lit
    JINRLIB

OSCILATORYFUN_INTEGRATION -

D133

: .., .. :

OSCILATORYFUN_INTEGRATION c pa d133_1 a o acoo opoc. poaac, o F(X) ocp ooo pa [A,B], o oo oopoo o , -1020 1020.
, F(X) P(T):

F(X) = Q(X) * P(T),   T = (FRQ * X + PHASE).

P(T). ZF - NZ. P(T) F(X) , [A,B]. ZF . NZ 20-30.
V - (NZ,2). < A.
,
- ERROR.

:

: - FUNCTION
: - OSCILATORYFUN_INTEGRATION
: - GAUSS_INTEGRATION(D132),
F - /-

:

GINT=OSCILATORYFUN_INTEGRATION(A,B,F,FRQ,PHASE,NZ,ZF,V,ERROR), :

A,B - p ppoa (a A = -cooc opc co, -1020, a a B = +cooc opc co, +1020);
F - opopa-, cocao ooa c a opao F(X).
F oa ocaa opaopo EXTERNAL popa;
FRQ,PHASE - oca pca P(T);
NZ - papoc aaoo acca ZF;
ZF - opoo o opaca oco P(T). o acc ooo coppoa p opa OSCILATORYFUN_INTEGRATION.
popaa OSCILATORYRUN_INTEGRATION caa pca P(T) F(X) op oco , copac pa [A,B]. coo oco ZF copac ;
V - pao acc papoc 2*NZ.
apoc acco ZF(NZ) V(NZ,2) ooo oca a popa;
ERROR - .

OSCILATORYFUN_INTEGRATION GAUSS_INTEGRATION(D132) c pa.

:

ooa d133_2 , (XL) - F(X), pca UL UL = ZL * VL.
pa c ( cooo pa):

d133_3

aco ca, oa Z = -1,

d133_4

oc oa o pca a pp, oa
B=+cooc Z=-1, oa

d133_5

c o p coc o, o oo p popaoa pa o coc p. oa o pocpoa coo p o oo pa, a ocac p popaoa. poa pa oo c c oo ppc, poo pao [5], .. p o aoac pac oaa copoc cooc.
o ca pa (1) oo pca c oo cooo, p pooo oao [4] oc aaoo oooo popaoa pa o c (c. [1]):

d133_9
d133_10(M,K,R < N), (M,K,R=0,1,2,...).

SMR sigmaKR cpa ppp op:
SMR+1 = P*SM+1R + Q*SMR,    sigmaKR+1 = Q*sigmaRK+1 + P*sigmaKR.      (2).
poo p, o a GR caoc ppo ao p ocaoo o R. c ocoa ac c SM0 sigmaK0 (M,K=0,1,2,...), a a SMR sigmaKR (R=0,1,2,...), c oo cooo (2) c opo opoc o a

d133_12

o oc po ooc o a po a paa c

IRM,K = SMR+1 + sigmaKR+1 ± ERROR ,
d133_14, sigmaL , - opoc c UL . c o po co, o o c o c SMR+1 sigmaKR+1.

:

  1. A B o co oopo.
  2. F(X) oooa pa [A,B]
    d133_11 opa ooo ocoaoc.
  3. popaa opoo paoa a ca, oa F(X) c oo c M<=N, N - co P(T) pa [A,B].
  4. F(X) oop co 2 3 oopo ac paa [A,B], copa ao oco P(T) p co ppoc F(X) o c pa [A,B].
  5. oc ca popaa a p pa, c poa o F(X) oo ppoc [2].
  6. ca 5, a a ca 4 oaa a opoc ERROR o oa .

:

  1. c pa [A,B] oac p P(T), o pa cc o popa GAUSS_INTEGRATION (D132), o c p o o po A B co, popaa o p SUM=0, ERROR=0 aa pa:
    " OSCILATORYFUN_INTEGRATION: not enough zeros in (A,B) !"
  2. c opa aa: A=-cooc B=+cooc oopo, popaa o p SUM=0, ERROR=0 aa pa:
    " OSCILATORYFUN_INTEGRATION: both A and B are indefinite !"

:

  1. a, oa A=-cooc B=+cooc, popa c ao . c c p o copaac po o.
  2. Papoc NZ acca ZF F(X) oooc pa o oo. c pa cc a ocoo pa, o ocaoo p 20-30 F(X) P(FRQ*X+PHASE).
    oa pa [A,B] o, oo aa c o 20-30 o aoo oa, poc ac cp paa. oa o aa oo acc , ppa c opoc p oopa opa OSCILATORYFUN_INTEGRATION c c apapa p apapa ZF NZ.

:

  1. M P.B. A .
    C . O, 10-7707, YHA, 1974.
  2. C .T. O .
    T . B.A.C, LXVI, 1962, .166-181.
  3. Longman J.M. Note on a method for computing infinite integrals of oscilatory functions. Proc. Cambridge Philos. Soc., V.52, N4, 1956, p.764-768.
  4. Longman J.M. A method for numerical evaluation of finite integrals of oscilatory functions. Mathematics of computation, V.14, N69, January 1960, p.53-59.
  5. Wynn P. A note on generalised Euler transformation.
    The Computer Journal. V.14, N4, November 1971, p.437-441.

:

:

d133_13
       . . .
       IMPLICIT REAL*8 (A-H,O-Z)
       EXTERNAL FEX
       PARAMETER(NZ=902)
       DIMENSION ZF(NZ),V(NZ,2)
       DATA PI/3.1415926536D0/
       . . .
       DO I=1,NZ
          ZF(I)=(I-1)*PI
       ENDDO
       GINT=OSCILATORYFUN_INTEGRATION(0.D0,10.D0,FEX,200.D0,0.D0,
     * ZF,NZ,V,ERROR)
       WRITE(*,*) ' INTEGRAL=',GINT,' ERROR=',ERROR
       . . .
       DOUBLE PRECISION FUNCTION FEX(X)
       IMPLICIT REAL*8 (A-H,O-Z)
       DOUBLE PRECISION X
       PN=1.D0
       DO 1 I=1,9
    1  PN=PN*(X-I)
       FEX=PN*DSIN(200.D0*X)
       RETURN
       END
:
       OSCILATORYFUN_INTEGRATION: not enough zeros in (A,B) !
       USUAL GAUSS_INTEGRATION USED.
       INTEGRAL=   -1123.629579815686000 ERROR=  1.419104600747234E-008


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