err(|e| format!("Design matrix conversion failed: {}", e))?;

    // Get dimension from first RHS element
    let dim = rhs[0].to_flattened().len();

    // Check all elements have consistent dimension
    for (i, val) in rhs.iter().enumerate() {
        if val.to_flattened().len() != dim {
            return Err(format!(
                "Inconsistent dimension at index {}: expected {}, got {}",
                i,
                dim,
                val.to_flattened().len()
            ));
        }
    }

    // Solve each dimension separately
    let mut weights_matrix = Array2::zeros((n, dim));
    for d in 0..dim {
        // Extract column from RHS
        let rhs_column: Array1<f64> = Array1::from_iter(rhs.iter().map(|x| x.to_flattened()[d]));

        // Solve for this dimension
        let w = design_array
            .solve(&rhs_column)
            .map_err(|e| format!("Linear solve failed for dimension {}: {}", d, e))?;

        weights_matrix.column_mut(d).assign(&w);
    }

    // Convert back to output type
    weights_matrix
        .outer_iter()
        .map(|row| T::from_flattened(row.to_vec()))
        .collect()
}