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Implemented tridiag_solve for cubic spline #28

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Merged
pasabanov merged 4 commits into from
Feb 9, 2025
Merged

Implemented tridiag_solve for cubic spline #28

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d5c73a9
Added non-working `tridiag_solve`
pasabanov Feb 3, 2025
6e0399d
Unstable "Not-a-knot" cubic spline
pasabanov Feb 7, 2025
4e37fef
Renamed `tridiag_solve` to `tridiag_solve_unstable` and added new sta…
pasabanov Feb 7, 2025
fb51b66
Modified "Clamped", "FixedSecond" and "FixedThird" cubic splines to u…
pasabanov Feb 9, 2025
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281 changes: 105 additions & 176 deletions ALFI/ALFI/spline/cubic.h
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Original file line number Diff line number Diff line change
Expand Up @@ -115,66 +115,43 @@ namespace alfi::spline {
coeffs = util::spline::simple_spline<Number,Container>(X, Y, 3);
return;
}

const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

Container<Number> B(n);

Container<Number> diag(n - 2), right(n - 2);
diag[0] = ((dX[0] + dX[1]) * (dX[0] + 2 * dX[1])) / dX[1];
for (SizeT i = 1; i < n - 3; ++i) {
diag[i] = 2 * (dX[i] + dX[i+1]);
}
diag[n-3] = ((dX[n-2] + dX[n-3]) * (dX[n-2] + 2 * dX[n-3])) / dX[n-3];
for (SizeT i = 0; i < n - 2; ++i) {
right[i] = 3 * (dY[i+1]/dX[i+1] - dY[i]/dX[i]);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
Container<Number> lower(n), diag(n), upper(n), right(n);
for (SizeT i = 1; i < n - 1; ++i) {
lower[i] = dX[i-1];
diag[i] = 2 * (dX[i-1] + dX[i]);
upper[i] = dX[i];
right[i] = 3 * (dY[i]/dX[i] - dY[i-1]/dX[i-1]);
}

const auto m_1 = dX[1] / diag[0];
diag[1] -= m_1 * (dX[1] - dX[0]*dX[0]/dX[1]);
right[1] -= m_1 * right[0];
for (SizeT i = 2; i < n - 3; ++i) {
const auto m = dX[i] / diag[i-1];
diag[i] -= m * dX[i];
right[i] -= m * right[i-1];
}
const auto m_n_3 = (dX[n-3] - dX[n-2]*dX[n-2]/dX[n-3]) / diag[n-4];
diag[n-3] -= m_n_3 * dX[n-3];
right[n-3] -= m_n_3 * right[n-4];

B[n-2] = right[n-3] / diag[n-3];
for (SizeT i = n - 3; i >= 2; --i) {
B[i] = (right[i-1] - dX[i] * B[i+1]) / diag[i-1];
lower[0] = NAN;
diag[0] = dX[0] - dX[1];
upper[0] = 2*dX[0] + dX[1];
right[0] = dX[0] / (dX[0]+dX[1]) * right[1];
lower[n-1] = 2*dX[n-2] + dX[n-3];
diag[n-1] = dX[n-2] - dX[n-3];
upper[n-1] = NAN;
right[n-1] = dX[n-2] / (dX[n-2]+dX[n-3]) * right[n-2];
const auto C = util::linalg::tridiag_solve(lower, diag, upper, right);
for (SizeT i = 0, index = 0; i < n - 1; ++i) {
coeffs[index++] = (C[i+1] - C[i]) / (3*dX[i]);
coeffs[index++] = C[i];
coeffs[index++] = dY[i]/dX[i] - dX[i] * ((2*C[i] + C[i+1]) / 3);
coeffs[index++] = Y[i];
}
B[1] = (right[0] - (dX[1] - dX[0]*dX[0]/dX[1]) * B[2]) / diag[0];
B[0] = B[1] + dX[0]/dX[1] * (B[1] - B[2]);
B[n-1] = B[n-2] + dX[n-2]/dX[n-3] * (B[n-2] - B[n-3]);

Container<Number> A(n - 1);
for (SizeT i = 1; i < A.size() - 1; ++i) {
A[i] = (B[i+1] - B[i]) / (3*dX[i]);
}
A[0] = A[1];
A[n-2] = A[n-3];
Container<Number> C(n - 1);
for (SizeT i = 0; i < C.size(); ++i) {
C[i] = dY[i]/dX[i] - dX[i] * ((2*B[i] + B[i+1]) / 3);
}
util::spline::merge_coeffs(coeffs, {A, B, C, Y});
},
[&](const typename Types::Periodic&) {
if (n <= 2) {
coeffs = util::spline::simple_spline<Number,Container>(X, Y, 3);
return;
}

const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

Container<Number> B(n);
Container<Number> C(n);
if (n == 3) {
B[0] = 3 * (dY[0]/dX[0] - dY[1]/dX[1]) / (dX[0] + dX[1]);
B[1] = -B[0];
B[2] = B[0];
C[0] = 3 * (dY[0]/dX[0] - dY[1]/dX[1]) / (dX[0] + dX[1]);
C[1] = -C[0];
C[2] = C[0];
} else {
Container<Number> diag(n - 1), right(n - 1);
for (SizeT i = 0; i < n - 2; ++i) {
Expand Down Expand Up @@ -204,170 +181,122 @@ namespace alfi::spline {
diag[n-2] -= last_row[n-3] / diag[n-3] * last_col[n-3];
right[n-2] -= last_row[n-3] / diag[n-3] * right[n-3];

B[n-1] = right[n-2] / diag[n-2];
B[n-2] = (right[n-3] - last_col[n-3] * B[n-1]) / diag[n-3];
C[n-1] = right[n-2] / diag[n-2];
C[n-2] = (right[n-3] - last_col[n-3] * C[n-1]) / diag[n-3];
for (SizeT i = n - 3; i >= 1; --i) {
B[i] = (right[i-1] - dX[i]*B[i+1] - last_col[i-1]*B[n-1]) / diag[i-1];
C[i] = (right[i-1] - dX[i]*C[i+1] - last_col[i-1]*C[n-1]) / diag[i-1];
}
B[0] = B[n-1];
C[0] = C[n-1];
}

Container<Number> A(n - 1);
for (SizeT i = 0; i < A.size(); ++i) {
A[i] = (B[i+1] - B[i]) / (3*dX[i]);
for (SizeT i = 0, index = 0; i < n - 1; ++i) {
coeffs[index++] = (C[i+1] - C[i]) / (3*dX[i]);
coeffs[index++] = C[i];
coeffs[index++] = dY[i]/dX[i] - dX[i] * ((2*C[i] + C[i+1]) / 3);
coeffs[index++] = Y[i];
}
Container<Number> C(n - 1);
for (SizeT i = 0; i < C.size(); ++i) {
C[i] = dY[i]/dX[i] - dX[i] * ((2*B[i] + B[i+1]) / 3);
}
util::spline::merge_coeffs(coeffs, {A, B, C, Y});
},
[&](const typename Types::ParabolicEnds&) {
coeffs = compute_coeffs(X, Y, typename Types::FixedThird{0, 0});
},
[&](const typename Types::Clamped& c) {
const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

Container<Number> B(n);
if (n == 2) {
B[0] = (3*dY[0]/dX[0] - c.df_n - 2*c.df_1) / dX[0];
B[1] = (c.df_1 + 2*c.df_n - 3*dY[0]/dX[0]) / dX[0];
} else if (n == 3) {
B[1] = (3*(dY[1]/dX[1] - dY[0]/dX[0]) + (c.df_1 - c.df_n)) / (dX[0] + dX[1]);
B[0] = (3*(dY[0]/dX[0] - c.df_1)/dX[0] - B[1]) / 2;
B[2] = (3*(c.df_n - dY[1]/dX[1])/dX[1] - B[1]) / 2;
} else {
Container<Number> diag(n - 2), right(n - 2);
diag[0] = 1.5*dX[0] + 2*dX[1];
right[0] = 3*dY[1]/dX[1] - 4.5*dY[0]/dX[0] + 1.5*c.df_1;
for (SizeT i = 1; i < n - 3; ++i) {
diag[i] = 2 * (dX[i] + dX[i+1]);
right[i] = 3 * (dY[i+1]/dX[i+1] - dY[i]/dX[i]);
}
diag[n-3] = 2*dX[n-3] + 1.5*dX[n-2];
right[n-3] = 4.5*dY[n-2]/dX[n-2] - 3*dY[n-3]/dX[n-3] - 1.5*c.df_n;

for (SizeT i = 1; i < n - 2; ++i) {
const auto m = dX[i] / diag[i-1];
diag[i] -= m * dX[i];
right[i] -= m * right[i-1];
}

B[n-2] = right[n-3] / diag[n-3];
for (SizeT i = n - 3; i >= 1; --i) {
B[i] = (right[i-1] - dX[i] * B[i+1]) / diag[i-1];
}
B[n-1] = 1.5*(c.df_n/dX[n-2] - dY[n-2]/dX[n-2]/dX[n-2]) - 0.5*B[n-2];
B[0] = 1.5*(dY[0]/dX[0]/dX[0] - c.df_1/dX[0]) - 0.5*B[1];
}

Container<Number> A(n - 1);
for (SizeT i = 0; i < A.size(); ++i) {
A[i] = (B[i+1] - B[i]) / (3*dX[i]);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
Container<Number> lower(n), diag(n), upper(n), right(n);
for (SizeT i = 1; i < n - 1; ++i) {
lower[i] = dX[i-1];
diag[i] = 2 * (dX[i-1] + dX[i]);
upper[i] = dX[i];
right[i] = 3 * (dY[i]/dX[i] - dY[i-1]/dX[i-1]);
}
Container<Number> C(n - 1);
for (SizeT i = 0; i < C.size(); ++i) {
C[i] = dY[i]/dX[i] - dX[i] * ((2*B[i] + B[i+1]) / 3);
lower[0] = NAN;
diag[0] = 2*dX[0];
upper[0] = dX[0];
right[0] = 3 * (dY[0]/dX[0] - c.df_1);
lower[n-1] = dX[n-2];
diag[n-1] = 2*dX[n-2];
upper[n-1] = NAN;
right[n-1] = 3 * (c.df_n - dY[n-2]/dX[n-2]);
const auto C = util::linalg::tridiag_solve(lower, diag, upper, right);
for (SizeT i = 0, index = 0; i < n - 1; ++i) {
coeffs[index++] = (C[i+1] - C[i]) / (3*dX[i]);
coeffs[index++] = C[i];
coeffs[index++] = dY[i]/dX[i] - dX[i] * ((2*C[i] + C[i+1]) / 3);
coeffs[index++] = Y[i];
}
util::spline::merge_coeffs(coeffs, {A, B, C, Y});
},
[&](const typename Types::FixedSecond& s) {
const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

Container<Number> B(n);
B[0] = s.d2f_1 / 2;
B[n-1] = s.d2f_n / 2;
if (n > 2) {
Container<Number> diag(n - 2), right(n - 2);
for (SizeT i = 0; i < n - 2; ++i) {
diag[i] = 2 * (dX[i] + dX[i+1]);
right[i] = 3 * (dY[i+1]/dX[i+1] - dY[i]/dX[i]);
}
right[0] -= B[0] * dX[0];
right[n-3] -= B[n-1] * dX[n-2];
for (SizeT i = 1; i < n - 2; ++i) {
const auto m = dX[i] / diag[i-1];
diag[i] -= m * dX[i];
right[i] -= m * right[i-1];
}
B[n-2] = right[n-3] / diag[n-3];
for (SizeT i = n - 3; i >= 1; --i) {
B[i] = (right[i-1] - dX[i] * B[i+1]) / diag[i-1];
}
}

Container<Number> A(n - 1);
for (SizeT i = 0; i < A.size(); ++i) {
A[i] = (B[i+1] - B[i]) / (3*dX[i]);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
Container<Number> lower(n), diag(n), upper(n), right(n);
for (SizeT i = 1; i < n - 1; ++i) {
lower[i] = dX[i-1];
diag[i] = 2 * (dX[i-1] + dX[i]);
upper[i] = dX[i];
right[i] = 3 * (dY[i]/dX[i] - dY[i-1]/dX[i-1]);
}
Container<Number> C(n - 1);
for (SizeT i = 0; i < C.size(); ++i) {
C[i] = dY[i]/dX[i] - dX[i] * ((2*B[i] + B[i+1]) / 3);
lower[0] = NAN;
diag[0] = 1;
upper[0] = 0;
right[0] = s.d2f_1 / 2;
lower[n-1] = 0;
diag[n-1] = 1;
upper[n-1] = NAN;
right[n-1] = s.d2f_n / 2;
const auto C = util::linalg::tridiag_solve(lower, diag, upper, right);
for (SizeT i = 0, index = 0; i < n - 1; ++i) {
coeffs[index++] = (C[i+1] - C[i]) / (3*dX[i]);
coeffs[index++] = C[i];
coeffs[index++] = dY[i]/dX[i] - dX[i] * ((2*C[i] + C[i+1]) / 3);
coeffs[index++] = Y[i];
}
util::spline::merge_coeffs(coeffs, {A, B, C, Y});
},
[&](const typename Types::FixedThird& t) {
const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

Container<Number> B(n);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
Container<Number> C(n);
if (n == 2) {
B[0] = -dX[0] * (t.d3f_1 + t.d3f_n) / 8;
B[1] = -B[0];
C[0] = -dX[0] * (t.d3f_1 + t.d3f_n) / 8;
C[1] = -C[0];
} else {
Container<Number> diag(n), right(n);
diag[0] = 1;
Container<Number> lower(n), diag(n), upper(n), right(n);
for (SizeT i = 1; i < n - 1; ++i) {
lower[i] = dX[i-1];
diag[i] = 2 * (dX[i-1] + dX[i]);
}
diag[n-1] = 1;
right[0] = -dX[0] * t.d3f_1 / 2;
for (SizeT i = 1; i < n - 1; ++i) {
upper[i] = dX[i];
right[i] = 3 * (dY[i]/dX[i] - dY[i-1]/dX[i-1]);
}
lower[0] = NAN;
diag[0] = 1;
upper[0] = -1;
right[0] = -dX[0] * t.d3f_1 / 2;
lower[n-1] = -1;
diag[n-1] = 1;
upper[n-1] = NAN;
right[n-1] = dX[n-2] * t.d3f_n / 2;

const auto m_1 = dX[0] / diag[0];
diag[1] -= m_1 * (-1);
right[1] -= m_1 * right[0];
for (SizeT i = 2; i < n - 1; ++i) {
const auto m = dX[i-1] / diag[i-1];
diag[i] -= m * dX[i-1];
right[i] -= m * right[i-1];
}
const auto m_n_1 = (-1) / diag[n-2];
diag[n-1] -= m_n_1 * dX[n-2];
right[n-1] -= m_n_1 * right[n-2];

B[n-1] = right[n-1] / diag[n-1];
for (SizeT i = n - 2; i >= 1; --i) {
B[i] = (right[i] - dX[i] * B[i+1]) / diag[i];
}
B[0] = right[0] + B[1];
C = util::linalg::tridiag_solve<Number,Container>(lower, diag, upper, right);
}

Container<Number> A(n - 1);
for (SizeT i = 1; i < A.size() - 1; ++i) {
A[i] = (B[i+1] - B[i]) / (3*dX[i]);
Container<Number> D(n - 1);
for (SizeT i = 1; i < D.size() - 1; ++i) {
D[i] = (C[i+1] - C[i]) / (3*dX[i]);
}
if (n == 2) {
A[0] = (t.d3f_1 + t.d3f_n) / 12;
D[0] = (t.d3f_1 + t.d3f_n) / 12;
} else {
A[0] = t.d3f_1 / 6;
A[n-2] = t.d3f_n / 6;
D[0] = t.d3f_1 / 6;
D[n-2] = t.d3f_n / 6;
}
Container<Number> C(n - 1);
for (SizeT i = 0; i < C.size(); ++i) {
C[i] = dY[i]/dX[i] - dX[i] * ((2*B[i] + B[i+1]) / 3);
for (SizeT i = 0, index = 0; i < n - 1; ++i) {
coeffs[index++] = D[i];
coeffs[index++] = C[i];
coeffs[index++] = dY[i]/dX[i] - dX[i] * ((2*C[i] + C[i+1]) / 3);
coeffs[index++] = Y[i];
}
util::spline::merge_coeffs(coeffs, {A, B, C, Y});
},
[&](const typename Types::NotAKnotStart&) {
if (n <= 4) {
coeffs = util::spline::simple_spline<Number,Container>(X, Y, 3);
return;
}

const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

{ // first four points
const auto h1 = dX[0], h2 = dX[1];
Expand Down Expand Up @@ -414,7 +343,7 @@ namespace alfi::spline {
return;
}

const Container<Number> dX = util::arrays::diff(X), dY = util::arrays::diff(Y);
const auto dX = util::arrays::diff(X), dY = util::arrays::diff(Y);

{ // last four points
const auto h1 = dX[n-4], h2 = dX[n-3];
Expand Down
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