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Renamed linsolve to lup_solve, added docs, and refactored #29

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Merged
pasabanov merged 1 commit into from
Feb 9, 2025
Merged

Renamed linsolve to lup_solve, added docs, and refactored #29

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4 changes: 2 additions & 2 deletions ALFI/ALFI/spline/cubic.h
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Original file line number Diff line number Diff line change
Expand Up @@ -303,7 +303,7 @@ namespace alfi::spline {
const auto h12 = X[2]-X[0], h13 = X[3]-X[0];
const auto d1 = Y[1]-Y[0], d12 = Y[2]-Y[0], d13 = Y[3]-Y[0];

const auto abc = util::linalg::linsolve(
const auto abc = util::linalg::lup_solve(
{{std::pow(h1, 3), std::pow(h1, 2), h1},
{std::pow(h12, 3), std::pow(h12, 2), h12},
{std::pow(h13, 3), std::pow(h13, 2), h13}},
Expand Down Expand Up @@ -350,7 +350,7 @@ namespace alfi::spline {
const auto h12 = X[n-2]-X[n-4], h13 = X[n-1]-X[n-4];
const auto d1 = dY[n-4], d12 = Y[n-2] - Y[n-4], d13 = Y[n-1] - Y[n-4];

const auto abc = util::linalg::linsolve(
const auto abc = util::linalg::lup_solve(
{{std::pow(h1, 3), std::pow(h1, 2), h1},
{std::pow(h12, 3), std::pow(h12, 2), h12},
{std::pow(h13, 3), std::pow(h13, 2), h13}},
Expand Down
63 changes: 37 additions & 26 deletions ALFI/ALFI/util/linalg.h
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Expand Up @@ -5,62 +5,73 @@
#include "../config.h"

namespace alfi::util::linalg {
template <typename Number = DefaultNumber, template <typename> class Container = DefaultContainer>
Container<Number> linsolve(Container<Container<Number>>&& A, const Container<Number>& B, Number epsilon = std::numeric_limits<Number>::epsilon()) {
const auto N = B.size();
/**
@brief Solves a system of linear equations using LUP decomposition.

Container<SizeT> P(N);
for (SizeT i = 0; i < N; ++i) {
P[i] = i;
}
This function solves the system of linear equations \f(AX = B\f), where \f(A\f) is a square matrix,
by performing LUP decomposition and forward-backward substitution.\n
The input matrix \f(A\f) is decomposed into lower \f(L\f) and upper \f(U\f) triangular matrices,
such that \f(PA = LU\f), transforming the equation into \f(PLUX = B\f) or \f(LUX = PB\f).

The solution is found in two stages:
1. The first stage solves the system \f(LY = PB\f) for the intermediate vector \f(Y\f) using forward substitution.
2. The second stage solves the system \f(UX = Y\f) for the final solution \f(X\f) using backward substitution.

If the pivot element in a given column is too small (less than the specified \f(epsilon\f)),
the function returns an empty container, indicating that the matrix is degenerate,
and prints the corresponding message to `std::cerr`.

for (SizeT i = 0; i < N; ++i) {
The input matrix \f(A\f) and vector \f(B\f) are modified in place.

@param A the matrix (2D container of size \f(n \times n\f))
@param B the right-hand side vector (1D container of size \f(n\f))
@param epsilon a small threshold value to detect degeneracy (default is machine epsilon)
@return the solution vector or an empty container if the matrix is degenerate
*/
template <typename Number = DefaultNumber, template <typename> class Container = DefaultContainer>
Container<Number> lup_solve(Container<Container<Number>>&& A, Container<Number>&& B, Number epsilon = std::numeric_limits<Number>::epsilon()) {
const auto n = B.size();
assert(n == A.size());
for (SizeT i = 0; i < n; ++i) {
assert(n == A[i].size());
Number max_a = std::abs(A[i][i]);
SizeT i_max = i;

for (SizeT k = i + 1; k < N; ++k) {
for (SizeT k = i + 1; k < n; ++k) {
const auto cur = std::abs(A[k][i]);
if (cur > max_a) {
max_a = cur;
i_max = k;
}
}

if (max_a < epsilon) {
std::cerr << "Error in function " << __FUNCTION__ << ": Matrix A is degenerate. Returning an empty array..." << std::endl;
return {};
}

if (i_max != i) {
std::swap(P[i], P[i_max]);
std::swap(A[i], A[i_max]);
std::swap(B[i], B[i_max]);
}

for (SizeT j = i + 1; j < N; ++j) {
for (SizeT j = i + 1; j < n; ++j) {
A[j][i] /= A[i][i];
for (SizeT k = i + 1; k < N; ++k) {
for (SizeT k = i + 1; k < n; ++k) {
A[j][k] -= A[j][i] * A[i][k];
}
}
}

Container<Number> X(N);

for (SizeT i = 0; i < N; ++i) {
X[i] = B[P[i]];
Container<Number> X(n);
for (SizeT i = 0; i < n; ++i) {
X[i] = B[i];
for (SizeT k = 0; k < i; ++k) {
X[i] -= A[i][k] * X[k];
}
}

for (SizeT iter = 0; iter < N; ++iter) {
const auto i = N - 1 - iter;
for (SizeT k = i + 1; k < N; ++k) {
for (SizeT iter = 0; iter < n; ++iter) {
const auto i = n - 1 - iter;
for (SizeT k = i + 1; k < n; ++k) {
X[i] -= A[i][k] * X[k];
}
X[i] /= A[i][i];
}

return X;
}

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2 changes: 1 addition & 1 deletion ALFI/ALFI/util/spline.h
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Original file line number Diff line number Diff line change
Expand Up @@ -65,7 +65,7 @@ namespace alfi::util::spline {
const auto h12 = X[2]-X[0], h13 = X[3]-X[0];
const auto d1 = Y[1]-Y[0], d12 = Y[2]-Y[0], d13 = Y[3]-Y[0];

const auto abc = util::linalg::linsolve(
const auto abc = util::linalg::lup_solve(
{{std::pow(h1, 3), std::pow(h1, 2), h1},
{std::pow(h12, 3), std::pow(h12, 2), h12},
{std::pow(h13, 3), std::pow(h13, 2), h13}},
Expand Down