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Diffstat (limited to 'intern/cycles/util/util_math_matrix.h')
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diff --git a/intern/cycles/util/util_math_matrix.h b/intern/cycles/util/util_math_matrix.h new file mode 100644 index 00000000000..31ea10f18a8 --- /dev/null +++ b/intern/cycles/util/util_math_matrix.h @@ -0,0 +1,379 @@ +/* + * Copyright 2011-2017 Blender Foundation + * + * Licensed under the Apache License, Version 2.0 (the "License"); + * you may not use this file except in compliance with the License. + * You may obtain a copy of the License at + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ + +#ifndef __UTIL_MATH_MATRIX_H__ +#define __UTIL_MATH_MATRIX_H__ + +CCL_NAMESPACE_BEGIN + +#define MAT(A, size, row, col) A[(row)*(size)+(col)] + +/* Variants that use a constant stride on GPUS. */ +#ifdef __KERNEL_GPU__ +#define MATS(A, n, r, c, s) A[((r)*(n)+(c))*(s)] +/* Element access when only the lower-triangular elements are stored. */ +#define MATHS(A, r, c, s) A[((r)*((r)+1)/2+(c))*(s)] +#define VECS(V, i, s) V[(i)*(s)] +#else +#define MATS(A, n, r, c, s) MAT(A, n, r, c) +#define MATHS(A, r, c, s) A[(r)*((r)+1)/2+(c)] +#define VECS(V, i, s) V[i] +#endif + +/* Zeroing helpers. */ + +ccl_device_inline void math_vector_zero(float *v, int n) +{ + for(int i = 0; i < n; i++) + v[i] = 0.0f; +} + +ccl_device_inline void math_matrix_zero(float *A, int n) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MAT(A, n, row, col) = 0.0f; +} + +/* Elementary vector operations. */ + +ccl_device_inline void math_vector_add(float *a, float ccl_restrict_ptr b, int n) +{ + for(int i = 0; i < n; i++) + a[i] += b[i]; +} + +ccl_device_inline void math_vector_mul(float *a, float ccl_restrict_ptr b, int n) +{ + for(int i = 0; i < n; i++) + a[i] *= b[i]; +} + +ccl_device_inline void math_vector_mul_strided(ccl_global float *a, float ccl_restrict_ptr b, int astride, int n) +{ + for(int i = 0; i < n; i++) + a[i*astride] *= b[i]; +} + +ccl_device_inline void math_vector_scale(float *a, float b, int n) +{ + for(int i = 0; i < n; i++) + a[i] *= b; +} + +ccl_device_inline void math_vector_max(float *a, float ccl_restrict_ptr b, int n) +{ + for(int i = 0; i < n; i++) + a[i] = max(a[i], b[i]); +} + +ccl_device_inline void math_vec3_add(float3 *v, int n, float *x, float3 w) +{ + for(int i = 0; i < n; i++) + v[i] += w*x[i]; +} + +ccl_device_inline void math_vec3_add_strided(ccl_global float3 *v, int n, float *x, float3 w, int stride) +{ + for(int i = 0; i < n; i++) + v[i*stride] += w*x[i]; +} + +/* Elementary matrix operations. + * Note: TriMatrix refers to a square matrix that is symmetric, and therefore its upper-triangular part isn't stored. */ + +ccl_device_inline void math_trimatrix_add_diagonal(ccl_global float *A, int n, float val, int stride) +{ + for(int row = 0; row < n; row++) + MATHS(A, row, row, stride) += val; +} + +/* Add Gramian matrix of v to A. + * The Gramian matrix of v is vt*v, so element (i,j) is v[i]*v[j]. */ +ccl_device_inline void math_matrix_add_gramian(float *A, + int n, + float ccl_restrict_ptr v, + float weight) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MAT(A, n, row, col) += v[row]*v[col]*weight; +} + +/* Add Gramian matrix of v to A. + * The Gramian matrix of v is vt*v, so element (i,j) is v[i]*v[j]. */ +ccl_device_inline void math_trimatrix_add_gramian_strided(ccl_global float *A, + int n, + float ccl_restrict_ptr v, + float weight, + int stride) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MATHS(A, row, col, stride) += v[row]*v[col]*weight; +} + +/* Transpose matrix A inplace. */ +ccl_device_inline void math_matrix_transpose(ccl_global float *A, int n, int stride) +{ + for(int i = 0; i < n; i++) { + for(int j = 0; j < i; j++) { + float temp = MATS(A, n, i, j, stride); + MATS(A, n, i, j, stride) = MATS(A, n, j, i, stride); + MATS(A, n, j, i, stride) = temp; + } + } +} + + + + +/* Solvers for matrix problems */ + +/* In-place Cholesky-Banachiewicz decomposition of the square, positive-definite matrix A + * into a lower triangular matrix L so that A = L*L^T. A is being overwritten by L. + * Also, only the lower triangular part of A is ever accessed. */ +ccl_device void math_trimatrix_cholesky(ccl_global float *A, int n, int stride) +{ + for(int row = 0; row < n; row++) { + for(int col = 0; col <= row; col++) { + float sum_col = MATHS(A, row, col, stride); + for(int k = 0; k < col; k++) { + sum_col -= MATHS(A, row, k, stride) * MATHS(A, col, k, stride); + } + if(row == col) { + sum_col = sqrtf(max(sum_col, 0.0f)); + } + else { + sum_col /= MATHS(A, col, col, stride); + } + MATHS(A, row, col, stride) = sum_col; + } + } +} + +/* Solve A*S=y for S given A and y, where A is symmetrical positive-semidefinite and both inputs are destroyed in the process. + * + * We can apply Cholesky decomposition to find a lower triangular L so that L*Lt = A. + * With that we get (L*Lt)*S = L*(Lt*S) = L*b = y, defining b as Lt*S. + * Since L is lower triangular, finding b is relatively easy since y is known. + * Then, the remaining problem is Lt*S = b, which again can be solved easily. + * + * This is useful for solving the normal equation S=inv(Xt*W*X)*Xt*W*y, since Xt*W*X is + * symmetrical positive-semidefinite by construction, so we can just use this function with A=Xt*W*X and y=Xt*W*y. */ +ccl_device_inline void math_trimatrix_vec3_solve(ccl_global float *A, ccl_global float3 *y, int n, int stride) +{ + math_trimatrix_add_diagonal(A, n, 1e-4f, stride); /* Improve the numerical stability. */ + math_trimatrix_cholesky(A, n, stride); /* Replace A with L so that L*Lt = A. */ + + /* Use forward substitution to solve L*b = y, replacing y by b. */ + for(int row = 0; row < n; row++) { + float3 sum = VECS(y, row, stride); + for(int col = 0; col < row; col++) + sum -= MATHS(A, row, col, stride) * VECS(y, col, stride); + VECS(y, row, stride) = sum / MATHS(A, row, row, stride); + } + + /* Use backward substitution to solve Lt*S = b, replacing b by S. */ + for(int row = n-1; row >= 0; row--) { + float3 sum = VECS(y, row, stride); + for(int col = row+1; col < n; col++) + sum -= MATHS(A, col, row, stride) * VECS(y, col, stride); + VECS(y, row, stride) = sum / MATHS(A, row, row, stride); + } +} + + + + + +/* Perform the Jacobi Eigenvalue Methon on matrix A. + * A is assumed to be a symmetrical matrix, therefore only the lower-triangular part is ever accessed. + * The algorithm overwrites the contents of A. + * + * After returning, A will be overwritten with D, which is (almost) diagonal, + * and V will contain the eigenvectors of the original A in its rows (!), + * so that A = V^T*D*V. Therefore, the diagonal elements of D are the (sorted) eigenvalues of A. + */ +ccl_device void math_matrix_jacobi_eigendecomposition(float *A, ccl_global float *V, int n, int v_stride) +{ + const float singular_epsilon = 1e-9f; + + for (int row = 0; row < n; row++) + for (int col = 0; col < n; col++) + MATS(V, n, row, col, v_stride) = (col == row) ? 1.0f : 0.0f; + + for (int sweep = 0; sweep < 8; sweep++) { + float off_diagonal = 0.0f; + for (int row = 1; row < n; row++) + for (int col = 0; col < row; col++) + off_diagonal += fabsf(MAT(A, n, row, col)); + if (off_diagonal < 1e-7f) { + /* The matrix has nearly reached diagonal form. + * Since the eigenvalues are only used to determine truncation, their exact values aren't required - a relative error of a few ULPs won't matter at all. */ + break; + } + + /* Set the threshold for the small element rotation skip in the first sweep: + * Skip all elements that are less than a tenth of the average off-diagonal element. */ + float threshold = 0.2f*off_diagonal / (n*n); + + for(int row = 1; row < n; row++) { + for(int col = 0; col < row; col++) { + /* Perform a Jacobi rotation on this element that reduces it to zero. */ + float element = MAT(A, n, row, col); + float abs_element = fabsf(element); + + /* If we're in a later sweep and the element already is very small, just set it to zero and skip the rotation. */ + if (sweep > 3 && abs_element <= singular_epsilon*fabsf(MAT(A, n, row, row)) && abs_element <= singular_epsilon*fabsf(MAT(A, n, col, col))) { + MAT(A, n, row, col) = 0.0f; + continue; + } + + if(element == 0.0f) { + continue; + } + + /* If we're in one of the first sweeps and the element is smaller than the threshold, skip it. */ + if(sweep < 3 && (abs_element < threshold)) { + continue; + } + + /* Determine rotation: The rotation is characterized by its angle phi - or, in the actual implementation, sin(phi) and cos(phi). + * To find those, we first compute their ratio - that might be unstable if the angle approaches 90°, so there's a fallback for that case. + * Then, we compute sin(phi) and cos(phi) themselves. */ + float singular_diff = MAT(A, n, row, row) - MAT(A, n, col, col); + float ratio; + if (abs_element > singular_epsilon*fabsf(singular_diff)) { + float cot_2phi = 0.5f*singular_diff / element; + ratio = 1.0f / (fabsf(cot_2phi) + sqrtf(1.0f + cot_2phi*cot_2phi)); + if (cot_2phi < 0.0f) ratio = -ratio; /* Copy sign. */ + } + else { + ratio = element / singular_diff; + } + + float c = 1.0f / sqrtf(1.0f + ratio*ratio); + float s = ratio*c; + /* To improve numerical stability by avoiding cancellation, the update equations are reformulized to use sin(phi) and tan(phi/2) instead. */ + float tan_phi_2 = s / (1.0f + c); + + /* Update the singular values in the diagonal. */ + float singular_delta = ratio*element; + MAT(A, n, row, row) += singular_delta; + MAT(A, n, col, col) -= singular_delta; + + /* Set the element itself to zero. */ + MAT(A, n, row, col) = 0.0f; + + /* Perform the actual rotations on the matrices. */ +#define ROT(M, r1, c1, r2, c2, stride) \ + { \ + float M1 = MATS(M, n, r1, c1, stride); \ + float M2 = MATS(M, n, r2, c2, stride); \ + MATS(M, n, r1, c1, stride) -= s*(M2 + tan_phi_2*M1); \ + MATS(M, n, r2, c2, stride) += s*(M1 - tan_phi_2*M2); \ + } + + /* Split into three parts to ensure correct accesses since we only store the lower-triangular part of A. */ + for(int i = 0 ; i < col; i++) ROT(A, col, i, row, i, 1); + for(int i = col+1; i < row; i++) ROT(A, i, col, row, i, 1); + for(int i = row+1; i < n ; i++) ROT(A, i, col, i, row, 1); + + for(int i = 0 ; i < n ; i++) ROT(V, col, i, row, i, v_stride); +#undef ROT + } + } + } + + /* Sort eigenvalues and the associated eigenvectors. */ + for (int i = 0; i < n - 1; i++) { + float v = MAT(A, n, i, i); + int k = i; + for (int j = i; j < n; j++) { + if (MAT(A, n, j, j) >= v) { + v = MAT(A, n, j, j); + k = j; + } + } + if (k != i) { + /* Swap eigenvalues. */ + MAT(A, n, k, k) = MAT(A, n, i, i); + MAT(A, n, i, i) = v; + /* Swap eigenvectors. */ + for (int j = 0; j < n; j++) { + float v = MATS(V, n, i, j, v_stride); + MATS(V, n, i, j, v_stride) = MATS(V, n, k, j, v_stride); + MATS(V, n, k, j, v_stride) = v; + } + } + } +} + +#ifdef __KERNEL_SSE3__ + +ccl_device_inline void math_vector_zero_sse(__m128 *A, int n) +{ + for(int i = 0; i < n; i++) + A[i] = _mm_setzero_ps(); +} +ccl_device_inline void math_matrix_zero_sse(__m128 *A, int n) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MAT(A, n, row, col) = _mm_setzero_ps(); +} + +/* Add Gramian matrix of v to A. + * The Gramian matrix of v is v^T*v, so element (i,j) is v[i]*v[j]. */ +ccl_device_inline void math_matrix_add_gramian_sse(__m128 *A, int n, __m128 ccl_restrict_ptr v, __m128 weight) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MAT(A, n, row, col) = _mm_add_ps(MAT(A, n, row, col), _mm_mul_ps(_mm_mul_ps(v[row], v[col]), weight)); +} + +ccl_device_inline void math_vector_add_sse(__m128 *V, int n, __m128 ccl_restrict_ptr a) +{ + for(int i = 0; i < n; i++) + V[i] = _mm_add_ps(V[i], a[i]); +} + +ccl_device_inline void math_vector_mul_sse(__m128 *V, int n, __m128 ccl_restrict_ptr a) +{ + for(int i = 0; i < n; i++) + V[i] = _mm_mul_ps(V[i], a[i]); +} + +ccl_device_inline void math_vector_max_sse(__m128 *a, __m128 ccl_restrict_ptr b, int n) +{ + for(int i = 0; i < n; i++) + a[i] = _mm_max_ps(a[i], b[i]); +} + +ccl_device_inline void math_matrix_hsum(float *A, int n, __m128 ccl_restrict_ptr B) +{ + for(int row = 0; row < n; row++) + for(int col = 0; col <= row; col++) + MAT(A, n, row, col) = _mm_hsum_ss(MAT(B, n, row, col)); +} +#endif + +#undef MAT + +CCL_NAMESPACE_END + +#endif /* __UTIL_MATH_MATRIX_H__ */ |