Second Kind. ACM Trans. Math. Software 2, 2 (June 1976), 196-199. Key Words
and Phrases: numerical analysis, linear integral equations, automatic algorithm,.
An Automatic Program for Linear Fredholm Integral Equations of the Second Kind KENDALL ATKINSON University of Iowa
Two automatic programs for solving linear Fredholm integral equations of the second kind are described and illustrated. It is assumed that the kernel function and solution are smooth and that they are given analytically, not as discrete data. The numerical method is based on the Nystr6m method, with an lteratlve technique to solve the resulting linear systems. The main discussion centers on Simpson's method as the numerical integration rule. A powerful variant based on Gaussian quadrature m also discussed, and tests for ill-conditioned problems have been incorporated into the program. Modifiability of the Simpson program is also discussed. The Algorithm: Algorithm 503, An Automatic Program for Fredhohn Integral Equations of the Second Kind. ACM Trans. Math. Software 2, 2 (June 1976), 196-199. Key Words and Phrases: numerical analysis, linear integral equations, automatic algorithm, Nystrbm method CR Categories: 5.18
1. INTRODUCTION I n this paper two automatic programs for solving Fredholm integral equations of the second kind, b
X(S) -- Ja K ( s , t ) x ( t ) d t = y ( s ) ,
a eupsjstem
TN(1),
WN(I),I=I .....
N.
~
STAGE A
R A T I O ; and for R2 > 1 in eq. (1.1). Define R E L M I N = max {R E L 1 , R E L 2 }, (5.3) with REL1
= COND*UNITRD*M~v/M,
REL2
= (M/N)312*][ z (1) - z (°) I[/11
z(l> 1[. (5.4)
In this definition, M is the order of the linear system currently being examined, and N is the order of the LU decomposition currently being used in solving the order M system. Always, M >__N; and M = N means that iteration is not being used. The numbers z ¢°~and z ~ were described earlier. The number U N I T R D gives the machine precision; it is the smallest number u for which 1 q- u > u > 0 in the computer. In double precision arithmetic on an I B M 360 machine, U N I T R D = 2.22 X 10-~6. The n u m b e r R E L 2 has been useful in detecting limitations on the accuracy as X approaches a characteristic value, in eq. (4.1). The construction of R E L M I N is a mixture of mathematical intuition plus empirical testing. It has worked reliably, as some of the following examples will show. Nonetheless, much more work needs to be done on this problem of detecting and quantifying ill-conditioning. Because of the greater complexity of the program, two new subprograms are used besides those used in I E S I M P . In the earlier list, delete T W I C E and add the following: (g) C O N E W is a function used in computing the condition number C O N D for computing R E L M I N . (h) L E A V E is a subroutine for setting parameters when leaving I E G A U S . Since the Legendre nodes are not a convenient set of node points, the user can specify his own node points at which he would like solution values. These are computed in L E A V E by NystrSm interpolation. 6. NUMERICAL EXAMPLES For I E G A U S the first four equations are the same as for the program I E S I M P , and four additional equations are included so as to explore more completely the behavior of the program. In all cases, N U P P E R = 32 and M U P P E R = 256. In the program, if the desired error E P is too small compared to R E L M I N , then E P is reset accordingly; an error indicator is set with I E R . See the listing of I E G A U S in [-4~ for the meanings of each value of I E R . ACM Transactions on Mathematmal Software, Vol 2, No. 2, June 1976.
166
Kendall Atkinson Table VI. IEGA US Results for Case (1) Error X
b 1.4
1.4 1.4 1.4 -2000.0 -1.42 1.48
Desired
Final
Predicted
Actual
N
M
1.0
1.0E-3
5.3E-11
1 . 6 E - 14
8
16
2.0 3 0 4.0 1.0 2.0 3.0
1 0E-3 1.0E-3 1.0E-3 1.0E-5 1.0E-5 1.0E-8
6.3E-14 2.4E-4 3.0E - 1 3 4.8E - 8 6 . 2 E - 14 3 . 4E - 13
3.3E-14 9.4E-14 5 . 7 E - 14 9 . 6 E - 13 2.7E - 1 4 5.5E - 1 4
16 32 32 8 16 32
32 32 64 16 32 64
Case (i) : K(s,t)
=cos(~st),
x(t) = etcos(7t),
O
