Forward/back substitution for triangular systems. More...
#include "config.h"#include "utilities/expected.h"#include "matrix_traits.h"#include "simple_type_traits.h"#include "math/math_utils.h"
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Namespaces | |
| namespace | matrix_algorithms |
Functions | |
| template<bool UnitDiag = false, typename T , my_size_t N> | |
| Expected< FusedVector< T, N >, MatrixStatus > | matrix_algorithms::forward_substitute (const FusedMatrix< T, N, N > &L, const FusedVector< T, N > &b) |
| Solve the lower-triangular system Lx = b by forward substitution. | |
| template<bool UnitDiag = false, typename T , my_size_t N> | |
| Expected< FusedVector< T, N >, MatrixStatus > | matrix_algorithms::back_substitute (const FusedMatrix< T, N, N > &U, const FusedVector< T, N > &b) |
| Solve the upper-triangular system Ux = b by back substitution. | |
| template<bool UnitDiag = false, typename T , my_size_t N, my_size_t Ncols> | |
| Expected< FusedMatrix< T, N, Ncols >, MatrixStatus > | matrix_algorithms::forward_substitute (const FusedMatrix< T, N, N > &L, const FusedMatrix< T, N, Ncols > &B) |
| Solve the lower-triangular system LX = B for multiple right-hand sides. | |
| template<bool UnitDiag = false, typename T , my_size_t N, my_size_t Ncols> | |
| Expected< FusedMatrix< T, N, Ncols >, MatrixStatus > | matrix_algorithms::back_substitute (const FusedMatrix< T, N, N > &U, const FusedMatrix< T, N, Ncols > &B) |
| Solve the upper-triangular system UX = B for multiple right-hand sides. | |
Detailed Description
Forward/back substitution for triangular systems.
Provides solvers for triangular linear systems Lx = b and Ux = b, with compile-time dispatch for fixed-size unrolled paths (N ∈ {3, 4, 6}) and a UnitDiag template parameter for LU-style implicit unit diagonals.
Also includes multi-RHS overloads (LX = B, UX = B) for matrix right-hand sides, used by matrix inverse and related algorithms.
ALGORITHMS
Forward substitution (Lx = b, L lower-triangular):
For i = 0 … N−1:
x(i) = ( b(i) − Σ_{k=0}^{i-1} L(i,k)·x(k) ) / L(i,i)
(division skipped when UnitDiag = true)
Back substitution (Ux = b, U upper-triangular):
For i = N−1 … 0:
x(i) = ( b(i) − Σ_{k=i+1}^{N-1} U(i,k)·x(k) ) / U(i,i)
(division skipped when UnitDiag = true)
Complexity: O(N²/2) per substitution.
FAILURE MODES
- MatrixStatus::DimensionMismatch — matrix rows != vector size
- MatrixStatus::Singular — zero diagonal encountered during substitution
