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Compute Determinant Numerically

To compute the determinant of a square matrix numerically, use the numeric::det function. For example, compute the determinant of the 5 ×5 Pascal matrix:

numeric::det(linalg::pascal(5))

When you use numeric functions, the result can be extremely sensitive to roundoff errors. For example, the determinant of a Pascal matrix of any size is 1. However, if you use the numeric::det function to compute the determinant of a 25×25 Pascal matrix, you get the following incorrect result:

numeric::det(linalg::pascal(15))

When computing determinants numerically, you can use the HardwareFloats and SoftwareFloats options to employ the hardware or software float arithmetic, respectively. (You can use the short names for these options: Hard and Soft.)

When you use the HardwareFloats option, MuPAD® converts all input data to hardware floating-point numbers, and passes the data for processing by compiled C code outside of the MuPAD session. Then, the results get back into the MuPAD session. Hardware arithmetic often allows you to perform computations much faster than software arithmetic, which uses the MuPAD kernel for performing computations.

The precision of hardware arithmetic is limited to about 15 digits. By default, the numeric::det function uses the HardwareFloats option. The function switches to software arithmetic under one or more of the following conditions:

  • You use the SoftwareFloats option or the MinorExpansion option explicitly.

  • The current value of DIGITS is larger than 15.

  • The input data or computed data involves numbers that are larger than 10308 or smaller than 10- 308. Hardware floats cannot represent such numbers.

The precision of hardware and software arithmetic can differ. Therefore, the results obtained with the HardwareFloats and SoftwareFloats options also can differ. For example, compute the determinant of the 25 ×25 Pascal matrix using each of the options. Both numeric results are several orders larger than the correct answer because of the roundoff errors. However, the result obtained using software arithmetic is several orders closer to the correct answer:

P := linalg::pascal(25):
detP := det(P):
float(detP);
numeric::det(P, SoftwareFloats);
numeric::det(P, HardwareFloats)

Another example of numerically ill-conditioned matrices is the Hilbert matrices. For example, create the 20×20 Hilbert matrix, compute its determinant symbolically, and then approximate the result numerically:

H := linalg::hilbert(15):
detH := det(H):
float(detH)

Now, use the numeric::det function to compute the determinant numerically:

numeric::det(H)

The numeric result obtained with the SoftwareFloats option is closer to the correct result:

numeric::det(linalg::hilbert(15), SoftwareFloats)

To prevent the conversion of input data to floating-point numbers while using the numeric::det function, use the Symbolic option. This option allows you to compute the determinant exactly (without roundoff errors). For matrices over particular rings and fields, the determinant computed by numeric::det with the Symbolic option can differ from the determinant computed by det. The reason is that det performs computations over the component domain of the input matrix. The numeric::det function with the Symbolic option always performs computations over the field of arbitrary MuPAD expressions. For example, create the following matrix over the domain Dom::IntegerMod(5):

A := Dom::Matrix(Dom::IntegerMod(5))([[1, 2], [3, 4]])

The det function computes the determinant over the component domain Dom::IntegerMod(5):

det(A)

The numeric::det function with the Symbolic option computes the determinant of the following matrix instead of the original matrix A:

expr(A)

The determinant of this matrix is an integer number:

numeric::det(A, Symbolic)

The numeric::det function switches to the Symbolic option under one or more of the following conditions:

  • You use the Symbolic option explicitly.

  • The input data contains symbolic objects.

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