NUMAL Section 6.9.2
BEGIN SECTION : 6.9.2 (December, 1978)
AUTHORS: M. BAKKER AND N.M. TEMME.
INSTITUTE: MATHEMATICAL CENTRE.
RECEIVED: 750201.
BRIEF DESCRIPTION:
THIS SECTION CONTAINS THE FOLLOWING PROCEDURES:
BESS I0;
COMPUTES THE MODIFIED BESSEL FUNCTION OF THE FIRST KIND
OF ORDER ZERO WITH ARGUMENT X;
BESS I1;
COMPUTES THE MODIFIED BESSEL FUNCTION OF THE FIRST KIND
OF ORDER ONE WITH ARGUMENT X;
BESS I;
GENERATES AN ARRAY OF MODIFIED BESSEL FUNCTIONS OF THE
FIRST KIND OF ORDER L (L = 0, ..., N) WITH ARGUMENT X;
BESS K01;
COMPUTES THE MODIFIED BESSEL FUNCTIONS OF THE THIRD KIND
OF ORDERS ZERO AND ONE WITH ARGUMENT X; X > 0;
BESS K;
GENERATES AN ARRAY OF MODIFIED BESSEL FUNCTIONS OF THE THIRD
KIND OF ORDER L ( L = 0, ..., N) WITH ARGUMENT X; X > 0;
NONEXP BESS I0;
DOES THE SAME AS BESS I0, BUT THE RESULT IS MULTIPLIED
BY EXP(-ABS(X));
NONEXP BESS I1;
DOES THE SAME AS BESS I1, BUT THE RESULT IS MULTIPLIED
BY EXP(-ABS(X));
NONEXP BESS I;
DOES THE SAME AS BESS I, BUT THE ARRAY ELEMENTS ARE
MULTIPLIED BY EXP(-ABS(X));
NONEXP BESS K01;
DOES THE SAME AS BESS K01, BUT THE RESULTS ARE MULTIPLIED
BY EXP(X);
NONEXP BESS K;
DOES THE SAME AS BESS K, BUT THE ARRAY ELEMENTS ARE
MULTIPLIED BY EXP(X).
KEYWORDS: BESSEL FUNCTIONS,
MODIFIED BESSEL FUNCTIONS,
INTEGER ORDER.
REFERENCES:
[1] M.ABRAMOWITZ AND I.A. STEGUN,
HANDBOOK OF MATHEMATICAL FUNCTIONS,
DOVER PUBLICATIONS, INC., NEW YORK, 1968.
[2] D.B.HUNTER,
THE CALCULATION OF SOME BESSEL FUNCTIONS,
MATHEMATICS OF COMPUTATION (1964), P. 123.
[3] YUDELL LUKE,
THE SPECIAL FUNCTIONS AND THEIR APPROXIMATIONS, VOLUME 2,
ACADEMIC PRESS, NEW YORK AND LONDON (1969).
[4] C.W.CLENSHAW,
CHEBYSHEV SERIES FOR MATHEMATICAL FUNCTIONS,
NAT. PHYS. LAB. MATH. TABLES, VOLUME 5,
HER MAJESTY,S STATIONARY OFFICE, LONDON (1962).
[5] W.GAUTSCHI,
COMPUTATIONAL ASPECTS OF THREE TERM RECURRENCE RELATIONS,
SIAM REVIEWS, VOLUME 9 (1967), P. 24.
[6] J.M.BLAIR,
RATIONAL CHEBYSHEV APPROXIMATIONS FOR THE MODIFIED
BESSEL FUNCTIONS I0(X) AND I1(X);
MATHEMATICS OF COMPUTATIONS,VOLUME 28,
NR 126, APRIL 1974, P. 581-583.
SUBSECTION: BESS I0.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" BESS I0(X); "VALUE" X; "REAL" X;
"CODE" 35170;
BESS I0 DELIVERS THE MODIFIED BESSEL FUNCTION OF THE
FIRST KIND OF ORDER ZERO WITH ARGUMENT X;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTION.
PROCEDURES USED:
NONEXP BESS I0 = CP35175.
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME:
FOR X = 0 BESS I0 IS ASSIGNED ITS VALUE IMMEDIATELY;
FOR 0 < ABS(X) <= 15.0 17 MULTIPLICATIONS AND ONE DIVISION
ARE REQUIRED;
FOR ABS(X) > 15.0 11 MULTIPLICATIONS, 3 DIVISIONS, ONE
EVALUATION OF THE SQUARE ROOT AND ONE EVALUATION OF THE
EXPONENNTIAL FUNCTION ARE REQUIRED.
METHOD AND PERFORMANCE: RATIONAL APPROXIMATION, SEE [6].
EXAMPLE OF USE:
THE PROGRAM
"BEGIN" "REAL" X;
X:= 1; OUTPUT(61,"("/,D,6B-.14D"-ZD")",
X, BESS I0(X))
"END"
PRINTS THE FOLLOWING RESULTS:
1 .12660658777520" 1
SUBSECTION: BESS I1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" BESS I1(X); "VALUE" X; "REAL" X;
"CODE" 35171;
BESS I1 DELIVERS THE MODIFIED BESSEL FUNCTION OF THE
FIRST KIND OF ORDER ONE WITH ARGUMENT X;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTION.
PROCEDURES USED:
NONEXP BESS I1 = CP35176.
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME:
FOR X = 0 BESS I1 IS ASSIGNED ITS VALUE IMMEDIATELY;
FOR 0 < ABS(X) <= 15.0 17 MULTIPLICATIONS AND ONE DIVISION
ARE REQUIRED;
FOR ABS(X) > 15.0 12 MULTIPLICATIONS, 3 DIVISIONS, ONE EVALUATION
OF THE SQUARE ROOT AND ONE EVALUATION OF THE EXPONENTIAL FUNCTION
ARE REQUIRED.
METHOD AND PERFORMANCE: RATIONAL APPROXIMATION, SEE [6].
EXAMPLE OF USE:
THE PROGRAM
"BEGIN" "REAL" X;
X:= 1; OUTPUT(61,"("/,D,6B-.14D"-ZD")",
X, BESS I1(X))
"END"
PRINTS THE FOLLOWING RESULTS:
1 .56515910399252" 0
SUBSECTION: BESS I.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" BESS I(X, N, I); "VALUE" X, N;
"INTEGER" N; "REAL" X; "ARRAY" I;
"CODE" 35172;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS;
N: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE INDICES OF THE ARRAY I;
I: <ARRAY IDENTIFIER>;
"ARRAY" I[0 : N];
EXIT: I[L] POSSESSES THE VALUE OF THE MODIFIED BESSEL FUNCTION
OF THE FIRST KIND OF ORDER L (0 <= L <= N).
METHOD AND PERFORMANCE: SEE NON EXP BESS I (THIS SECTION).
PROCEDURES USED :
NONEXP BESS I = CP 35177.
REQUIRED CENTRAL MEMORY:
NO AUXILIARY ARRAYS ARE USED.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO THE MAXIMUM OF
1.359 * X + 72 AND N + 18.
EXAMPLE OF USE : THE FOLLOWING PROGRAM CHECKS FOR X = 1 (1) 20
THE WRONSKIAN RELATION
X * (I[N - 1] * K[N] + I[N] * K[N - 1]) - 1 = 0
FOR N = 1 (1) 5; THE PROGRAM READS:
"BEGIN" "REAL" X; "INTEGER" N; "ARRAY" I, K[0:5];
"FOR" X:= 1 "STEP" 1 "UNTIL" 20 "DO"
"BEGIN" OUTPUT(61,"("/ZD")", X);
BESS I(X, 5, I); BESS K(X, 5, K);
"FOR" N:= 1, 2, 3, 4, 5 "DO"
OUTPUT(61,"("BB-.D"-ZD")",
X * (I[N] * K[N - 1] + I[N - 1] * K[N]) - 1)
"END"
"END"
THE RESULTS ARE:
1 .0" 0 .0" 0 -.7"-14 -.7"-14 -.7"-14
2 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
3 .7"-14 .7"-14 .0" 0 .0" 0 .0" 0
4 .7"-14 .0" 0 .0" 0 .0" 0 .0" 0
5 .0" 0 .7"-14 .7"-14 .0" 0 .0" 0
6 .0" 0 .0" 0 .0" 0 .0" 0 -.7"-14
7 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
8 -.1"-13 -.1"-13 -.1"-13 -.1"-13 -.1"-13
9 .0" 0 .0" 0 .0" 0 -.7"-14 -.7"-14
10 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
11 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
12 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
13 .7"-14 .7"-14 .0" 0 .7"-14 .0" 0
14 .0" 0 .7"-14 .0" 0 .0" 0 .0" 0
15 .0" 0 .0" 0 .0" 0 .0" 0 .0" 0
16 .0" 0 .0" 0 .0" 0 .0" 0 -.7"-14
17 .7"-14 .0" 0 .0" 0 .0" 0 .0" 0
18 .7"-14 .0" 0 .0" 0 .0" 0 -.7"-14
19 .7"-14 .0" 0 .0" 0 .0" 0 .0" 0
20 .0" 0 .0" 0 .0" 0 .0" 0 -.7"-14
SUBSECTION: BESS K01.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" BESS K01(X, K0, K1); "VALUE" X; "REAL" X, K0, K1;
"CODE" 35173;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS; X > 0;
K0: <VARIABLE>;
EXIT: K0 HAS THE VALUE OF THE MODIFIED BESSEL FUNCTION
OF THE THIRD KIND OF ORDER 0 WITH ARGUMENT X;
K1: <VARIABLE>;
EXIT: K1 HAS THE VALUE OF THE MODIFIED BESSEL FUNCTION
OF THE THIRD KIND OF ORDER ONE.
PROCEDURES USED:
NONEXP BESS K01 = CP35178
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME: DEPENDS ON THE VALUE OF X;
THE GLOBAL VALUES IN MILLISECONDS ARE:
0 < X <= 1.5 : 2.2 MS,
1.5 < X <= 5.0 : 5.5 MS,
5.0 < X : 2.3 MS, ON THE CYBER 73/28.
METHOD AND PERFORMANCE:
FOR THE COMPUTATION OF K0 AND K1 THREE DIFFERENT METHODS
ARE USED DEPENDING ON THE VALUE OF X:
FOR 0 < X <= 1.5 K0 AND K1 ARE EVALUATED BY MEANS OF TAYLOR SERIES
EXPANSIONS (SEE [1], P. 375, FORMULA 9.6.13);
FOR X > 1.5 K0 AND K1 ARE COMPUTED BY MEANS OF A CALL
OF THE CODE PROCEDURE NONEXP BESS K01 (SEE DESCRIPTION AHEAD)
AND MULTIPLICATION BY EXP(- X).
EXAMPLE OF USE: THE PROGRAM
"BEGIN" "REAL" X, K0, K1;
"FOR" X:= .5, 1.5, 2.5 "DO"
"BEGIN" BESS K01(X, K0, K1);
OUTPUT(61,"("/,4BD.D,2(B-.14D"-ZD)")",X,K0,K1)
"END"
"END"
PRINTS THE FOLLOWING RESULTS:
0.5 .92441907122766" 0 .16564411200033" 1
1.5 .21380556264754" 0 .27738780045683" 0
2.5 .62347553200366" -1 .73890816347746" -1
SUBSECTION: BESS K.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" BESS K(X, N, K); "VALUE" X, N;
"INTEGER" N; "REAL" X; "ARRAY" K;
"CODE" 35174;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS; X > 0;
N: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE INDICES OF THE ARRAY K; N >= 0;
K: <ARRAY IDENTIFIER>;
"ARRAY" K[0 : N];
EXIT: K[I] POSSESSES THE VALUE OF THE MODIFIED BESSEL FUNCTION
OF THE THIRD KIND OF ORDER I (0 <= I <= N).
PROCEDURES USED: BESS K01 = CP 35173.
REQUIRED CENTRAL MEMORY:
NO AUXILIARY ARRAYS ARE USED.
RUNNING TIME :
DEPENDS ON THE VALUE OF X (SEE TABLE BELONGING TO BESS K01)
AND N.
METHOD AND PERFORMANCE:
K[0], ..., K[N] ARE COMPUTED ACCORDING TO THE RECURRENCE RELATION
K[I + 1] = K[I - 1] + (2 * I / X) * K[I], I = 2, ..., N,
(SEE [1], P. 376, FORMULA 9.6.26).
EXAMPLE OF USE: THE PROGRAM
"BEGIN" "ARRAY" K[0 : 2]; "REAL" X;
"FOR" X:= .5, 1.0, 1.5, 2.0 "DO"
"BEGIN" BESS K(X, 2, K);
OUTPUT(61,"("/D.D,3(BB.12D"-D)")",X,K)
"END"
"END"
PRINTS THE FOLLOWING RESULTS:
0.5 .924419071228"0 .165644112000"1 .755018355124"1
1.0 .421024438241"0 .601907230197"0 .162483889864"1
1.5 .213805562648"0 .277387800457"0 .583655963257"0
2.0 .113893872750"0 .139865881817"0 .253759754566"0
SUBSECTION: NONEXP BESS I0.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" NONEXP BESS I0(X); "VALUE" X; "REAL" X;
"CODE" 35175;
NONEXP BESS I0 DELIVERS THE MODIFIED BESSEL FUNCTION OF THE
FIRST KIND OF ORDER 0 WITH ARGUMENT X MULTIPLIED BY EXP(-ABS(X)).
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTION.
PROCEDURES USED:
BESS I0 = CP35170.
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME:
FOR X = 0 NONEXP BESS I0 IS ASSIGNED ITS VALUE IMMEDIATELY;
FOR 0 < ABS(X) <= 15.0 18 MULTIPLICATIONS, ONE DIVISION AND
ONE EVALUATION OF THE EXPONENTIAL FUNCTION ARE REQUIRED;
FOR ABS(X) > 15.0 10 MULTIPLICATIONS, 3 DIVISIONS AND ONE
EVALUATION OF THE SQUARE ROOT ARE REQUIRED.
METHOD AND PERFORMANCE:
SEE [6].
EXAMPLE OF USE:
THE PROGRAM
"BEGIN" "REAL" X;
X:= 1; OUTPUT(61,"("/,D,6B-.14D"-ZD")",
X, NONEXP BESS I0(X))
"END"
PRINTS THE FOLLOWING RESULTS:
1 .46575960759364" 0
SUBSECTION: NONEXP BESS I1.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"REAL" "PROCEDURE" NONEXP BESS I1(X); "VALUE" X; "REAL" X;
"CODE" 35176;
NONEXP BESS I1 DELIVERS THE MODIFIED BESSEL FUNCTION OF THE
FIRST KIND OF ORDER 1 WITH ARGUMENT X MULTIPLIED BY EXP(-ABS(X)).
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTION.
PROCEDURES USED:
BESS I1 = CP35171.
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME:
FOR X = 0 NONEXP BESS I1 IS ASSIGNED ITS VALUE IMMEDIATELY;
FOR 0 < ABS(X) <= 15.0 18 MULTIPLICATIONS, ONE DIVISION AND ONE
EVALUATION OF THE EXPONENTIAL FUNCTION ARE REQUIRED;
FOR X > 15.0 11 MULTIPLICATIONS, 3 DIVISIONS AND ONE
EVALUATION OF THE SQUARE ROOT ARE REQUIRED.
METHOD AND PERFORMANCE:
SEE [6].
EXAMPLE OF USE:
THE PROGRAM
"BEGIN" "REAL" X;
X:= 1; OUTPUT(61,"("/,D,6B-.14D"-ZD")",
X, NONEXP BESS I1(X))
"END"
DELIVERS THE FOLLOWING RESULTS:
1 .20791041534972" 0
SUBSECTION: NONEXP BESS I.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" NONEXP BESS I(X, N, I); "VALUE" X, N;
"INTEGER" N; "REAL" X; "ARRAY" I;
"CODE" 35177;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS;
N: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE INDICES OF THE ARRAY I; N >= 0;
I: <ARRAY IDENTIFIER>;
"ARRAY" I[0:N];
EXIT: I[L] POSSESSES THE VALUE OF THE MODIFIED
BESSEL FUNCTION OF THE FIRST KIND OF ORDER L (L=0,..,N)
MULTIPLIED BY EXP (- ABS(X)).
PROCEDURES USED: START = CP 35185;
REQUIRED CENTRAL MEMORY:
NO AUXILIARY ARRAYS ARE USED.
RUNNING TIME:
ROUGHLY PROPORTIONAL TO THE MAXIMUM OF 1.359*X + 72 AND N+18.
METHOD AND PERFORMANCE: SEE [5].
EXAMPLE OF USE: THE PROGRAM
"BEGIN" "REAL" X; "ARRAY" I[0:2];
"FOR" X:= .5, 1.0, 1.5, 2.0, 2.5 "DO"
"BEGIN" NONEXP BESS I(X, 2, I);
OUTPUT(61, "("/,4BZ.D,3(B-.12D"-D)")",X,
I[0], I[1], I[2])
"END"
"END"
PRINTS THE FOLLOWING RESULTS:
.5 .645035270449" 0 .156420803185" 0 .193520577097"-1
1.0 .465759607594" 0 .207910415350" 0 .499387768942"-1
1.5 .367433609054" 0 .219039387421" 0 .753810924929"-1
2.0 .308508322554" 0 .215269289249" 0 .932390333047"-1
2.5 .270046441612" 0 .206584649531" 0 .104778721987" 0
SUBSECTION: NONEXP BESS K01.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" NONEXP BESS K01(X, K0, K1);
"VALUE" X; "REAL" X, K0, K1;
"CODE" 35178;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS; X > 0;
K0: <VARIABLE>;
EXIT: K0 HAS THE VALUE OF THE MODIFIED BESSEL FUNCTION
OF THE THIRD KIND OF ORDER 0 WITH ARGUMENT X MULTIPLIED
BY EXP(X);
K1: <VARIABLE>;
EXIT: K1 HAS THE VALUE OF THE MODIFIED BESSEL FUNCTION OF
THE THIRD KIND OF ORDER 1 MULTIPLIED BY EXP(X).
PROCEDURES USED:
BESS K01 = CP35173.
REQUIRED CENTRAL MEMORY:
NO ARRAYS ARE USED.
RUNNING TIME:
DEPENDS ON THE VALUE OF X; BECAUSE OF THE STRONG
INTERDEPENDENCE OF THE BESS K01 ( = CP35173) AND NONEXP BESS K01
THE READER IS REFERRED TO THE TABLE OF RUNNING TIMES BELONGING
TO BESS K01.
METHOD AND PERFORMANCE:
FOR THE COMPUTATION OF K0 AND K1 THREE
DIFFERENT METHODS ARE USED DEPENDING ON THE VALUE OF X:
FOR 0 < X <= 1.5 K0 AND K1 ARE COMPUTED BY MEANS OF
MULTIPLICATION OF THE MODIFIED BESSEL FUNCTIONS OF ORDER
ZERO AND ONE (SEE DESCRIPTION OF K0) BY EXP(X);
FOR 1.5 < X <= 5 K0 AND K1 ARE COMPUTED BY
THE EVALUATION OF THEIR INTEGRAL REPRESENTATIONS (SEE [1],
P. 376, FORMULA 9.6.23) BY MEANS OF THE TRAPEZOIDAL RULE (SEE [2]);
FOR X > 5 K0 AND K1 ARE COMPUTED BY MEANS OF
A FINITE CHEBYSHEV SERIES EXPANSION (SEE [3], P. 339 AND [4]).
EXAMPLE OF USE: THE PROGRAM
"BEGIN" "REAL" X, K0, K1;
"FOR" X:= .5, 1.0, 1.5, 2.0, 2.5 "DO"
"BEGIN" NON EXP BESS K01(X, K0, K1);
OUTPUT(61,"("/,4BZ.D,2(5B-.14D"-ZD)")",
X, K0, K1)
"END"
"END"
PRINTS THE FOLLOWING RESULTS:
.5 .15241093857739" 1 .27310097082118" 1
1.0 .11444630798069" 1 .16361534862633" 1
1.5 .95821005329496" 0 .12431658735525" 1
2.0 .84156821507078" 0 .10334768470687" 1
2.5 .75954869032810" 0 .90017442390788" 0
SUBSECTION: NONEXP BESS K.
CALLING SEQUENCE:
THE HEADING OF THE PROCEDURE READS:
"PROCEDURE" NONEXP BESS K(X, N,K); "VALUE" X, N;
"INTEGER" N; "REAL" X; "ARRAY" K;
"CODE" 35179;
THE MEANING OF THE FORMAL PARAMETERS IS:
X: <ARITHMETIC EXPRESSION>;
THE ARGUMENT OF THE BESSEL FUNCTIONS; X > 0;
N: <ARITHMETIC EXPRESSION>;
THE UPPER BOUND OF THE INDICES OF THE ARRAY K; N >= 0;
K: <ARRAY IDENTIFIER>;
"ARRAY" K[0:N];
EXIT: K[I] POSSESSES THE VALUE OF THE MODIFIED BESSEL
FUNCTION OF THE THIRD KIND OF ORDER I (I = 0, ..., N)
MULTIPLIED BY EXP(X).
PROCEDURES USED:
NONEXP BESS K01 = CP 35178.
REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE USED.
METHOD AND PERFORMANCE:
K[0] AND K[1] ARE COMPUTED BY USING NONEXP BESS K01 (CP 35178),
WHILE K[2], ..., K[N] ARE COMPUTED ACCORDING TO THE
RECURRENCE RELATION
K[I+1]=K[I]+(2*I/X)*K[I], I>=2
(SEE [1], P. 376, FORMULA 9.6.26).
EXAMPLE OF USE:
THE PROGRAM
"BEGIN" "REAL" X; "ARRAY" K[0:2];
"FOR" X:= .5, 1.0, 1.5, 2.0 "DO"
"BEGIN" NONEXP BESS K(X, 2, K);
OUTPUT(61, "("/,Z.D,3(5B.14D"D)")",X,K)
"END"
"END"
PRINTS THE FOLLOWING RESULTS:
.5 .15241093857739"1 .27310097082118"1 .12448148218621"2
1.0 .11444630798069"1 .16361534862633"1 .44167700523334"1
1.5 .95821005329496"0 .12431658735525"1 .26157645513649"1
2.0 .84156821507078"0 .10334768470687"1 .18750450621395"1
SOURCE TEXT(S):
"CODE" 35170;
"CODE" 35171;
"CODE" 35172;
"CODE" 35173;
"CODE" 35174;
"CODE" 35175;
"CODE" 35176;
"CODE" 35177;
"CODE" 35178;
"CODE" 35179;