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Mathematics Wolfram Language Symbolic Computations

Simplify a 12+ order symbolic determinant?

Lee KC

Posted 8 years ago

How can I simplify a 12+ order symbolic determination to its simplest form in mma? The second example(In[2]) below should be a polynomial of x with maximal order of 12. But, mma gives a polynomial of x^192. In[1]: b1 = RandomReal[{0, 1}, {11, 11}]; c1 = Table[b1[[i, j]] + x, {i, 1, 11}, {j, 1, 11}]; Print[c1 // MatrixForm]; d = Simplify[Expand[Det[c1]]] Out[1]: 0.033469 + 0.076227 x - 2.2204510^-16 x^2 + 5.9967510^-17 x^3 - 4.1908210^-31 x^4 + 1.1261610^-31 x^5 - 5.4738210^-47 x^6 + 1.6421510^-47 x^7 In[2]: b1 = RandomReal[{0, 1}, {12, 12}]; c1 = Table[b1[[i, j]] + x, {i, 1, 12}, {j, 1, 12}]; d = Simplify[Expand[Det[c1]]] Out[2]: `......` -8.1786710^9 x^184 - 6.5129710^8 x^185 - 4.4996910^7 x^186 - 2.6421210^6 x^187 - 128194. x^188 - 4934.19 x^189 - 141.247 x^190 - 2.67311 x^191 - 0.0250846 x^192)/(((0.214925 + x)^10) ((0.647147 + 1. x)^9) ((0.289144 + 1.56025 x + 1. x^2)^8) ......

How can I simplify a 12+ order symbolic determination to its simplest form in mma?
The second example(In[2]) below should be a polynomial of x with maximal order of 12. But, mma gives a polynomial of x^192.

In[1]:    
b1 = RandomReal[{0, 1}, {11, 11}];    
c1 = Table[b1[[i, j]] + x, {i, 1, 11}, {j, 1, 11}];    
Print[c1 // MatrixForm];    
d = Simplify[Expand[Det[c1]]]   

Out[1]:    
0.033469 + 0.076227 x - 2.22045*10^-16 x^2 + 5.99675*10^-17 x^3 -  4.19082*10^-31 x^4 + 1.12616*10^-31 x^5 - 5.47382*10^-47 x^6 + 1.64215*10^-47 x^7

enter image description here

In[2]:    
b1 = RandomReal[{0, 1}, {12, 12}];    
c1 = Table[b1[[i, j]] + x, {i, 1, 12}, {j, 1, 12}];    
d = Simplify[Expand[Det[c1]]]    

Out[2]:    
    `......` -8.17867*10^9 x^184 - 6.51297*10^8 x^185 - 4.49969*10^7 x^186 - 
    2.64212*10^6 x^187 - 128194. x^188 - 4934.19 x^189 - 
    141.247 x^190 - 2.67311 x^191 - 
    0.0250846 x^192)/(((0.214925 + x)^10) ((0.647147 + 
    1. x)^9) ((0.289144 + 1.56025 x + 1. x^2)^8) ......

POSTED BY: Lee KC

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 8 years ago

I have no idea why we get a rational function instead of a polynomial. However, if we `Rationalize` the coefficients, the determinant becomes a polynomial of degree 1: b1 = RandomReal[{0, 1}, {12, 12}]; c1 = b1 + x; Det[Rationalize[c1, .000001]]