blender/intern/cycles/util/util_math_matrix.h

/*
 * 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, const float *ccl_restrict b, int n)
{
	for(int i = 0; i < n; i++) {
		a[i] += b[i];
	}
}

ccl_device_inline void math_vector_mul(float *a, const float *ccl_restrict 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, const float *ccl_restrict 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, const float *ccl_restrict 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++) {
		ccl_global float *elem = (ccl_global float*) (v + i*stride);
		atomic_add_and_fetch_float(elem+0, w.x*x[i]);
		atomic_add_and_fetch_float(elem+1, w.y*x[i]);
		atomic_add_and_fetch_float(elem+2, w.z*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,
                                                  const float *ccl_restrict 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,
                                                          const float *ccl_restrict v,
                                                          float weight,
                                                          int stride)
{
	for(int row = 0; row < n; row++) {
		for(int col = 0; col <= row; col++) {
			atomic_add_and_fetch_float(&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)
{
	/* Since the first entry of the design row is always 1, the upper-left element of XtWX is a good
	 * heuristic for the amount of pixels considered (with weighting), therefore the amount of correction
	 * is scaled based on it. */
	math_trimatrix_add_diagonal(A, n, 3e-7f*A[0], 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(float4 *A, int n)
{
	for(int i = 0; i < n; i++) {
		A[i] = make_float4(0.0f);
	}
}

ccl_device_inline void math_matrix_zero_sse(float4 *A, int n)
{
	for(int row = 0; row < n; row++) {
		for(int col = 0; col <= row; col++) {
			MAT(A, n, row, col) = make_float4(0.0f);
		}
	}
}

/* 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(float4 *A, int n, const float4 *ccl_restrict v, float4 weight)
{
	for(int row = 0; row < n; row++) {
		for(int col = 0; col <= row; col++) {
			MAT(A, n, row, col) = MAT(A, n, row, col) + v[row] * v[col] * weight;
		}
	}
}

ccl_device_inline void math_vector_add_sse(float4 *V, int n, const float4 *ccl_restrict a)
{
	for(int i = 0; i < n; i++) {
		V[i] += a[i];
	}
}

ccl_device_inline void math_vector_mul_sse(float4 *V, int n, const float4 *ccl_restrict a)
{
	for(int i = 0; i < n; i++) {
		V[i] *= a[i];
	}
}

ccl_device_inline void math_vector_max_sse(float4 *a, const float4 *ccl_restrict b, int n)
{
	for(int i = 0; i < n; i++) {
		a[i] = max(a[i], b[i]);
	}
}

ccl_device_inline void math_matrix_hsum(float *A, int n, const float4 *ccl_restrict B)
{
	for(int row = 0; row < n; row++) {
		for(int col = 0; col <= row; col++) {
			MAT(A, n, row, col) = reduce_add(MAT(B, n, row, col))[0];
		}
	}
}
#endif

#undef MAT

CCL_NAMESPACE_END

#endif  /* __UTIL_MATH_MATRIX_H__ */