det determinant of a square matrix

## Syntax

                d = det(X)
                [e,m] = det(X)

## Arguments

X
  square matrix of real or complex numbers, polynomials, or rationals. Sparse-encoded matrices accepted.

d
  Scalar of the `X`'s type: the determinant of `X`. If `X` is sparse-encoded, `d` is dense.

m
  real or complex number: the determinant base 10 mantissa, with `abs(m) ∈ [1,10)`. Not supported for `X` polynomial or rational.

e
  integer: the determinant base 10 exponent, such that `d = m * 10e`. Not supported for `X` polynomial or rational.

## Description

d = det(X) yields the determinant of the matrix `X`.

For a polynomial or rational matrix, `d=det(X)` uses `determ(..)` whose algorithm is based on the FFT. `d=detr(X)` can be alternatively used, based on the Leverrier algorithm. Both methods yield equivalent results. For rational matrices, turning off `simp_mode(%f)` might be required to get identical results.

[e, m] = det(X) can be used only for a matrix of numbers. This syntax allows to overcome computation's underflow or overflow, when `abs(d)` is smaller than `number_properties("tiny")` ≈ 2.23 10-308 or bigger than `number_properties("huge")` ≈ 1.80 10308.

For denses matrices, `det(..)` is based on the Lapack routines DGETRF for real matrices and ZGETRF for the complex case.

For sparse matrices, the determinant is obtained from LU factorization thanks to the umfpack library.

## Examples

    A = rand(3,3)*5;
    det(A)
    [e, m] = det(A)

    // Matrix of complex numbers:
    // A = grand(3,3,"uin",0,10) + grand(3,3,"uin",0,10)*%i
    A = [3+%i, 9+%i*3, 9+%i ; 8+%i*8, 4+%i*3, 7+%i*7 ; 4, 6+%i*2, 6+%i*9]
    det(A)
    [e, m] = det(A)
    abs(m)  // in [1, 10)

    --> A = rand(3,3)*5;
    --> det(A)
     ans  =
      -10.805163

    --> [e, m] = det(A)
     e  =
       1.
     m  =
      -1.0805163

    --> // Matrix of complex numbers:
    --> A = [3+%i, 9+%i*3, 9+%i ; 8+%i*8, 4+%i*3, 7+%i*7 ; 4, 6+%i*2, 6+%i*9]
     A  =
       3. + i     9. + 3.i   9. + i
       8. + 8.i   4. + 3.i   7. + 7.i
       4. + 0.i   6. + 2.i   6. + 9.i

    --> det(A)
     ans  =
       745. - 225.i

    --> [e, m] = det(A)
     e  =
       2.
     m  =
       7.45 - 2.25i

    --> abs(m)  // in [1, 10)
     ans  =
       7.7823518

Very big or small determinants: underflow and overflow handling:

    // Very big determinant:
    n = 1000;
    A = rand(n, n);
    det(A)
    [e, m] = det(A)

    // Very small determinant (of a sparse-encoded matrix):
    A = (triu(sprand(n,n,1)) + diag(rand(1,n)))/1.5;
    det(A)
    prod(diag(A))
    [e, m] = det(A)
    A = A/2;
    det(A)
    [e, m] = det(A)

    --> // Very big determinant:
    --> A = rand(n, n);
    --> det(A)
     ans  =
      -Inf

    --> [e, m] = det(A)  // -3.1199e743
     e  =
       743.
     m  =
      -3.1198687

    --> // Very small determinant (of a sparse-encoded matrix):
    --> n = 1000;
    --> A = (triu(sprand(n,n,1)) + diag(rand(1,n)))/1.5;
    --> det(A)
     ans  =
       5.21D-236

    --> prod(diag(A))
     ans  =
       5.21D-236

    --> [e, m] = det(A)
     e  =
      -236.
     m  =
       5.2119757

    --> A = A/2;
    --> det(A)
     ans  =
       0.

    --> [e, m] = det(A)
     e  =
      -537.
     m  =
       4.8641473

Determinant of a polynomial matrix:

    s = %s;
    det([s, 1+s ; 2-s, s^2])

    w = ssrand(2,2,4);
    roots(det(systmat(w))),trzeros(w)   //zeros of linear system

    --> det([s, 1+s ; 2-s, s^2])
     ans  =
      -2 -s +s² +s³

    --> w = ssrand(2,2,4);
    --> roots(det(systmat(w))),trzeros(w)
     ans  =
      -3.1907522 + 0.i
       2.3596502 + 0.i

     ans  =
       2.3596502 + 0.i
      -3.1907522 + 0.i

## See also

- detr
- determ
- simp_mode

## History

6.1.1 [e,m]=det(X) syntax extended to sparse matrices.