forked from bartvdbraak/blender
449 lines
15 KiB
C
449 lines
15 KiB
C
/*
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* Copyright 2011-2017 Blender Foundation
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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#ifndef __UTIL_MATH_MATRIX_H__
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#define __UTIL_MATH_MATRIX_H__
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CCL_NAMESPACE_BEGIN
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#define MAT(A, size, row, col) A[(row) * (size) + (col)]
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/* Variants that use a constant stride on GPUS. */
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#ifdef __KERNEL_GPU__
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# define MATS(A, n, r, c, s) A[((r) * (n) + (c)) * (s)]
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/* Element access when only the lower-triangular elements are stored. */
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# define MATHS(A, r, c, s) A[((r) * ((r) + 1) / 2 + (c)) * (s)]
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# define VECS(V, i, s) V[(i) * (s)]
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#else
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# define MATS(A, n, r, c, s) MAT(A, n, r, c)
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# define MATHS(A, r, c, s) A[(r) * ((r) + 1) / 2 + (c)]
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# define VECS(V, i, s) V[i]
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#endif
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/* Zeroing helpers. */
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ccl_device_inline void math_vector_zero(float *v, int n)
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{
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for (int i = 0; i < n; i++) {
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v[i] = 0.0f;
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}
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}
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ccl_device_inline void math_matrix_zero(float *A, int n)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MAT(A, n, row, col) = 0.0f;
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}
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}
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}
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/* Elementary vector operations. */
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ccl_device_inline void math_vector_add(float *a, const float *ccl_restrict b, int n)
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{
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for (int i = 0; i < n; i++) {
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a[i] += b[i];
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}
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}
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ccl_device_inline void math_vector_mul(float *a, const float *ccl_restrict b, int n)
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{
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for (int i = 0; i < n; i++) {
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a[i] *= b[i];
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}
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}
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ccl_device_inline void math_vector_mul_strided(ccl_global float *a,
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const float *ccl_restrict b,
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int astride,
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int n)
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{
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for (int i = 0; i < n; i++) {
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a[i * astride] *= b[i];
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}
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}
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ccl_device_inline void math_vector_scale(float *a, float b, int n)
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{
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for (int i = 0; i < n; i++) {
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a[i] *= b;
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}
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}
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ccl_device_inline void math_vector_max(float *a, const float *ccl_restrict b, int n)
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{
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for (int i = 0; i < n; i++) {
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a[i] = max(a[i], b[i]);
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}
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}
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ccl_device_inline void math_vec3_add(float3 *v, int n, float *x, float3 w)
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{
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for (int i = 0; i < n; i++) {
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v[i] += w * x[i];
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}
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}
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ccl_device_inline void math_vec3_add_strided(
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ccl_global float3 *v, int n, float *x, float3 w, int stride)
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{
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for (int i = 0; i < n; i++) {
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ccl_global float *elem = (ccl_global float *)(v + i * stride);
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atomic_add_and_fetch_float(elem + 0, w.x * x[i]);
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atomic_add_and_fetch_float(elem + 1, w.y * x[i]);
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atomic_add_and_fetch_float(elem + 2, w.z * x[i]);
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}
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}
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/* Elementary matrix operations.
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* Note: TriMatrix refers to a square matrix that is symmetric,
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* and therefore its upper-triangular part isn't stored. */
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ccl_device_inline void math_trimatrix_add_diagonal(ccl_global float *A,
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int n,
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float val,
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int stride)
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{
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for (int row = 0; row < n; row++) {
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MATHS(A, row, row, stride) += val;
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}
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}
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/* Add Gramian matrix of v to A.
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* The Gramian matrix of v is vt*v, so element (i,j) is v[i]*v[j]. */
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ccl_device_inline void math_matrix_add_gramian(float *A,
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int n,
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const float *ccl_restrict v,
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float weight)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MAT(A, n, row, col) += v[row] * v[col] * weight;
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}
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}
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}
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/* Add Gramian matrix of v to A.
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* The Gramian matrix of v is vt*v, so element (i,j) is v[i]*v[j]. */
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ccl_device_inline void math_trimatrix_add_gramian_strided(
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ccl_global float *A, int n, const float *ccl_restrict v, float weight, int stride)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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atomic_add_and_fetch_float(&MATHS(A, row, col, stride), v[row] * v[col] * weight);
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}
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}
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}
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ccl_device_inline void math_trimatrix_add_gramian(ccl_global float *A,
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int n,
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const float *ccl_restrict v,
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float weight)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MATHS(A, row, col, 1) += v[row] * v[col] * weight;
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}
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}
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}
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/* Transpose matrix A inplace. */
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ccl_device_inline void math_matrix_transpose(ccl_global float *A, int n, int stride)
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{
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for (int i = 0; i < n; i++) {
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for (int j = 0; j < i; j++) {
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float temp = MATS(A, n, i, j, stride);
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MATS(A, n, i, j, stride) = MATS(A, n, j, i, stride);
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MATS(A, n, j, i, stride) = temp;
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}
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}
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}
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/* Solvers for matrix problems */
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/* In-place Cholesky-Banachiewicz decomposition of the square, positive-definite matrix A
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* into a lower triangular matrix L so that A = L*L^T. A is being overwritten by L.
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* Also, only the lower triangular part of A is ever accessed. */
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ccl_device void math_trimatrix_cholesky(ccl_global float *A, int n, int stride)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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float sum_col = MATHS(A, row, col, stride);
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for (int k = 0; k < col; k++) {
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sum_col -= MATHS(A, row, k, stride) * MATHS(A, col, k, stride);
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}
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if (row == col) {
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sum_col = sqrtf(max(sum_col, 0.0f));
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}
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else {
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sum_col /= MATHS(A, col, col, stride);
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}
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MATHS(A, row, col, stride) = sum_col;
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}
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}
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}
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/* Solve A*S=y for S given A and y,
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* where A is symmetrical positive-semi-definite and both inputs are destroyed in the process.
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*
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* We can apply Cholesky decomposition to find a lower triangular L so that L*Lt = A.
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* With that we get (L*Lt)*S = L*(Lt*S) = L*b = y, defining b as Lt*S.
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* Since L is lower triangular, finding b is relatively easy since y is known.
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* Then, the remaining problem is Lt*S = b, which again can be solved easily.
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*
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* This is useful for solving the normal equation S=inv(Xt*W*X)*Xt*W*y, since Xt*W*X is
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* symmetrical positive-semidefinite by construction,
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* so we can just use this function with A=Xt*W*X and y=Xt*W*y. */
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ccl_device_inline void math_trimatrix_vec3_solve(ccl_global float *A,
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ccl_global float3 *y,
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int n,
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int stride)
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{
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/* Since the first entry of the design row is always 1, the upper-left element of XtWX is a good
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* heuristic for the amount of pixels considered (with weighting),
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* therefore the amount of correction is scaled based on it. */
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math_trimatrix_add_diagonal(A, n, 3e-7f * A[0], stride); /* Improve the numerical stability. */
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math_trimatrix_cholesky(A, n, stride); /* Replace A with L so that L*Lt = A. */
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/* Use forward substitution to solve L*b = y, replacing y by b. */
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for (int row = 0; row < n; row++) {
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float3 sum = VECS(y, row, stride);
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for (int col = 0; col < row; col++)
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sum -= MATHS(A, row, col, stride) * VECS(y, col, stride);
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VECS(y, row, stride) = sum / MATHS(A, row, row, stride);
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}
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/* Use backward substitution to solve Lt*S = b, replacing b by S. */
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for (int row = n - 1; row >= 0; row--) {
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float3 sum = VECS(y, row, stride);
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for (int col = row + 1; col < n; col++)
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sum -= MATHS(A, col, row, stride) * VECS(y, col, stride);
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VECS(y, row, stride) = sum / MATHS(A, row, row, stride);
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}
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}
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/* Perform the Jacobi Eigenvalue Method on matrix A.
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* A is assumed to be a symmetrical matrix, therefore only the lower-triangular part is ever
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* accessed. The algorithm overwrites the contents of A.
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*
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* After returning, A will be overwritten with D, which is (almost) diagonal,
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* and V will contain the eigenvectors of the original A in its rows (!),
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* so that A = V^T*D*V. Therefore, the diagonal elements of D are the (sorted) eigenvalues of A.
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*/
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ccl_device void math_matrix_jacobi_eigendecomposition(float *A,
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ccl_global float *V,
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int n,
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int v_stride)
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{
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const float singular_epsilon = 1e-9f;
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for (int row = 0; row < n; row++) {
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for (int col = 0; col < n; col++) {
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MATS(V, n, row, col, v_stride) = (col == row) ? 1.0f : 0.0f;
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}
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}
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for (int sweep = 0; sweep < 8; sweep++) {
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float off_diagonal = 0.0f;
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for (int row = 1; row < n; row++) {
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for (int col = 0; col < row; col++) {
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off_diagonal += fabsf(MAT(A, n, row, col));
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}
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}
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if (off_diagonal < 1e-7f) {
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/* The matrix has nearly reached diagonal form.
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* Since the eigenvalues are only used to determine truncation, their exact values aren't
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* required - a relative error of a few ULPs won't matter at all. */
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break;
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}
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/* Set the threshold for the small element rotation skip in the first sweep:
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* Skip all elements that are less than a tenth of the average off-diagonal element. */
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float threshold = 0.2f * off_diagonal / (n * n);
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for (int row = 1; row < n; row++) {
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for (int col = 0; col < row; col++) {
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/* Perform a Jacobi rotation on this element that reduces it to zero. */
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float element = MAT(A, n, row, col);
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float abs_element = fabsf(element);
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/* If we're in a later sweep and the element already is very small,
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* just set it to zero and skip the rotation. */
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if (sweep > 3 && abs_element <= singular_epsilon * fabsf(MAT(A, n, row, row)) &&
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abs_element <= singular_epsilon * fabsf(MAT(A, n, col, col))) {
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MAT(A, n, row, col) = 0.0f;
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continue;
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}
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if (element == 0.0f) {
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continue;
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}
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/* If we're in one of the first sweeps and the element is smaller than the threshold,
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* skip it. */
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if (sweep < 3 && (abs_element < threshold)) {
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continue;
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}
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/* Determine rotation: The rotation is characterized by its angle phi - or,
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* in the actual implementation, sin(phi) and cos(phi).
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* To find those, we first compute their ratio - that might be unstable if the angle
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* approaches 90°, so there's a fallback for that case.
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* Then, we compute sin(phi) and cos(phi) themselves. */
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float singular_diff = MAT(A, n, row, row) - MAT(A, n, col, col);
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float ratio;
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if (abs_element > singular_epsilon * fabsf(singular_diff)) {
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float cot_2phi = 0.5f * singular_diff / element;
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ratio = 1.0f / (fabsf(cot_2phi) + sqrtf(1.0f + cot_2phi * cot_2phi));
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if (cot_2phi < 0.0f)
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ratio = -ratio; /* Copy sign. */
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}
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else {
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ratio = element / singular_diff;
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}
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float c = 1.0f / sqrtf(1.0f + ratio * ratio);
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float s = ratio * c;
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/* To improve numerical stability by avoiding cancellation, the update equations are
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* reformulized to use sin(phi) and tan(phi/2) instead. */
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float tan_phi_2 = s / (1.0f + c);
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/* Update the singular values in the diagonal. */
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float singular_delta = ratio * element;
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MAT(A, n, row, row) += singular_delta;
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MAT(A, n, col, col) -= singular_delta;
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/* Set the element itself to zero. */
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MAT(A, n, row, col) = 0.0f;
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/* Perform the actual rotations on the matrices. */
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#define ROT(M, r1, c1, r2, c2, stride) \
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{ \
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float M1 = MATS(M, n, r1, c1, stride); \
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float M2 = MATS(M, n, r2, c2, stride); \
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MATS(M, n, r1, c1, stride) -= s * (M2 + tan_phi_2 * M1); \
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MATS(M, n, r2, c2, stride) += s * (M1 - tan_phi_2 * M2); \
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}
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/* Split into three parts to ensure correct accesses since we only store the
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* lower-triangular part of A. */
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for (int i = 0; i < col; i++)
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ROT(A, col, i, row, i, 1);
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for (int i = col + 1; i < row; i++)
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ROT(A, i, col, row, i, 1);
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for (int i = row + 1; i < n; i++)
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ROT(A, i, col, i, row, 1);
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for (int i = 0; i < n; i++)
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ROT(V, col, i, row, i, v_stride);
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#undef ROT
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}
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}
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}
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/* Sort eigenvalues and the associated eigenvectors. */
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for (int i = 0; i < n - 1; i++) {
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float v = MAT(A, n, i, i);
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int k = i;
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for (int j = i; j < n; j++) {
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if (MAT(A, n, j, j) >= v) {
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v = MAT(A, n, j, j);
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k = j;
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}
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}
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if (k != i) {
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/* Swap eigenvalues. */
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MAT(A, n, k, k) = MAT(A, n, i, i);
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MAT(A, n, i, i) = v;
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/* Swap eigenvectors. */
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for (int j = 0; j < n; j++) {
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float v = MATS(V, n, i, j, v_stride);
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MATS(V, n, i, j, v_stride) = MATS(V, n, k, j, v_stride);
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MATS(V, n, k, j, v_stride) = v;
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}
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}
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}
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}
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#ifdef __KERNEL_SSE3__
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ccl_device_inline void math_vector_zero_sse(float4 *A, int n)
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{
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for (int i = 0; i < n; i++) {
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A[i] = make_float4(0.0f);
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}
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}
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ccl_device_inline void math_matrix_zero_sse(float4 *A, int n)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MAT(A, n, row, col) = make_float4(0.0f);
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}
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}
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}
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/* Add Gramian matrix of v to A.
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* The Gramian matrix of v is v^T*v, so element (i,j) is v[i]*v[j]. */
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ccl_device_inline void math_matrix_add_gramian_sse(float4 *A,
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int n,
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const float4 *ccl_restrict v,
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float4 weight)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MAT(A, n, row, col) = MAT(A, n, row, col) + v[row] * v[col] * weight;
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}
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}
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}
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ccl_device_inline void math_vector_add_sse(float4 *V, int n, const float4 *ccl_restrict a)
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{
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for (int i = 0; i < n; i++) {
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V[i] += a[i];
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}
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}
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ccl_device_inline void math_vector_mul_sse(float4 *V, int n, const float4 *ccl_restrict a)
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{
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for (int i = 0; i < n; i++) {
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V[i] *= a[i];
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}
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}
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ccl_device_inline void math_vector_max_sse(float4 *a, const float4 *ccl_restrict b, int n)
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{
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for (int i = 0; i < n; i++) {
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a[i] = max(a[i], b[i]);
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}
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}
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ccl_device_inline void math_matrix_hsum(float *A, int n, const float4 *ccl_restrict B)
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{
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for (int row = 0; row < n; row++) {
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for (int col = 0; col <= row; col++) {
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MAT(A, n, row, col) = reduce_add(MAT(B, n, row, col))[0];
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}
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}
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}
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#endif
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#undef MAT
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CCL_NAMESPACE_END
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#endif /* __UTIL_MATH_MATRIX_H__ */
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