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solve system of matrices

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Hajar Alshaikh
Hajar Alshaikh 2023-2-17
编辑： Torsten 2023-2-19
% I want to solve this system
f'(A,B)*[dA dB]'=[g(C))+B B*A-eye(n-1)]'
but when I found the derivative of f in the left side, I found that
f'(A,B)*[dA dB]'= [g(dA))-dB A*dB+B*dA]'
where A,B are (n-1)×(n-1) matrices
and C is n×n matrices
and I define a function chi from (n-1)×(n-1) matrices to the n×n matrices as
function chi= g(X)
chi= 3*trace X;
end
I want to solve this system for dA and dB
the problem for me is in the derivative part here
f'(A,B)*[dA dB]'= [g(dA))-dB A*dB+B*dA]'
can I said
[g(dA))-dB A*dB+B*dA]'= [g -eye(n-1);B A]*[dA dB]'
if not can I solve this system for dA and dB if i write it as
[g(dA)-dB A*dB+B*dA]'=[g(C)+B B*A-eye(n-1)]'
  9 个评论
显示 7更早的评论
Hajar Alshaikh
Hajar Alshaikh 2023-2-18
first of all I really want to thank you about your help and i appritiate your time, I learend from you many functions that I dont know about them before.
Now I tried your way, but I got this error
Warning: Solution does not exist because the system is inconsistent.
Error using vertcat
Dimensions of arrays being concatenated are not consistent.
Torsten
Torsten 2023-2-18
My example worked - so I cannot tell you what went wrong with your code.

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采纳的回答

Torsten
Torsten 2023-2-19
Ran in:
Should be faster than the symbolic solution.
rng("default")
n = 50;
A = rand(n-1);
B = rand(n-1);
C = rand(n-1);
x0 = zeros(2*(n-1)^2,1);
x = fsolve(@(x)fun(x,A,B,C,n),x0);
Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
dA = reshape(x(1:(n-1)^2),[n-1,n-1])
dA = 49×49
1.0e+03 * -0.9697 -1.2841 -1.1100 -1.0445 -1.1586 -1.2281 -1.0768 -0.9605 -1.0724 -1.0369 -1.1764 -0.9773 -0.9512 -1.0230 -1.1881 -0.9808 -1.0192 -1.0678 -1.2432 -1.0212 -1.0383 -0.9359 -0.9486 -1.1359 -1.0584 -1.1338 -1.1257 -0.9812 -0.9628 -0.9208 -0.5211 -0.6925 -0.5981 -0.5635 -0.6226 -0.6578 -0.5814 -0.5162 -0.5780 -0.5564 -0.6303 -0.5278 -0.5112 -0.5491 -0.6402 -0.5259 -0.5471 -0.5732 -0.6705 -0.5500 -0.5594 -0.5001 -0.5119 -0.6114 -0.5700 -0.6076 -0.6053 -0.5272 -0.5172 -0.4967 0.2252 0.2944 0.2581 0.2441 0.2652 0.2793 0.2498 0.2234 0.2489 0.2389 0.2685 0.2307 0.2179 0.2377 0.2720 0.2269 0.2369 0.2441 0.2848 0.2372 0.2407 0.2138 0.2232 0.2611 0.2435 0.2586 0.2588 0.2268 0.2248 0.2153 0.2900 0.3845 0.3341 0.3153 0.3454 0.3672 0.3232 0.2882 0.3204 0.3103 0.3511 0.2943 0.2830 0.3059 0.3565 0.2920 0.3050 0.3196 0.3719 0.3056 0.3105 0.2781 0.2845 0.3404 0.3153 0.3384 0.3377 0.2935 0.2883 0.2761 0.9613 1.2734 1.1008 1.0342 1.1489 1.2161 1.0665 0.9516 1.0624 1.0265 1.1667 0.9681 0.9430 1.0141 1.1791 0.9709 1.0102 1.0580 1.2321 1.0116 1.0289 0.9269 0.9393 1.1246 1.0487 1.1223 1.1157 0.9738 0.9525 0.9124 0.4635 0.6062 0.5273 0.4976 0.5500 0.5843 0.5120 0.4595 0.5122 0.4952 0.5603 0.4662 0.4534 0.4860 0.5625 0.4661 0.4895 0.5086 0.5910 0.4872 0.4929 0.4485 0.4539 0.5406 0.5028 0.5393 0.5354 0.4676 0.4623 0.4404 0.3057 0.4060 0.3501 0.3310 0.3651 0.3872 0.3408 0.3036 0.3384 0.3259 0.3707 0.3096 0.3003 0.3228 0.3775 0.3085 0.3202 0.3362 0.3932 0.3226 0.3280 0.2935 0.2991 0.3589 0.3339 0.3559 0.3557 0.3093 0.3034 0.2922 0.6562 0.8683 0.7499 0.7020 0.7852 0.8329 0.7274 0.6510 0.7249 0.7032 0.8006 0.6565 0.6466 0.6907 0.8043 0.6631 0.6925 0.7231 0.8428 0.6889 0.7014 0.6374 0.6407 0.7674 0.7176 0.7690 0.7614 0.6654 0.6508 0.6224 1.4197 1.8876 1.6310 1.5331 1.6967 1.7968 1.5786 1.4065 1.5696 1.5174 1.7234 1.4345 1.3923 1.5008 1.7445 1.4359 1.4941 1.5624 1.8221 1.4944 1.5230 1.3674 1.3917 1.6608 1.5518 1.6598 1.6481 1.4376 1.4084 1.3491 -0.3479 -0.4612 -0.3958 -0.3722 -0.4170 -0.4424 -0.3865 -0.3433 -0.3847 -0.3716 -0.4258 -0.3469 -0.3432 -0.3660 -0.4280 -0.3510 -0.3667 -0.3843 -0.4484 -0.3650 -0.3720 -0.3375 -0.3386 -0.4079 -0.3806 -0.4080 -0.4035 -0.3528 -0.3436 -0.3303
dB = reshape(x((n-1)^2+1:2*(n-1)^2),[n-1,n-1])
dB = 49×49
-0.7671 -1.4794 -1.1889 -0.6632 -0.6091 -0.9174 -0.9572 -0.8564 -1.2919 -1.5003 -1.4821 -1.1780 -1.2001 -1.4399 -0.9749 -1.0941 -1.3005 -1.1523 -1.2406 -0.8458 -0.8755 -1.2904 -1.1953 -0.7815 -1.1088 -0.8290 -0.7059 -1.4871 -1.0271 -0.7387 -0.7014 -0.9972 -1.4334 -0.6450 -1.4131 -0.8650 -1.0542 -0.9570 -0.8857 -1.1241 -1.2580 -0.6627 -1.5752 -1.0206 -1.0571 -0.5852 -1.2458 -1.4416 -1.3939 -1.4102 -0.7027 -1.0503 -0.7914 -1.5710 -0.5791 -0.6178 -0.8306 -1.5393 -0.8213 -1.0100 -1.3987 -0.9358 -1.2485 -1.4661 -1.4134 -0.7486 -1.4896 -1.4423 -1.2488 -1.0516 -0.9565 -1.5000 -1.0950 -1.1257 -1.1427 -1.2666 -0.7565 -0.7762 -1.3715 -1.5730 -0.9660 -1.2863 -1.2233 -0.6713 -1.4615 -0.8110 -1.4616 -1.1483 -1.4464 -1.0828 -1.2156 -1.0667 -1.1013 -0.8108 -0.6275 -0.9769 -0.5925 -0.9976 -1.2301 -1.0742 -1.2096 -0.6313 -1.5682 -1.1445 -1.0673 -1.5237 -1.1282 -1.2501 -1.0468 -1.2274 -1.3954 -1.5357 -0.9583 -1.2283 -0.9820 -0.9388 -0.7739 -1.1405 -1.1063 -0.9530 -0.5938 -0.8336 -0.8765 -1.4393 -1.1236 -1.2753 -0.7344 -0.8175 -1.1087 -0.8866 -0.8209 -1.1047 -0.8042 -1.2581 -0.8475 -1.4512 -1.5375 -1.4795 -0.8872 -1.2816 -1.5588 -1.0834 -0.6814 -0.7928 -0.8789 -1.2111 -0.6990 -0.7543 -1.4918 -1.0580 -1.4736 -1.5068 -1.2816 -1.2894 -1.5208 -0.7813 -1.0492 -1.1753 -1.2928 -1.5285 -1.1491 -0.6965 -0.9757 -0.9490 -1.5674 -0.6909 -1.1737 -0.7768 -1.2652 -1.5100 -1.4397 -0.8827 -0.9552 -0.8215 -1.5282 -1.5638 -1.1214 -1.0913 -1.5516 -0.9201 -1.0930 -1.0444 -0.9593 -1.4505 -0.8991 -1.2440 -1.1207 -1.0571 -1.0825 -1.5597 -1.5594 -0.9578 -1.2742 -0.6559 -0.7613 -0.9322 -1.3862 -0.8759 -1.5645 -1.2653 -0.6615 -1.3675 -0.8405 -0.9174 -1.0383 -0.7848 -0.8923 -1.1261 -1.1631 -1.3548 -1.1222 -0.8317 -1.1453 -1.5157 -1.3841 -1.0207 -0.6374 -1.4762 -1.0657 -1.0912 -1.4273 -1.3905 -0.6423 -1.0340 -1.4393 -0.8196 -1.5622 -1.0742 -1.3476 -1.1460 -0.9154 -0.8141 -0.8190 -0.7755 -0.8653 -1.3347 -0.9597 -0.7429 -0.6966 -0.9616 -1.1841 -1.0089 -1.4655 -0.7174 -1.1791 -1.0110 -1.2357 -1.5124 -1.0755 -1.5703 -0.8611 -0.8218 -1.3253 -0.6255 -0.6103 -1.1380 -1.4636 -1.4676 -1.4072 -0.9585 -0.8138 -0.8120 -1.2006 -1.0845 -0.6623 -1.4640 -1.3692 -1.0716 -1.5042 -1.2892 -1.3381 -1.2802 -1.4206 -0.9716 -1.3673 -0.7529 -1.4673 -1.3956 -1.5136 -1.0335 -1.2601 -1.4621 -0.9981 -1.3159 -0.9096 -1.1904 -0.7915 -1.0791 -1.2837 -1.2122 -0.8955 -1.0424 -1.1006 -1.5284 -1.1598 -1.0499 -1.4168 -1.1128 -1.1712 -1.0586
[3*trace(dA)-dB A*dB+B*dA]'-[3*trace(C)+B B*A-eye(n-1)]'
ans = 98×49
1.0e-10 * 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2775 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2773 0.2774 0.2773 0.2773 0.2774 0.2774 0.2774 0.2774 0.2773 0.2774 0.2774 0.2773 0.2774 0.2773 0.2774 0.2774 0.2773 0.2775 0.2774 0.2774 0.2773 0.2773 0.2774
function res = fun(x,A,B,C,n)
dA = reshape(x(1:(n-1)^2),[n-1,n-1]);
dB = reshape(x((n-1)^2+1:2*(n-1)^2),[n-1,n-1]);
res = [3*trace(dA)-dB A*dB+B*dA]' - [3*trace(C)+B B*A-eye(n-1)]';
res = res(:);
end

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