Matvec Users’ Guide

9.3. SPECIAL MATRIX PACKAGE 55
i = 1 i = 2 i = 3 i = 4
2 3 5 7
> prime_next(4)
5
> prime_next(888)
907
> prime_less(8)
i = 1 i = 2 i = 3 i = 4
2 3 5 7
9.3 Special Matrix Package
The macro package special matrix contains following special matrices:
hadamard, pascal, toeplitz.
hilb, invhilb, hankel, vander,
9.3.1 hilb
hilb(n) creates the Hilbert matrix of order n. The i, j element H(i,j) of a Hilbert matrix is defined as H(i,j)
= 1 / (i + j - 1). For example,
> package("special_matrix");
> hilb(3)
Col 1 Col 2 Col 3
Row 1 1 0.5 0.333333
Row 2 0.5 0.333333 0.25
Row 3 0.333333 0.25 0.2
9.3.2 invhilb
invhilb(n) returns the inverse of the Hilbert matrix of order n. This is the exat results. See and compare
invhilb(n) with hilb(n).inv(). For example,
> package("special_matrix");
> invhilb(3)
9.3.3 hankel
Col 1 Col 2 Col 3
Row 1 9 -36 30
Row 2 -36 192 -180
Row 3 30 -180 180
hankel(C) returns a square Hankel matrix whose first column is C and whose elements are zero below the
first anti-diagonal; whereas hankel(C,R) returns a Hankel matrix whose first column is C and whose last
row is R. Hankel matrices are symmetric, constant across the anti-diagonals, and have elements H(i,j) =
R(i+j-1). For example,
> package("special_matrix");
> hankel([1,2,3])
Col 1 Col 2 Col 3