1.6 Bigradients and Hadamard's Formula

 

Hons are needed in the presence of degeneracy as an exercise. With these conventions, and in the absence of degeneracy, determinant

The determinants in (6.3) and (6.4) are called polynomial bigradients. An formed from the coefficients a, arranged in its first L columns with a negative gradient and from the coefficients gl arranged in the next M columns with a positive gradient. Specifically, define

Trudi's theorem is probably the best-known result which uses bigradients explicitly. It also identifies the nature of degeneracies encountered with bigradients.

Theorem 1.6.2 [Trudi, 1862]. Let g(z) be a polynomial of degree /, and let d(z) be a polynomial of degree m. Define the bigradient (6.6). If as in and then d(z) and g(z) have a common polynomial divisor of order j.

of Section 4, this Pade approximant is also the [l/m] Pade approximant of f(z), and so the theorem follows.

For further properties of bigradients, we refer to Householder [1970, 1971], Householder and Stewart [1969], and Pade [1900].

Exercise If are both polynomials, under what circumstances is the problem of determining and in (6.2) well posed?