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272 lines
9.7 KiB
272 lines
9.7 KiB
3 years ago
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/*
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* Copyright 2022 Collabora, Ltd.
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*
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* Permission is hereby granted, free of charge, to any person obtaining
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* a copy of this software and associated documentation files (the
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* "Software"), to deal in the Software without restriction, including
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* without limitation the rights to use, copy, modify, merge, publish,
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* distribute, sublicense, and/or sell copies of the Software, and to
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* permit persons to whom the Software is furnished to do so, subject to
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* the following conditions:
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*
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* The above copyright notice and this permission notice (including the
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* next paragraph) shall be included in all copies or substantial
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* portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
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* BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
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* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
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* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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* SOFTWARE.
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*/
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#include "config.h"
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#include <math.h>
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#include <libweston/matrix.h>
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#include "weston-test-client-helper.h"
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/*
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* A helper to lay out a matrix in the natural writing order in code
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* instead of needing to transpose in your mind every time you read it.
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* The matrix is laid out as written:
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* ⎡ a11 a12 a13 a14 ⎤
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* ⎢ a21 a22 a23 a24 ⎥
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* ⎢ a31 a32 a33 a34 ⎥
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* ⎣ a41 a42 a43 a44 ⎦
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* where the first digit is row and the second digit is column.
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*
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* The type field is set to the most pessimistic case possible so that if
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* weston_matrix_invert() ever gets special-case code paths, we don't take
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* them.
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*/
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#define MAT(a11, a12, a13, a14, \
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a21, a22, a23, a24, \
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a31, a32, a33, a34, \
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a41, a42, a43, a44) ((struct weston_matrix) \
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{ \
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.d[0] = a11, .d[4] = a12, .d[ 8] = a13, .d[12] = a14, \
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.d[1] = a21, .d[5] = a22, .d[ 9] = a23, .d[13] = a24, \
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.d[2] = a31, .d[6] = a32, .d[10] = a33, .d[14] = a34, \
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.d[3] = a41, .d[7] = a42, .d[11] = a43, .d[15] = a44, \
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.type = WESTON_MATRIX_TRANSFORM_TRANSLATE | \
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WESTON_MATRIX_TRANSFORM_SCALE | \
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WESTON_MATRIX_TRANSFORM_ROTATE | \
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WESTON_MATRIX_TRANSFORM_OTHER, \
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})
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static const struct weston_matrix IDENTITY =
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MAT(1, 0, 0, 0,
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0, 1, 0, 0,
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0, 0, 1, 0,
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0, 0, 0, 1);
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static void
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subtract_matrix(struct weston_matrix *from, const struct weston_matrix *what)
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{
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unsigned i;
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for (i = 0; i < ARRAY_LENGTH(from->d); i++)
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from->d[i] -= what->d[i];
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}
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static void
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print_matrix(const struct weston_matrix *m)
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{
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unsigned r, c;
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for (r = 0; r < 4; ++r) {
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for (c = 0; c < 4; ++c)
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testlog(" %14.6e", m->d[r + c * 4]);
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testlog("\n");
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}
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}
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/*
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* Matrix infinity norm
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* http://www.netlib.org/lapack/lug/node75.html
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*/
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static double
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matrix_inf_norm(const struct weston_matrix *mat)
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{
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unsigned row;
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double infnorm = -1.0;
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for (row = 0; row < 4; row++) {
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unsigned col;
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double sum = 0.0;
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for (col = 0; col < 4; col++)
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sum += fabs(mat->d[col * 4 + row]);
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if (infnorm < sum)
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infnorm = sum;
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}
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return infnorm;
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}
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struct test_matrix {
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/* the matrix to test */
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struct weston_matrix M;
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/*
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* Residual error limit; inf norm(M * inv(M) - I) < err_limit
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* The residual error as calculated here represents the relative
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* error added by transforming a vector with inv(M).
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*
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* Since weston_matrix stores the inverse matrix in 32-bit floats,
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* that limits the precision considerably.
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*/
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double err_limit;
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};
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static const struct test_matrix matrices[] = {
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/* A very trivial case. */
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{
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.M = MAT(1, 0, 0, 0,
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0, 2, 0, 0,
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0, 0, 3, 0,
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0, 0, 0, 4),
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.err_limit = 0.0,
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},
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/*
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* A very likely case in a compositor, being a matrix applying
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* just a translation. Surprisingly, fourbyfour-analyze says:
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*
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* -------------------------------------------------------------------
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* $ ./fourbyfour-analyse 1 0 0 1980 0 1 0 1080
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* Your input matrix A is
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* 1 0 0 1980
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* 0 1 0 1080
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* 0 0 1 0
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* 0 0 0 1
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*
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* The singular values of A are: 2255.39, 1, 1, 0.000443382
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* The condition number according to 2-norm of A is 5.087e+06.
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*
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* This means that if you were to solve the linear system Ax=b for vector x,
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* in the worst case you would lose 6.7 digits (22.3 bits) of precision.
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* The condition number is how much errors in vector b would be amplified
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* when solving x even with infinite computational precision.
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*
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* Compare this to the precision of vectors b and x:
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*
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* - Single precision floating point has 7.2 digits (24 bits) of precision,
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* leaving your result with no correct digits.
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* Single precision, matrix A has rank 3 which means that the solution space
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* for x has 1 dimension and therefore has many solutions.
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*
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* - Double precision floating point has 16.0 digits (53 bits) of precision,
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* leaving your result with 9.2 correct digits (30 correct bits).
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* Double precision, matrix A has full rank which means the solution x is
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* unique.
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*
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* NOTE! The above gives you only an upper limit on errors.
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* If the upper limit is low, you can be confident of your computations. But,
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* if the upper limit is high, it does not necessarily imply that your
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* computations will be doomed.
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* -------------------------------------------------------------------
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*
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* This is one example where the condition number is highly pessimistic,
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* while the actual inversion results in no error at all.
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*
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* https://gitlab.freedesktop.org/pq/fourbyfour
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*/
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{
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.M = MAT(1, 0, 0, 1980,
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0, 1, 0, 1080,
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0, 0, 1, 0,
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0, 0, 0, 1),
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.err_limit = 0.0,
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},
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/*
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* The following matrices have been generated with
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* fourbyfour-generate using parameters out of a hat as listed below.
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*
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* If you want to verify the matrices in Octave, type this:
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* M = [ <paste the series of numbers> ]
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* mat = reshape(M, 4, 4)
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* det(mat)
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* cond(mat)
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*/
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/* cond = 1e3 */
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{
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.M = MAT(-4.12798022231678357619e-02, -7.93301899046665176529e-02, 2.49367040174418935772e-01, -2.22400462135059429070e-01,
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2.02416121867255743849e-01, -2.25754422240346010187e-02, -2.91283152417864787953e-01, 1.49354988316431153139e-01,
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6.18473094065821293874e-01, 5.81511312950217934548e-02, -1.18363610818063924590e+00, 8.00087538947595322547e-01,
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1.25723127083294305972e-01, 7.72723720984487272290e-02, -3.76023220287807879991e-01, 2.82473279931768073148e-01),
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.err_limit = 1e-5,
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},
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/* cond = 1e3, abs det = 15 */
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{
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.M = MAT(6.84154939885726509630e+00, -6.87241565273813304060e+00, -2.56772939909334070308e+01, -2.52185055099662420730e+01,
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2.04511561406330022450e+00, -3.67551043874248994925e+00, -1.96421641406619129633e+00, -2.40644091603848320204e+00,
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5.83631095663641819016e+00, -9.31051765621826277197e+00, -1.80402129629135217215e+01, -1.78475057662460052654e+01,
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-9.88588496379959025262e+00, 1.49790516545410774540e+01, 2.64975800675967363418e+01, 2.65795891678410747261e+01),
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.err_limit = 1e-4,
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},
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/* cond = 700, abs det = 1e-6, invertible regardless of det */
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{
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.M = MAT(1.32125189257677579449e-03, -1.67411409720826992453e-01, 1.07940907587735196449e-01, -1.22163309792902186057e-01,
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-5.42113793774764013422e-02, 5.30455105336593901733e-01, -2.59607412684229155175e-01, 4.36480803188117993940e-01,
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2.88175168292948129939e-03, -1.85262537685181277736e-01, 1.46265858042118279680e-01, -9.41398969709369287662e-02,
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-2.88900393087768159184e-03, 1.57987202530630227448e-01, -1.20781192010860280450e-01, 8.95194304475115387731e-02),
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.err_limit = 1e-4,
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},
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/* cond = 1e6, this is a little more challenging */
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{
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.M = MAT(-4.41851445093878913983e-01, -5.16386185043831491548e-01, 2.86186055948129847160e-01, -5.79440137716940473211e-01,
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2.49798696238173301154e-01, 2.84965614532234345901e-01, -1.65729639683955931595e-01, 3.12568045963485974248e-01,
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3.15253213984537428161e-01, 3.71270066781250074328e-01, -2.02675623845341434937e-01, 4.19969870491003371971e-01,
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5.60818677658178832424e-01, 6.45373659426444201692e-01, -3.68902466471524526082e-01, 7.13785795079988516498e-01),
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.err_limit = 0.02,
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},
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/* cond = 15, abs det = 1e-9, should be well invertible */
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{
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.M = MAT(-5.37536200142514660589e-05, 7.92552373388843642288e-03, -3.90554524958281433500e-03, 2.68892064500873568395e-03,
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-9.72329428437283989350e-03, 8.32075145342783470404e-03, 6.52648485926096092596e-03, 1.06707947887298994737e-03,
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1.04453728969657322345e-02, -1.03627268579679666927e-02, -3.56835980207569763989e-03, -3.95935925157862422114e-03,
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5.37160838929722633805e-03, 6.13466744624343262009e-05, -1.23695935407398946090e-04, 8.21231194921675112380e-04),
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.err_limit = 1e-6,
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},
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};
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TEST_P(matrix_inversion_precision, matrices)
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{
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const struct test_matrix *tm = data;
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struct weston_matrix rr;
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double err;
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/* Compute rr = M * inv(M) */
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weston_matrix_invert(&rr, &tm->M);
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weston_matrix_multiply(&rr, &tm->M);
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/* Residual: subtract identity matrix (expected result) */
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subtract_matrix(&rr, &IDENTITY);
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/*
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* Infinity norm of the residual is our measure.
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* See https://gitlab.freedesktop.org/pq/fourbyfour/-/blob/master/README.d/precision_testing.md
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*/
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err = matrix_inf_norm(&rr);
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testlog("Residual error %g (%.1f bits precision), limit %g.\n",
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err, -log2(err), tm->err_limit);
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if (err > tm->err_limit) {
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testlog("Error is too high for matrix\n");
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print_matrix(&tm->M);
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assert(0);
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}
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}
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