'**************************************************** '* Program to demonstrate Hermite coefficients * '* ------------------------------------------------ * '* Reference: BASIC Scientific Subroutines, Vol. II * '* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].* '* * '* ------------------------------------------------ * '* SAMPLE RUN: * '* * '* Hermite polynomial coefficients for order 2 * '* * '* A( 0) = -2 * '* A( 1) = 0 * '* A( 2) = 4 * '* * '* Hermite polynomial coefficients for order 3 * '* * '* A( 0) = 0 * '* A( 1) = -12 * '* A( 2) = 0 * '* A( 3) = 8 * '* * '* Hermite polynomial coefficients for order 4 * '* * '* A( 0) = 12 * '* A( 1) = 0 * '* A( 2) = -48 * '* A( 3) = 0 * '* A( 4) = 16 * '* * '* Hermite polynomial coefficients for order 5 * '* * '* A( 0) = 0 * '* A( 1) = 120 * '* A( 2) = 0 * '* A( 3) = -160 * '* A( 4) = 0 * '* A( 5) = 32 * '* * '* Hermite polynomial coefficients for order 6 * '* * '* A( 0) = -120 * '* A( 1) = 0 * '* A( 2) = 720 * '* A( 3) = 0 * '* A( 4) = -480 * '* A( 5) = 0 * '* A( 6) = 64 * '* * '* Hermite polynomial coefficients for order 7 * '* * '* A( 0) = 0 * '* A( 1) = -1680 * '* A( 2) = 0 * '* A( 3) = 3360 * '* A( 4) = 0 * '* A( 5) = -1344 * '* A( 6) = 0 * '* A( 7) = 128 * '* * '* Hermite polynomial coefficients for order 8 * '* * '* A( 0) = 1680 * '* A( 1) = 0 * '* A( 2) = -13440 * '* A( 3) = 0 * '* A( 4) = 13440 * '* A( 5) = 0 * '* A( 6) = -3584 * '* A( 7) = 0 * '* A( 8) = 256 * '* * '* Hermite polynomial coefficients for order 9 * '* * '* A( 0) = 0 * '* A( 1) = 30240 * '* A( 2) = 0 * '* A( 3) = -80640 * '* A( 4) = 0 * '* A( 5) = 48384 * '* A( 6) = 0 * '* A( 7) = -9216 * '* A( 8) = 0 * '* A( 9) = 512 * '* * '* Hermite polynomial coefficients for order 10 * '* * '* A( 0) = -30240 * '* A( 1) = 0 * '* A( 2) = 302400 * '* A( 3) = 0 * '* A( 4) = -403200 * '* A( 5) = 0 * '* A( 6) = 161280 * '* A( 7) = 0 * '* A( 8) = -23040 * '* A( 9) = 0 * '* A(10) = 1024 * '* * '**************************************************** ' Explanations ' ------------ ' ' Hermite polynomials are defined over the range -inf. < x < inf. The weight- ' ing function is w(x) = e^(-x^2): ' ' inf. ' Sum e^(-x^2) H (x) H (x) dx = 0 for n <> m (3.9.8) ' -inf. n m ' = f(n) for n = m ' ' The corresponding recursion relation is ' ' H (x) = 2x H (x) - 2n H (x) (3.9.9) ' n+1 n n-1 ' ' where H (x) = 1 and H (x) = 2x ' 0 1 ' ' As with the other polynomials, a simple subroutine for evaluating the coefficients ' be written (see program HERMITE). ' ' Note that Hermite polynomials are either even or odd, depending on N, and the co- ' efficients are integers. '----------------------------------------------------------------------------------- defint i-n defdbl a-h,o-z dim A(10),B(10,10) cls for n=2 to 10 print print " Hermite polynomial coefficients for order ";n print gosub 1000 for k=0 to n print " A(";k;") = ";A(k) next k print if n<10 then input r\$ next n print end '************************************************** '* Hermite polynomial coefficients evaluation by * '* means of recursion relation. The order of the * '* polynomial is n. The coefficients are returned * '* in A(i). * '************************************************** 1000 'Establish h0 and h1 coefficients B(0,0)=1# : B(1,0)=0# : B(1,1)=2# 'Return if order is less than two if n<2 then return for i=2 to n B(i,0)=-2*(i-1)*B(i-2,0) for j=1 to i 'Basic recursion relation B(i,j)=2*B(i-1,j-1)-2*(i-1)*B(i-2,j) next j next i for i=0 to n A(i)=B(n,i) next i return 'End of file Hermite.bas