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
354
355
356
357
358
359
360
361
362
363
364
365
#[cfg(feature = "serde-serialize")]
use serde::{Deserialize, Serialize};

use num::One;
use simba::scalar::ComplexField;
use simba::simd::SimdComplexField;

use crate::allocator::Allocator;
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, Vector};
use crate::constraint::{SameNumberOfRows, ShapeConstraint};
use crate::dimension::{Dim, DimAdd, DimDiff, DimSub, DimSum, U1};
use crate::storage::{Storage, StorageMut};

/// The Cholesky decomposition of a symmetric-definite-positive matrix.
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(serialize = "DefaultAllocator: Allocator<N, D>,
         MatrixN<N, D>: Serialize"))
)]
#[cfg_attr(
    feature = "serde-serialize",
    serde(bound(deserialize = "DefaultAllocator: Allocator<N, D>,
         MatrixN<N, D>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Cholesky<N: SimdComplexField, D: Dim>
where
    DefaultAllocator: Allocator<N, D, D>,
{
    chol: MatrixN<N, D>,
}

impl<N: SimdComplexField, D: Dim> Copy for Cholesky<N, D>
where
    DefaultAllocator: Allocator<N, D, D>,
    MatrixN<N, D>: Copy,
{
}

impl<N: SimdComplexField, D: Dim> Cholesky<N, D>
where
    DefaultAllocator: Allocator<N, D, D>,
{
    /// Computes the Cholesky decomposition of `matrix` without checking that the matrix is definite-positive.
    ///
    /// If the input matrix is not definite-positive, the decomposition may contain trash values (Inf, NaN, etc.)
    pub fn new_unchecked(mut matrix: MatrixN<N, D>) -> Self {
        assert!(matrix.is_square(), "The input matrix must be square.");

        let n = matrix.nrows();

        for j in 0..n {
            for k in 0..j {
                let factor = unsafe { -*matrix.get_unchecked((j, k)) };

                let (mut col_j, col_k) = matrix.columns_range_pair_mut(j, k);
                let mut col_j = col_j.rows_range_mut(j..);
                let col_k = col_k.rows_range(j..);
                col_j.axpy(factor.simd_conjugate(), &col_k, N::one());
            }

            let diag = unsafe { *matrix.get_unchecked((j, j)) };
            let denom = diag.simd_sqrt();

            unsafe {
                *matrix.get_unchecked_mut((j, j)) = denom;
            }

            let mut col = matrix.slice_range_mut(j + 1.., j);
            col /= denom;
        }

        Cholesky { chol: matrix }
    }

    /// Retrieves the lower-triangular factor of the Cholesky decomposition with its strictly
    /// upper-triangular part filled with zeros.
    pub fn unpack(mut self) -> MatrixN<N, D> {
        self.chol.fill_upper_triangle(N::zero(), 1);
        self.chol
    }

    /// Retrieves the lower-triangular factor of the Cholesky decomposition, without zeroing-out
    /// its strict upper-triangular part.
    ///
    /// The values of the strict upper-triangular part are garbage and should be ignored by further
    /// computations.
    pub fn unpack_dirty(self) -> MatrixN<N, D> {
        self.chol
    }

    /// Retrieves the lower-triangular factor of the Cholesky decomposition with its strictly
    /// uppen-triangular part filled with zeros.
    pub fn l(&self) -> MatrixN<N, D> {
        self.chol.lower_triangle()
    }

    /// Retrieves the lower-triangular factor of the Cholesky decomposition, without zeroing-out
    /// its strict upper-triangular part.
    ///
    /// This is an allocation-less version of `self.l()`. The values of the strict upper-triangular
    /// part are garbage and should be ignored by further computations.
    pub fn l_dirty(&self) -> &MatrixN<N, D> {
        &self.chol
    }

    /// Solves the system `self * x = b` where `self` is the decomposed matrix and `x` the unknown.
    ///
    /// The result is stored on `b`.
    pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<N, R2, C2, S2>)
    where
        S2: StorageMut<N, R2, C2>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
    {
        self.chol.solve_lower_triangular_unchecked_mut(b);
        self.chol.ad_solve_lower_triangular_unchecked_mut(b);
    }

    /// Returns the solution of the system `self * x = b` where `self` is the decomposed matrix and
    /// `x` the unknown.
    pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<N, R2, C2, S2>) -> MatrixMN<N, R2, C2>
    where
        S2: Storage<N, R2, C2>,
        DefaultAllocator: Allocator<N, R2, C2>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
    {
        let mut res = b.clone_owned();
        self.solve_mut(&mut res);
        res
    }

    /// Computes the inverse of the decomposed matrix.
    pub fn inverse(&self) -> MatrixN<N, D> {
        let shape = self.chol.data.shape();
        let mut res = MatrixN::identity_generic(shape.0, shape.1);

        self.solve_mut(&mut res);
        res
    }
}

impl<N: ComplexField, D: Dim> Cholesky<N, D>
where
    DefaultAllocator: Allocator<N, D, D>,
{
    /// Attempts to compute the Cholesky decomposition of `matrix`.
    ///
    /// Returns `None` if the input matrix is not definite-positive. The input matrix is assumed
    /// to be symmetric and only the lower-triangular part is read.
    pub fn new(mut matrix: MatrixN<N, D>) -> Option<Self> {
        assert!(matrix.is_square(), "The input matrix must be square.");

        let n = matrix.nrows();

        for j in 0..n {
            for k in 0..j {
                let factor = unsafe { -*matrix.get_unchecked((j, k)) };

                let (mut col_j, col_k) = matrix.columns_range_pair_mut(j, k);
                let mut col_j = col_j.rows_range_mut(j..);
                let col_k = col_k.rows_range(j..);

                col_j.axpy(factor.conjugate(), &col_k, N::one());
            }

            let diag = unsafe { *matrix.get_unchecked((j, j)) };
            if !diag.is_zero() {
                if let Some(denom) = diag.try_sqrt() {
                    unsafe {
                        *matrix.get_unchecked_mut((j, j)) = denom;
                    }

                    let mut col = matrix.slice_range_mut(j + 1.., j);
                    col /= denom;
                    continue;
                }
            }

            // The diagonal element is either zero or its square root could not
            // be taken (e.g. for negative real numbers).
            return None;
        }

        Some(Cholesky { chol: matrix })
    }

    /// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `v`,
    /// performs a rank one update such that we end up with the decomposition of `M + sigma * (v * v.adjoint())`.
    #[inline]
    pub fn rank_one_update<R2: Dim, S2>(&mut self, x: &Vector<N, R2, S2>, sigma: N::RealField)
    where
        S2: Storage<N, R2, U1>,
        DefaultAllocator: Allocator<N, R2, U1>,
        ShapeConstraint: SameNumberOfRows<R2, D>,
    {
        Self::xx_rank_one_update(&mut self.chol, &mut x.clone_owned(), sigma)
    }

    /// Updates the decomposition such that we get the decomposition of a matrix with the given column `col` in the `j`th position.
    /// Since the matrix is square, an identical row will be added in the `j`th row.
    pub fn insert_column<R2, S2>(
        &self,
        j: usize,
        col: Vector<N, R2, S2>,
    ) -> Cholesky<N, DimSum<D, U1>>
    where
        D: DimAdd<U1>,
        R2: Dim,
        S2: Storage<N, R2, U1>,
        DefaultAllocator: Allocator<N, DimSum<D, U1>, DimSum<D, U1>> + Allocator<N, R2>,
        ShapeConstraint: SameNumberOfRows<R2, DimSum<D, U1>>,
    {
        let mut col = col.into_owned();
        // for an explanation of the formulas, see https://en.wikipedia.org/wiki/Cholesky_decomposition#Updating_the_decomposition
        let n = col.nrows();
        assert_eq!(
            n,
            self.chol.nrows() + 1,
            "The new column must have the size of the factored matrix plus one."
        );
        assert!(j < n, "j needs to be within the bound of the new matrix.");

        // loads the data into a new matrix with an additional jth row/column
        let mut chol = unsafe {
            Matrix::new_uninitialized_generic(
                self.chol.data.shape().0.add(U1),
                self.chol.data.shape().1.add(U1),
            )
        };
        chol.slice_range_mut(..j, ..j)
            .copy_from(&self.chol.slice_range(..j, ..j));
        chol.slice_range_mut(..j, j + 1..)
            .copy_from(&self.chol.slice_range(..j, j..));
        chol.slice_range_mut(j + 1.., ..j)
            .copy_from(&self.chol.slice_range(j.., ..j));
        chol.slice_range_mut(j + 1.., j + 1..)
            .copy_from(&self.chol.slice_range(j.., j..));

        // update the jth row
        let top_left_corner = self.chol.slice_range(..j, ..j);

        let col_j = col[j];
        let (mut new_rowj_adjoint, mut new_colj) = col.rows_range_pair_mut(..j, j + 1..);
        assert!(
            top_left_corner.solve_lower_triangular_mut(&mut new_rowj_adjoint),
            "Cholesky::insert_column : Unable to solve lower triangular system!"
        );

        new_rowj_adjoint.adjoint_to(&mut chol.slice_range_mut(j, ..j));

        // update the center element
        let center_element = N::sqrt(col_j - N::from_real(new_rowj_adjoint.norm_squared()));
        chol[(j, j)] = center_element;

        // update the jth column
        let bottom_left_corner = self.chol.slice_range(j.., ..j);
        // new_colj = (col_jplus - bottom_left_corner * new_rowj.adjoint()) / center_element;
        new_colj.gemm(
            -N::one() / center_element,
            &bottom_left_corner,
            &new_rowj_adjoint,
            N::one() / center_element,
        );
        chol.slice_range_mut(j + 1.., j).copy_from(&new_colj);

        // update the bottom right corner
        let mut bottom_right_corner = chol.slice_range_mut(j + 1.., j + 1..);
        Self::xx_rank_one_update(
            &mut bottom_right_corner,
            &mut new_colj,
            -N::RealField::one(),
        );

        Cholesky { chol }
    }

    /// Updates the decomposition such that we get the decomposition of the factored matrix with its `j`th column removed.
    /// Since the matrix is square, the `j`th row will also be removed.
    pub fn remove_column(&self, j: usize) -> Cholesky<N, DimDiff<D, U1>>
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, DimDiff<D, U1>, DimDiff<D, U1>> + Allocator<N, D>,
    {
        let n = self.chol.nrows();
        assert!(n > 0, "The matrix needs at least one column.");
        assert!(j < n, "j needs to be within the bound of the matrix.");

        // loads the data into a new matrix except for the jth row/column
        let mut chol = unsafe {
            Matrix::new_uninitialized_generic(
                self.chol.data.shape().0.sub(U1),
                self.chol.data.shape().1.sub(U1),
            )
        };
        chol.slice_range_mut(..j, ..j)
            .copy_from(&self.chol.slice_range(..j, ..j));
        chol.slice_range_mut(..j, j..)
            .copy_from(&self.chol.slice_range(..j, j + 1..));
        chol.slice_range_mut(j.., ..j)
            .copy_from(&self.chol.slice_range(j + 1.., ..j));
        chol.slice_range_mut(j.., j..)
            .copy_from(&self.chol.slice_range(j + 1.., j + 1..));

        // updates the bottom right corner
        let mut bottom_right_corner = chol.slice_range_mut(j.., j..);
        let mut workspace = self.chol.column(j).clone_owned();
        let mut old_colj = workspace.rows_range_mut(j + 1..);
        Self::xx_rank_one_update(&mut bottom_right_corner, &mut old_colj, N::RealField::one());

        Cholesky { chol }
    }

    /// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `x`,
    /// performs a rank one update such that we end up with the decomposition of `M + sigma * (x * x.adjoint())`.
    ///
    /// This helper method is called by `rank_one_update` but also `insert_column` and `remove_column`
    /// where it is used on a square slice of the decomposition
    fn xx_rank_one_update<Dm, Sm, Rx, Sx>(
        chol: &mut Matrix<N, Dm, Dm, Sm>,
        x: &mut Vector<N, Rx, Sx>,
        sigma: N::RealField,
    ) where
        //N: ComplexField,
        Dm: Dim,
        Rx: Dim,
        Sm: StorageMut<N, Dm, Dm>,
        Sx: StorageMut<N, Rx, U1>,
    {
        // heavily inspired by Eigen's `llt_rank_update_lower` implementation https://eigen.tuxfamily.org/dox/LLT_8h_source.html
        let n = x.nrows();
        assert_eq!(
            n,
            chol.nrows(),
            "The input vector must be of the same size as the factorized matrix."
        );

        let mut beta = crate::one::<N::RealField>();

        for j in 0..n {
            // updates the diagonal
            let diag = N::real(unsafe { *chol.get_unchecked((j, j)) });
            let diag2 = diag * diag;
            let xj = unsafe { *x.get_unchecked(j) };
            let sigma_xj2 = sigma * N::modulus_squared(xj);
            let gamma = diag2 * beta + sigma_xj2;
            let new_diag = (diag2 + sigma_xj2 / beta).sqrt();
            unsafe { *chol.get_unchecked_mut((j, j)) = N::from_real(new_diag) };
            beta += sigma_xj2 / diag2;
            // updates the terms of L
            let mut xjplus = x.rows_range_mut(j + 1..);
            let mut col_j = chol.slice_range_mut(j + 1.., j);
            // temp_jplus -= (wj / N::from_real(diag)) * col_j;
            xjplus.axpy(-xj / N::from_real(diag), &col_j, N::one());
            if gamma != crate::zero::<N::RealField>() {
                // col_j = N::from_real(nljj / diag) * col_j  + (N::from_real(nljj * sigma / gamma) * N::conjugate(wj)) * temp_jplus;
                col_j.axpy(
                    N::from_real(new_diag * sigma / gamma) * N::conjugate(xj),
                    &xjplus,
                    N::from_real(new_diag / diag),
                );
            }
        }
    }
}