How to find the determinant of 15 X 15 symbolic matrix?

I am trying to solve the problem which involves manipulations of symbolic equations to get a 15 X 15 matrix ( Sylvester matrix). each term of matrix is a polynomial in one variable, say x. The determinant of this matrix suppose to give 40 degree polynomial equation in x.

As I tried to find determinant using ' det() ' function, computation exceeds the size of memory (8 GB)and computer hung in between.

Is there any other way to compute a determinant of such a big matrix in symbolic form?or else is there any way to get the polynomial (numerical methods)?

Thanks.

Answer 1

[Sergey Kasyanov]

Hi,

I encountered the similar problem recently and I wrote a simple script for this task. It's designed for parallel computing of Gauss elimination but it will be usefull for you too.

Check this.

You should use function GaussElminitaion:

ADet=det(GaussElimination(AMatrix));

[Vijay Maraviya]

Thank you for help. As I tried to run above command, I encounter following errors. I am using Matlab R2018a.

I am relatively new to matlab so I can't really comprehend the errors. I am greatful for your help.

Error using sym>convertChar (line 1448) Character vectors and strings in the first argument can only specify a variable or number. To evaluate character vectors and strings representing symbolic expressions, use 'str2sym'.

Error in sym>tomupad (line 1214) S = convertChar(x);

Error in sym (line 211) S.s = tomupad(x);

Error in sym/privsubsasgn (line 1065) R = sym®;

Error in sym/subsasgn (line 922) C = privsubsasgn(L,R,inds{:});

Error in GaussElimination (line 22) k(k==0)='';

[Sergey Kasyanov]

Can you append your matrix here? Screenshot or .mat file.

[Vijay Maraviya]

sorry for being late.(exams) here is the .mat file. 15X15 matrix is stored in a variable named 'syl'. all the elements of the matrix are polynomial in symbolic variable c3.

[Sergey Kasyanov]

You should load new version of the file and try this:

Det=prod(diag(GaussElimination(A,'')));

[Murthy MVVS]

I have a 13 x 13 matrix given as M. I am using Matlab 2021a

If I use your code as detM = prod(diag(GaussElimination(M,'NULL')))

I am getting error as,

Error: File: GaussElimination.m Line: 55 Column: 13

Unable to classify the variable 'A1' in the body of the parfor-loop. For more information, see Parallel for

Loops in MATLAB, "Solve Variable Classification Issues in parfor-Loops".

And

If I use the matlab 'det' as detM = det(M)

I am getting error as,

detM =

Warning: Unable to display symbolic object because 'symengine' was reset. Repeat commands to regenerate

result.

> In sym/disp (line 44)

In sym/display>displayVariable (line 89)

In sym/display (line 51)

In det_m (line 37)

The matlab script file is,

%%%%%%%%% det_m.m %%%%%%%%%

clear all;
clc;

syms a1 a3 a7 a9 a10 a11;
syms b2 b3 b8 b12 b13;
syms c1 c3 c4 c7 c9 c10 c11; 
syms d4 d6 d7 d9 d10 d11; 
syms e5 e6 e8 e12 e13; 
syms f4 f6 f7 f9 f10 f11; 
syms g1 g3 g4 g6 g7 g9 g10 g11 g12; 
syms h2 h5 h8 h9 h11 h12 h13; 
syms i1 i3 i4 i6 i7 i8 i9 i10 i11 i13; 
syms j1 j3 j4 j6 j7 j8 j9 j10 j11 j13; 
syms k1 k3 k4 k6 k7 k8 k9 k10 k11 k13; 
syms l2 l5 l7 l8 l10 l12 l13; 
syms m2 m5 m8 m9 m11 m12 m13;

M = [
    
[ a1,  0, a3,  0,  0,  0, a7,  0, a9, a10, a11,   0,   0]
[  0, b2, b3,  0,  0,  0,  0, b8,  0,   0,   0, b12, b13]
[ c1,  0, c3,  0,  0,  0, c7,  0, c9, c10, c11,   0,   0]
[  0,  0,  0, d4,  0, d6, d7,  0, d9, d10, d11,   0,   0]
[  0,  0,  0,  0, e5, e6,  0, e8,  0,   0,   0, e12, e13]
[  0,  0,  0, f4,  0, f6, f7,  0, f9, f10, f11,   0,   0]
[ g1,  0, g3, g4,  0, g6, g7,  0, g9, g10, g11, g12,   0]
[  0, h2,  0,  0, h5,  0,  0, h8, h9,   0, h11, h12, h13]
[ i1,  0, i3, i4,  0, i6, i7, i8, i9, i10, i11,   0, i13]
[ j1,  0, j3, j4,  0, j6, j7, j8, j9, j10, j11,   0, j13]
[ k1,  0, k3, k4,  0, k6, k7, k8, k9, k10, k11,   0, k13]
[  0, l2,  0,  0, l5,  0, l7, l8,  0, l10,   0, l12, l13]
[  0, m2,  0,  0, m5,  0,  0, m8, m9,   0, m11, m12, m13]
]

detM = prod(diag(GaussElimination(M,'NULL')))

% detM = det(M)

%%%%%%%%% End det_m.m %%%%%%%%%

[Walter Roberson]

You need about 8 terabytes of memory (or more) to store the determinant of that matrix.

Answer 2

[laith saadi]

clear all;
clc;
syms a1 a3 a7 a9 a10 a11;
syms b2 b3 b8 b12 b13;
syms c1 c3 c4 c7 c9 c10 c11; 
syms d4 d6 d7 d9 d10 d11; 
syms e5 e6 e8 e12 e13; 
syms f4 f6 f7 f9 f10 f11; 
syms g1 g3 g4 g6 g7 g9 g10 g11 g12; 
syms h2 h5 h8 h9 h11 h12 h13; 
syms i1 i3 i4 i6 i7 i8 i9 i10 i11 i13; 
syms j1 j3 j4 j6 j7 j8 j9 j10 j11 j13; 
syms k1 k3 k4 k6 k7 k8 k9 k10 k11 k13; 
syms l2 l5 l7 l8 l10 l12 l13; 
syms m2 m5 m8 m9 m11 m12 m13;
M = [
    
[ a1,  0, a3,  0,  0,  0, a7,  0, a9, a10, a11,   0,   0]
[  0, b2, b3,  0,  0,  0,  0, b8,  0,   0,   0, b12, b13]
[ c1,  0, c3,  0,  0,  0, c7,  0, c9, c10, c11,   0,   0]
[  0,  0,  0, d4,  0, d6, d7,  0, d9, d10, d11,   0,   0]
[  0,  0,  0,  0, e5, e6,  0, e8,  0,   0,   0, e12, e13]
[  0,  0,  0, f4,  0, f6, f7,  0, f9, f10, f11,   0,   0]
[ g1,  0, g3, g4,  0, g6, g7,  0, g9, g10, g11, g12,   0]
[  0, h2,  0,  0, h5,  0,  0, h8, h9,   0, h11, h12, h13]
[ i1,  0, i3, i4,  0, i6, i7, i8, i9, i10, i11,   0, i13]
[ j1,  0, j3, j4,  0, j6, j7, j8, j9, j10, j11,   0, j13]
[ k1,  0, k3, k4,  0, k6, k7, k8, k9, k10, k11,   0, k13]
[  0, l2,  0,  0, l5,  0, l7, l8,  0, l10,   0, l12, l13]
[  0, m2,  0,  0, m5,  0,  0, m8, m9,   0, m11, m12, m13]
]

M = 

detM = prod(diag(GaussElimination(M,'NULL')))

Unrecognized function or variable 'GaussElimination'.

% detM = det(M)

[Murthy MVVS]

Use that function provided by Sergey Kasyanov, which can be downloaded from the given link below.

https://in.mathworks.com/matlabcentral/fileexchange/90661-gausselimination

Answer 3

[laith saadi]

clear all;

clc;

syms a1 a3 a7 a9 a10 a11;

syms b2 b3 b8 b12 b13;

syms c1 c3 c4 c7 c9 c10 c11;

syms d4 d6 d7 d9 d10 d11;

syms e5 e6 e8 e12 e13;

syms f4 f6 f7 f9 f10 f11;

syms g1 g3 g4 g6 g7 g9 g10 g11 g12;

syms h2 h5 h8 h9 h11 h12 h13;

syms i1 i3 i4 i6 i7 i8 i9 i10 i11 i13;

syms j1 j3 j4 j6 j7 j8 j9 j10 j11 j13;

syms k1 k3 k4 k6 k7 k8 k9 k10 k11 k13;

syms l2 l5 l7 l8 l10 l12 l13;

syms m2 m5 m8 m9 m11 m12 m13;

M = [

[ a1, 0, a3, 0, 0, 0, a7, 0, a9, a10, a11, 0, 0]

[ 0, b2, b3, 0, 0, 0, 0, b8, 0, 0, 0, b12, b13]

[ c1, 0, c3, 0, 0, 0, c7, 0, c9, c10, c11, 0, 0]

[ 0, 0, 0, d4, 0, d6, d7, 0, d9, d10, d11, 0, 0]

[ 0, 0, 0, 0, e5, e6, 0, e8, 0, 0, 0, e12, e13]

[ 0, 0, 0, f4, 0, f6, f7, 0, f9, f10, f11, 0, 0]

[ g1, 0, g3, g4, 0, g6, g7, 0, g9, g10, g11, g12, 0]

[ 0, h2, 0, 0, h5, 0, 0, h8, h9, 0, h11, h12, h13]

[ i1, 0, i3, i4, 0, i6, i7, i8, i9, i10, i11, 0, i13]

[ j1, 0, j3, j4, 0, j6, j7, j8, j9, j10, j11, 0, j13]

[ k1, 0, k3, k4, 0, k6, k7, k8, k9, k10, k11, 0, k13]

[ 0, l2, 0, 0, l5, 0, l7, l8, 0, l10, 0, l12, l13]

[ 0, m2, 0, 0, m5, 0, 0, m8, m9, 0, m11, m12, m13]

]

detM = prod(diag(GaussElimination(M,'NULL')))

% detM = det(M)