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
use crate::storage::Storage;
use crate::{
    Allocator, Bidiagonal, Cholesky, ComplexField, DefaultAllocator, Dim, DimDiff, DimMin,
    DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, Schur, SymmetricEigen, SymmetricTridiagonal,
    LU, QR, SVD, U1,
};

/// # Rectangular matrix decomposition
///
/// This section contains the methods for computing some common decompositions of rectangular
/// matrices with real or complex components. The following are currently supported:
///
/// | Decomposition            | Factors             | Details |
/// | -------------------------|---------------------|--------------|
/// | QR                       | `Q * R`             | `Q` is an unitary matrix, and `R` is upper-triangular. |
/// | LU with partial pivoting | `P⁻¹ * L * U`       | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` is a permutation matrix. |
/// | LU with full pivoting    | `P⁻¹ * L * U ~ Q⁻¹` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` and `Q` are permutation matrices. |
/// | SVD                      | `U * Σ * Vᵀ`        | `U` and `V` are two orthogonal matrices and `Σ` is a diagonal matrix containing the singular values. |
impl<N: ComplexField, R: Dim, C: Dim, S: Storage<N, R, C>> Matrix<N, R, C, S> {
    /// Computes the bidiagonalization using householder reflections.
    pub fn bidiagonalize(self) -> Bidiagonal<N, R, C>
    where
        R: DimMin<C>,
        DimMinimum<R, C>: DimSub<U1>,
        DefaultAllocator: Allocator<N, R, C>
            + Allocator<N, C>
            + Allocator<N, R>
            + Allocator<N, DimMinimum<R, C>>
            + Allocator<N, DimDiff<DimMinimum<R, C>, U1>>,
    {
        Bidiagonal::new(self.into_owned())
    }

    /// Computes the LU decomposition with full pivoting of `matrix`.
    ///
    /// This effectively computes `P, L, U, Q` such that `P * matrix * Q = LU`.
    pub fn full_piv_lu(self) -> FullPivLU<N, R, C>
    where
        R: DimMin<C>,
        DefaultAllocator: Allocator<N, R, C> + Allocator<(usize, usize), DimMinimum<R, C>>,
    {
        FullPivLU::new(self.into_owned())
    }

    /// Computes the LU decomposition with partial (row) pivoting of `matrix`.
    pub fn lu(self) -> LU<N, R, C>
    where
        R: DimMin<C>,
        DefaultAllocator: Allocator<N, R, C> + Allocator<(usize, usize), DimMinimum<R, C>>,
    {
        LU::new(self.into_owned())
    }

    /// Computes the QR decomposition of this matrix.
    pub fn qr(self) -> QR<N, R, C>
    where
        R: DimMin<C>,
        DefaultAllocator: Allocator<N, R, C> + Allocator<N, R> + Allocator<N, DimMinimum<R, C>>,
    {
        QR::new(self.into_owned())
    }

    /// Computes the Singular Value Decomposition using implicit shift.
    pub fn svd(self, compute_u: bool, compute_v: bool) -> SVD<N, R, C>
    where
        R: DimMin<C>,
        DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
        DefaultAllocator: Allocator<N, R, C>
            + Allocator<N, C>
            + Allocator<N, R>
            + Allocator<N, DimDiff<DimMinimum<R, C>, U1>>
            + Allocator<N, DimMinimum<R, C>, C>
            + Allocator<N, R, DimMinimum<R, C>>
            + Allocator<N, DimMinimum<R, C>>
            + Allocator<N::RealField, DimMinimum<R, C>>
            + Allocator<N::RealField, DimDiff<DimMinimum<R, C>, U1>>,
    {
        SVD::new(self.into_owned(), compute_u, compute_v)
    }

    /// Attempts to compute the Singular Value Decomposition of `matrix` using implicit shift.
    ///
    /// # Arguments
    ///
    /// * `compute_u` − set this to `true` to enable the computation of left-singular vectors.
    /// * `compute_v` − set this to `true` to enable the computation of right-singular vectors.
    /// * `eps`       − tolerance used to determine when a value converged to 0.
    /// * `max_niter` − maximum total number of iterations performed by the algorithm. If this
    /// number of iteration is exceeded, `None` is returned. If `niter == 0`, then the algorithm
    /// continues indefinitely until convergence.
    pub fn try_svd(
        self,
        compute_u: bool,
        compute_v: bool,
        eps: N::RealField,
        max_niter: usize,
    ) -> Option<SVD<N, R, C>>
    where
        R: DimMin<C>,
        DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
        DefaultAllocator: Allocator<N, R, C>
            + Allocator<N, C>
            + Allocator<N, R>
            + Allocator<N, DimDiff<DimMinimum<R, C>, U1>>
            + Allocator<N, DimMinimum<R, C>, C>
            + Allocator<N, R, DimMinimum<R, C>>
            + Allocator<N, DimMinimum<R, C>>
            + Allocator<N::RealField, DimMinimum<R, C>>
            + Allocator<N::RealField, DimDiff<DimMinimum<R, C>, U1>>,
    {
        SVD::try_new(self.into_owned(), compute_u, compute_v, eps, max_niter)
    }
}

/// # Square matrix decomposition
///
/// This section contains the methods for computing some common decompositions of square
/// matrices with real or complex components. The following are currently supported:
///
/// | Decomposition            | Factors                   | Details |
/// | -------------------------|---------------------------|--------------|
/// | Hessenberg               | `Q * H * Qᵀ`             | `Q` is a unitary matrix and `H` an upper-Hessenberg matrix. |
/// | Cholesky                 | `L * Lᵀ`                 | `L` is a lower-triangular matrix. |
/// | Schur decomposition      | `Q * T * Qᵀ`             | `Q` is an unitary matrix and `T` a quasi-upper-triangular matrix. |
/// | Symmetric eigendecomposition | `Q ~ Λ ~ Qᵀ`   | `Q` is an unitary matrix, and `Λ` is a real diagonal matrix. |
/// | Symmetric tridiagonalization | `Q ~ T ~ Qᵀ`   | `Q` is an unitary matrix, and `T` is a tridiagonal matrix. |
impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> Matrix<N, D, D, S> {
    /// Attempts to compute the Cholesky decomposition of this 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 cholesky(self) -> Option<Cholesky<N, D>>
    where
        DefaultAllocator: Allocator<N, D, D>,
    {
        Cholesky::new(self.into_owned())
    }

    /// Computes the Hessenberg decomposition of this matrix using householder reflections.
    pub fn hessenberg(self) -> Hessenberg<N, D>
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, D, D> + Allocator<N, D> + Allocator<N, DimDiff<D, U1>>,
    {
        Hessenberg::new(self.into_owned())
    }

    /// Computes the Schur decomposition of a square matrix.
    pub fn schur(self) -> Schur<N, D>
    where
        D: DimSub<U1>, // For Hessenberg.
        DefaultAllocator: Allocator<N, D, DimDiff<D, U1>>
            + Allocator<N, DimDiff<D, U1>>
            + Allocator<N, D, D>
            + Allocator<N, D>,
    {
        Schur::new(self.into_owned())
    }

    /// Attempts to compute the Schur decomposition of a square matrix.
    ///
    /// If only eigenvalues are needed, it is more efficient to call the matrix method
    /// `.eigenvalues()` instead.
    ///
    /// # Arguments
    ///
    /// * `eps`       − tolerance used to determine when a value converged to 0.
    /// * `max_niter` − maximum total number of iterations performed by the algorithm. If this
    /// number of iteration is exceeded, `None` is returned. If `niter == 0`, then the algorithm
    /// continues indefinitely until convergence.
    pub fn try_schur(self, eps: N::RealField, max_niter: usize) -> Option<Schur<N, D>>
    where
        D: DimSub<U1>, // For Hessenberg.
        DefaultAllocator: Allocator<N, D, DimDiff<D, U1>>
            + Allocator<N, DimDiff<D, U1>>
            + Allocator<N, D, D>
            + Allocator<N, D>,
    {
        Schur::try_new(self.into_owned(), eps, max_niter)
    }

    /// Computes the eigendecomposition of this symmetric matrix.
    ///
    /// Only the lower-triangular part (including the diagonal) of `m` is read.
    pub fn symmetric_eigen(self) -> SymmetricEigen<N, D>
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, D, D>
            + Allocator<N, DimDiff<D, U1>>
            + Allocator<N::RealField, D>
            + Allocator<N::RealField, DimDiff<D, U1>>,
    {
        SymmetricEigen::new(self.into_owned())
    }

    /// Computes the eigendecomposition of the given symmetric matrix with user-specified
    /// convergence parameters.
    ///
    /// Only the lower-triangular part (including the diagonal) of `m` is read.
    ///
    /// # Arguments
    ///
    /// * `eps`       − tolerance used to determine when a value converged to 0.
    /// * `max_niter` − maximum total number of iterations performed by the algorithm. If this
    /// number of iteration is exceeded, `None` is returned. If `niter == 0`, then the algorithm
    /// continues indefinitely until convergence.
    pub fn try_symmetric_eigen(
        self,
        eps: N::RealField,
        max_niter: usize,
    ) -> Option<SymmetricEigen<N, D>>
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, D, D>
            + Allocator<N, DimDiff<D, U1>>
            + Allocator<N::RealField, D>
            + Allocator<N::RealField, DimDiff<D, U1>>,
    {
        SymmetricEigen::try_new(self.into_owned(), eps, max_niter)
    }

    /// Computes the tridiagonalization of this symmetric matrix.
    ///
    /// Only the lower-triangular part (including the diagonal) of `m` is read.
    pub fn symmetric_tridiagonalize(self) -> SymmetricTridiagonal<N, D>
    where
        D: DimSub<U1>,
        DefaultAllocator: Allocator<N, D, D> + Allocator<N, DimDiff<D, U1>>,
    {
        SymmetricTridiagonal::new(self.into_owned())
    }
}