Page 542 - Modelling in Transport Phenomena A Conceptual Approach

522 APPENDIX A. MATHEMATICAL PRELIMINARIES

A.8.4.3 Gauss-Hermite quadrature

The Gauss-Hermite quadrature can be used to evaluate integrals of the form

The weight factors and appropriate roots for the first few quadrature formulas are
given in Table A.5.

Table A.5 Roots and weight factors for GaussHermite quadrature (Abramowitz
and Stegun, 1970).

n Roots (xi) Weight Factors (wi)
1 f 0.70710 67811 0.88622 69255
f 1.22474 48714 0.29540 89752
0.00000 00000 1.18163 59006

f 1.65068 01239 0.08131 28354
k0.52464 76233 0.80491 40900
f 2.02018 28705 0.01995 32421
4 f0.95857 24646 0.39361 93232
0.00000 00000 0.94530 87205

A.9 MATMCES

A rectangular array of elements or functions is called a mat&. If the array has m
rows and n columns, it is called an m x n matrix and expressed in the form

r all a12 a13 ... aln 1
a21 a22 a23 --- a2n
. . . . . . . . . . . . . , . . . . . . . . . J (A.9-1)
am1 am2 am3 e.- amn
The numbers or functions aij are called the elements of a matrix. Equation (A.9-1)
is also expressed as
A = (aij) (A.9-2)
in which the subscripts i and j represent the row and the column of the matrix,
respectively.
A matrix having only one row is called a row mat& (or, row vector) while
a matrix having only one column is called a column matria: (or, column vector).
When the number of rows and the number of columns are the same, i.e., m = n,
the matrix is called a spare mat+ or a matrix of order n.