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I have to solve the equality between 2 matrixes 12x12 containing a lot of symbolic variables and with which I perform inversion of matrix.

My system is solved fastly when I take for example 2 matrices 2x2, the inversion is pretty direct.

Now, with the case of 2 matrixes 12x12, I need before actually to inverse a 31x31 matrix of symbolic variables (I marginalize after), since inversion takes a lot of time.

I would like to benefit from my GPU NVIDIA Card 24GB ( QUADRO RTX 6000) to achieve this inversion faster.

I have looked at the doc on mathworks but there is not a lot of informations about GPU matrix inversion with MATLAB.

Below the script where you will find the line of inversion :

COV_ALL = inv(FISH_SYM)

clear;

clc;

format long;

% 2 Fisher Matrixes symbolic : FISH_GCsp_SYM, : 1 cosmo params + 1 bias spectro put for common

% FISH_XC_SYM : 1 cosmo params + 2 bias photo correlated

% GCsp Fisher : 7 param cosmo and 5 bias spectro which will be summed

FISH_GCsp_SYM = sym('sp_', [17,17], 'positive');

% Force symmetry for GCsp

FISH_GCsp_SYM = tril(FISH_GCsp_SYM.') + triu(FISH_GCsp_SYM,1)

% GCph Fisher : 7 param cosmo + 3 I.A + 11 bias photo correlated

FISH_XC_SYM = sym('xc_', [21,21], 'positive');

% Force symmetry for GCph

FISH_XC_SYM = tril(FISH_XC_SYM.') + triu(FISH_XC_SYM,1)

% Brutal Common Bias : sum of 7 cosmo param ans 5 bias spectro : FISH_ALL1 = first left matrix

FISH_ALL1 = sym('xc_', [12,12], 'positive');

% Sum cosmo

FISH_ALL1(1:7,1:7) = FISH_GCsp_SYM(1:7,1:7) + FISH_XC_SYM(1:7,1:7);

% Brutal sum of bias

FISH_ALL1(7:12,7:12) = FISH_GCsp_SYM(7:12,7:12) + FISH_XC_SYM(15:20,15:20);

% Adding new observable "O" terms

FISH_O_SYM = sym('o_', [2,2], 'positive');

% Definition of sigma_o

SIGMA_O = sym('sigma_o', 'positive');

FISH_O_SYM = 1/(SIGMA_O*SIGMA_O) * FISH_O_SYM

% Force symmetry

FISH_O_SYM = (tril(FISH_O_SYM.') + triu(FISH_O_SYM,1))

FISH_O_SYM

FISH_SYM = sym('xc_', [31,31], 'positive');

FISH_BIG_GCsp = sym('sp_', [31,31], 'positive');

FISH_BIG_XC = sym('xc_', [31,31], 'positive');

% Block bias spectro + pshot and correlations

FISH_BIG_GCsp(1:7,1:7) = FISH_GCsp_SYM(1:7,1:7)

FISH_BIG_GCsp(7:17,7:17) = FISH_GCsp_SYM(7:17,7:17)

FISH_BIG_GCsp(1:7,7:17) = FISH_GCsp_SYM(1:7,7:17)

FISH_BIG_GCsp(7:17,1:7) = FISH_GCsp_SYM(7:17,1:7)

% Block bias photo and correlations

FISH_BIG_XC(1:7,1:7) = FISH_XC_SYM(1:7,1:7)

FISH_BIG_XC(21:31,21:31) = FISH_XC_SYM(11:21,11:21)

FISH_BIG_XC(1:7,21:31) = FISH_XC_SYM(1:7,11:21)

FISH_BIG_XC(21:31,1:7) = FISH_XC_SYM(11:21,1:7)

% Block I.A and correlations

FISH_BIG_XC(18:20,18:20) = FISH_XC_SYM(8:10,8:10)

FISH_BIG_XC(1:7,18:20) = FISH_XC_SYM(1:7,8:10)

FISH_BIG_XC(18:20,1:7) = FISH_XC_SYM(8:10,1:7)

% Final summation

FISH_SYM = FISH_BIG_GCsp + FISH_BIG_XC

% Add O observable

FISH_SYM(6,6) = FISH_SYM(6,6) + FISH_O_SYM(1,1);

FISH_SYM(6,26) = FISH_SYM(6,26) + FISH_O_SYM(2,2);

FISH_SYM(26,6) = FISH_SYM(26,6) + FISH_O_SYM(1,2);

FISH_SYM(26,26) = FISH_SYM(26,26) + FISH_O_SYM(2,1);

% Force symmetry

FISH_SYM = (tril(FISH_SYM.') + triu(FISH_SYM,1))

% Marginalize FISH_SYM2 in order to get back a 2x2 matrix

% Invert to marginalyze

COV_ALL = inv(FISH_SYM);

% Marginalize

COV_ALL([13:31],:) = [];

COV_ALL(:,[13:31]) = [];

FISH_ALL2 = inv(COV_ALL);

FISH_ALL1

FISH_ALL2

% Matricial equation to solve

eqn = FISH_ALL1 == FISH_ALL2;

% Solving : sigma_o unknown

[solx, parameters, conditions] = solve(eqn, SIGMA_O, 'ReturnConditions', true);

solx

Could anyone give me please clues/suggestions/modifications to optimize this Matlab script with GPU NVIDIA for inverse the matrix considered ?

Joss Knight
on 18 Apr 2021

Sorry, but Symbolic Math in MATLAB does not have GPU support.

Joss Knight
on 23 Apr 2021

You said "So, from what I have understood, you could help me to optimize the INV function with parfor or parfeval and containing symbolic variables."

What I said (with emphasis): "To answer your question to me, I DON'T KNOW how to optimize INV using parfor or parfeval."

Steven Lord
on 22 Apr 2021

To elaborate on what John D'Errico said:

for n = 2:6

A = sym('A', [n, n]);

tic

IA = inv(A);

t = toc;

fprintf("Inverting a %d by %d symbolic matrix " + ...

"took %g seconds and the result was %d characters long\n", ...

n, n, t, strlength(char(IA)));

end

I couldn't go to n = 7 without MATLAB Answers execution timing out because the code takes longer than 55 seconds. But you can see the time increases sharply for n = 6 and the length in characters of the result is increasing roughly by a factor of 9 or 10. No matter what you do, inverting a 12-by-12 symbolic matrix is going to take a lot of time and the result is going to be very complicated and long.

If you're doing this to solve a system of equations, you don't want the inverse. Try the \ operator instead, using subs to substitute values for the symbolic variables before calling \ if possible.

Steven Lord
on 23 Apr 2021

Are you absolutely, positively required to compute the explicit inverse for this matrix? If not, don't compute the inverse.

What exactly and specifically is the problem you're trying to solve where you believe you need the inverse? I would be willing to bet that likely what you actually need is to solve a system of equations, in which case the right solution would be to use the \ operator.

I'm about 95% sure GPU arrays or a parpool will not help you in any way with symbolic calculations.

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