Large substitution from symbolic matrix to numerical values while keeping 4 unknowns into the new matrix

I have to solve the equality between 2 matrixes 12x12 containing a lot of symbolic variables and with which I perform inversion of matrix. There are only one unknown called `SIGAM_O, and FISH_O_SYM(1,1), FISH_O_SYM(1,2) and FISH_O_SYM(2,2) (FISH_O_SYM(2,1) = FISH_O_SYM(1,2)`.

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 to achieve this inversion faster but the GPU optimization is not supported currently for Symbolic arrays.

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

COV_ALL = inv(FISH_SYM)

and the entire code :

 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');
    FISH_SYM = zeros(31,31,'sym');
    FISH_BIG_GCsp = zeros(31,31,'sym');
    FISH_BIG_XC = zeros(31,31,'sym');
    
    % 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 : take a long long time 
    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

Actually, this inversion of 31x31 size takes a long long time (I had to stop it).

So, now, the strategy is to replace almost all the symolic unknowns by numerical values : I want just to keep 4 unknown (`SIGAM_O, and FISH_O_SYM(1,1), FISH_O_SYM(1,2) and FISH_O_SYM(2,2) (FISH_O_SYM(2,1) = FISH_O_SYM(1,2)`)

So, I would like to know how to do a large substitution of arrays `FISH_XC_SYM` and `FISH_GCsp_SYM` by numerical values of the equivalent of these 2 matrixes.

I could do for example :

  FISG_GCsp_NUM = load('Fisher_GCsp_num.dat')
    FISG_XC_NUM = load('Fisher_XC_num.dat')

But how to assign quickly to the both arrays `FISH_GCsp_SYM` and `FISH_XC_SYM` the numerical values of arrays `FISH_GCsp_NUM` and `FISH_XC_NUM`, al of this while keeping the 4 unknown above ?

Thanks in advance,

Best regards

[Steven Lord]

Why did you start a new question rather than continuing in your previous question where you've already received some assistance?

[petit]

the strategy that I would like to apply is slightly different from the previous question. Indeed, here it is about a global substitution on symbolic variables by numerical values. So I thought this would deserve a new question.

Sorry if it is diturbing for you but it seems to be appropriate for me.

Best regards