Log1rho

Log1rho is not a widely standardized mathematical function. In practice, it is a label that may denote either the natural logarithm of 1 plus a real number rho, or the matrix logarithm of the matrix I plus rho when rho is treated as a matrix. The exact meaning depends on context and the notation used in a given field or software library.

For scalars: log1rho(rho) commonly means log(1 + rho), using the natural logarithm. The domain is rho > -1. Examples: rho = 0.5 gives log1rho(0.5) = ln(1.5) ≈ 0.4055; rho = -0.5 gives ln(0.5) ≈ -0.6931. This interpretation is equivalent to the standard log1p(rho) notation used in numerical analysis to improve accuracy for small rho.

For matrices: log1rho(rho) is sometimes used to denote the matrix logarithm of I + rho, written log(I + rho), where log is the principal matrix logarithm. In this case, rho should be such that I + rho is invertible and, for a real-valued logarithm, I + rho should have no eigenvalues on the nonpositive real axis. In computation, log(I + rho) may be found via eigen decomposition or scalable methods such as scaling and squaring with Padé approximants, or via dedicated matrix logarithm routines (often denoted logm in software).

Properties and caveats: scalar log1rho inherits the standard properties of the natural log, including monotonicity and the derivative 1/(1 + rho). For matrices, the derivative is a Frechet derivative of the matrix logarithm, and numerical stability depends on conditioning of I + rho. Branch choices and complex values may arise if rho leads to eigenvalues crossing problematic regions.

See also: log1p, natural logarithm, matrix logarithm, log.