slogdet

slogdet(x: array, /) → Tuple[array, array]

Returns the sign and the natural logarithm of the absolute value of the determinant of a square matrix (or a stack of square matrices) x.

Note

The purpose of this function is to calculate the determinant more accurately when the determinant is either very small or very large, as calling det may overflow or underflow.

The sign of the determinant is given by

\[\begin{split}\operatorname{sign}(\det x) = \begin{cases} 0 & \textrm{if } \det x = 0 \\ \frac{\det x}{|\det x|} & \textrm{otherwise} \end{cases}\end{split}\]

where \(|\det x|\) is the absolute value of the determinant of x.

When x is a stack of matrices, the function must compute the sign and natural logarithm of the absolute value of the determinant for each matrix in the stack.

Special Cases

For real-valued floating-point operands,

  • If the determinant is zero, the sign should be 0 and logabsdet should be -infinity.

For complex floating-point operands,

  • If the determinant is 0 + 0j, the sign should be 0 + 0j and logabsdet should be -infinity + 0j.

Note

Depending on the underlying algorithm, when the determinant is zero, the returned result may differ from -infinity (or -infinity + 0j). In all cases, the determinant should be equal to sign * exp(logabsdet) (although, again, the result may be subject to numerical precision errors).

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out (Tuple[array, array]) – a namedtuple (sign, logabsdet) whose

  • first element must have the field name sign and must be an array containing a number representing the sign of the determinant for each square matrix. Must have the same data type as x.

  • second element must have the field name logabsdet and must be an array containing the natural logarithm of the absolute value of the determinant for each square matrix. If x is real-valued, the returned array must have a real-valued floating-point data type determined by Type Promotion Rules. If x is complex, the returned array must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, logabsdet must have a float32 data type).

Each returned array must have shape shape(x)[:-2].

Notes

Changed in version 2022.12: Added complex data type support.