Computation of the Approximate Symmetric Chordal Metric for Complex Numbers

The basic theoretical properties of the approximate symmetric chordal metric (ASCM) for two real or complex numbers are studied, and reliable, accurate, and efficient algorithms are proposed for its computation. ASCM is defined as the minimum between the moduli of the differences of the two numbers and of their reciprocals. It differs from the chordal metric by including the modulus of the difference of the numbers. ASCM is not a true mathematical distance, but is a useful replacement for a distance in some applications. For instance, sensitivity analysis or block diagonalization of matrix pencils benefit from a measure of closeness of eigenvalues and also of their reciprocals; ASCM is ideal for this purpose. The proposed algorithms can be easily implemented on various architectures and compilers. Extensive numerical tests were performed to assess the performance of the associated implementation. The results were compared to those obtained in MATLAB, but with appropriate modifications for numbers very close to the bounds of the range of representable values, where the usual formulas give wrong results.