{**************************************************************** * EVALUATE A I-BESSEL FUNCTION OF COMPLEX ARGUMENT (MODIFIED * * FIRST KIND) * * ------------------------------------------------------------- * * SAMPLE RUN: * * (Evaluate I0 to I4 for argument Z=(1.0,2.0) ). * * * * zr(0) = 0.187854 * * zi(0) = 0.646169 * * zr(1) = -0.079933 * * zi(1) = 0.790623 * * zr(2) = -0.412672 * * zi(2) = 0.265974 * * zr(3) = -0.175353 * * zi(3) = -0.082431 * * zr(4) = -0.004414 * * zi(4) = -0.055957 * * NZ = 0 * * Error code: 0 * * * * ------------------------------------------------------------- * * Ref.: From Numath Library By Tuan Dang Trong in Fortran 77. * * * * TPW Release 1.0 By J-P Moreau, Paris * * (www.jpmoreau.fr) * ***************************************************************** Note: Used files: CBess0,CBess00,CBess1,CBess2,Complex. ------------------------------------------------------- } PROGRAM TEST_ZBESI; Uses WinCrt, Complex, CBess1 {for ZBINU}; Var zr,zi: double; cyr, cyi: VEC; i,n,nz,ierr: integer; Procedure ZBESI(ZR, ZI, FNU:double; KODE, N: integer; Var CYR, CYI: VEC; var NZ, IERR:integer); {***BEGIN PROLOGUE ZBESI !***DATE WRITTEN 830501 (YYMMDD) !***REVISION DATE 830501 (YYMMDD) !***CATEGORY NO. B5K !***KEYWORDS I-BESSEL FUNCTION,COMPLEX BESSEL FUNCTION, ! MODIFIED BESSEL FUNCTION OF THE FIRST KIND !***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES !***PURPOSE TO COMPUTE I-BESSEL FUNCTIONS OF COMPLEX ARGUMENT !***DESCRIPTION ! ! ***A DOUBLE PRECISION ROUTINE*** ! ON KODE=1, ZBESI COMPUTES AN N MEMBER SEQUENCE OF COMPLEX ! BESSEL FUNCTIONS CY(J)=I(FNU+J-1,Z) FOR REAL, NONNEGATIVE ! ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z IN THE CUT PLANE ! -PI.LT.ARG(Z).LE.PI. ON KODE=2, ZBESI RETURNS THE SCALED ! FUNCTIONS ! ! CY(J)=EXP(-ABS(X))*I(FNU+J-1,Z) J = 1,...,N , X=REAL(Z) ! ! WITH THE EXPONENTIAL GROWTH REMOVED IN BOTH THE LEFT AND ! RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION ! ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS ! (REF. 1). ! ! INPUT ZR,ZI,FNU ARE DOUBLE PRECISION ! ZR,ZI - Z=CMPLX(ZR,ZI), -PI.LT.ARG(Z).LE.PI ! FNU - ORDER OF INITIAL I FUNCTION, FNU.GE.0.0D0 ! KODE - A PARAMETER TO INDICATE THE SCALING OPTION ! KODE= 1 RETURNS ! CY(J)=I(FNU+J-1,Z), J=1,...,N ! = 2 RETURNS ! CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)), J=1,...,N ! N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1 ! ! OUTPUT CYR,CYI ARE DOUBLE PRECISION ! CYR,CYI- DOUBLE PRECISION VECTORS WHOSE FIRST N COMPONENTS ! CONTAIN REAL AND IMAGINARY PARTS FOR THE SEQUENCE ! CY(J)=I(FNU+J-1,Z) OR ! CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)) J=1,...,N ! DEPENDING ON KODE, X=REAL(Z) ! NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW, ! NZ= 0 , NORMAL RETURN ! NZ.GT.0 , LAST NZ COMPONENTS OF CY SET TO ZERO ! TO UNDERFLOW, CY(J)=CMPLX(0.0D0,0.0D0) ! J = N-NZ+1,...,N ! IERR - ERROR FLAG ! IERR=0, NORMAL RETURN - COMPUTATION COMPLETED ! IERR=1, INPUT ERROR - NO COMPUTATION ! IERR=2, OVERFLOW - NO COMPUTATION, REAL(Z) TOO ! LARGE ON KODE=1 ! IERR=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE ! BUT LOSSES OF SIGNIFCANCE BY ARGUMENT ! REDUCTION PRODUCE LESS THAN HALF OF MACHINE ! ACCURACY ! IERR=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA- ! TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI- ! CANCE BY ARGUMENT REDUCTION ! IERR=5, ERROR - NO COMPUTATION, ! ALGORITHM TERMINATION CONDITION NOT MET ! !***LONG DESCRIPTION ! ! THE COMPUTATION IS CARRIED OUT BY THE POWER SERIES FOR ! SMALL CABS(Z), THE ASYMPTOTIC EXPANSION FOR LARGE CABS(Z), ! THE MILLER ALGORITHM NORMALIZED BY THE WRONSKIAN AND A ! NEUMANN SERIES FOR IMTERMEDIATE MAGNITUDES, AND THE ! UNIFORM ASYMPTOTIC EXPANSIONS FOR I(FNU,Z) AND J(FNU,Z) ! FOR LARGE ORDERS. BACKWARD RECURRENCE IS USED TO GENERATE ! SEQUENCES OR REDUCE ORDERS WHEN NECESSARY. ! ! THE CALCULATIONS ABOVE ARE DONE IN THE RIGHT HALF PLANE AND ! CONTINUED INTO THE LEFT HALF PLANE BY THE FORMULA ! ! I(FNU,Z*EXP(M*PI)) = EXP(M*PI*FNU)*I(FNU,Z) REAL(Z).GT.0.0 ! M = +I OR -I, I**2=-1 ! ! FOR NEGATIVE ORDERS,THE FORMULA ! ! I(-FNU,Z) = I(FNU,Z) + (2/PI)*SIN(PI*FNU)*K(FNU,Z) ! ! CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE ! THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE ! INTEGER,THE MAGNITUDE OF I(-FNU,Z)=I(FNU,Z) IS A LARGE ! NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT AN INTEGER, ! K(FNU,Z) DOMINATES IN MAGNITUDE WITH A LARGE POSITIVE POWER OF ! TEN AND THE MOST THAT THE SECOND TERM CAN BE REDUCED IS BY ! UNIT ROUNDOFF FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN ! OCCUR WITHIN UNIT ROUNDOFF OF A LARGE INTEGER FOR FNU. HERE, ! LARGE MEANS FNU.GT.CABS(Z). ! ! IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE- ! MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS ! LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. ! CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN ! LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG ! IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS ! DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION. ! IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS ! LOST AND IERR=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS ! MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE ! INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS ! RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3 ! ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION ! ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION ! ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN ! THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT ! TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS ! IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC. ! SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES. ! ! THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX ! BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT ! ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE- ! SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE ! ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))), ! ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF ! CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY ! HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN ! ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY ! SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER ! THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K, ! 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS ! THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER ! COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY ! BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER ! COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE ! MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES, ! THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P, ! OR -PI/2+P. ! !***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ ! AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF ! COMMERCE, 1955. ! ! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT ! BY D. E. AMOS, SAND83-0083, MAY, 1983. ! ! COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT ! AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983 ! ! A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ! ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85- ! 1018, MAY, 1985 ! ! A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX ! ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS. ! MATH. SOFTWARE, 1986 ! !***ROUTINES CALLED ZBINU,I1MACH,D1MACH !***END PROLOGUE ZBESI ! COMPLEX CONE,CSGN,CW,CY,CZERO,Z,ZN } Label 40, 120, 130, 260, Return; Var AA, ALIM, ARG, CONEI, CONER, CSGNI, CSGNR, DIG, ELIM, FNUL, PI, RL, R1M5, STR, TOL, ZNI, ZNR, AZ, BB, FN: Double; I, INU, K, K1, K2, NN: Integer; Begin {***FIRST EXECUTABLE STATEMENT ZBESI } CONER:=1.0; CONEI:=0.0; IERR := 0; NZ:=0; IF FNU < 0.0 Then IERR:=1; IF (KODE < 1) OR (KODE > 2) Then IERR:=1; IF N < 1 Then IERR:=1; IF IERR <> 0 Then goto Return; {----------------------------------------------------------------------- ! SET PARAMETERS RELATED TO MACHINE CONSTANTS. ! TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1E-18. ! ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT. ! EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL AND ! EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR ! UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE. ! RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z. ! DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10^(-DIG). ! FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU. !----------------------------------------------------------------------} TOL := DMAX(D1MACH(4),1E-18); K1 := I1MACH(15); K2 := I1MACH(16); R1M5 := D1MACH(5); K := IMIN(ABS(K1), ABS(K2)); ELIM := 2.303*(K*R1M5-3.0); K1 := I1MACH(14) - 1; AA := R1M5*K1; DIG := DMIN(AA,18.0); AA := AA*2.303; ALIM := ELIM + DMAX(-AA,-41.45); RL := 1.2*DIG + 3.0; FNUL := 10.0 + 6.0*(DIG-3.0); {----------------------------------------------------------------------- ! TEST FOR PROPER RANGE !----------------------------------------------------------------------} AZ := ZABS(ZR,ZI); FN := FNU+1.0*(N-1); AA := 0.5/TOL; BB:=I1MACH(9)*0.5; AA := DMIN(AA,BB); IF AZ > AA Then GOTO 260; IF FN > AA Then GOTO 260; AA := SQRT(AA); IF AZ > AA Then IERR:=3; IF FN > AA Then IERR:=3; ZNR := ZR; ZNI := ZI; CSGNR := CONER; CSGNI := CONEI; IF ZR >= 0.0 Then GOTO 40; ZNR := -ZR; ZNI := -ZI; {----------------------------------------------------------------------- ! CALCULATE CSGN=EXP(FNU*PI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE ! WHEN FNU IS LARGE !----------------------------------------------------------------------} INU := Round(FNU); ARG := (FNU-1.0*INU)*PI; IF ZI < 0.0 Then ARG := -ARG; CSGNR := COS(ARG); CSGNI := SIN(ARG); IF (INU MOD 2) = 0 Then GOTO 40; CSGNR := -CSGNR; CSGNI := -CSGNI; 40: ZBINU(ZNR, ZNI, FNU, KODE, N, CYR, CYI, NZ, RL, FNUL, TOL, ELIM, ALIM); IF NZ < 0 Then GOTO 120; IF ZR >= 0.0 Then goto RETURN; {----------------------------------------------------------------------- ! ANALYTIC CONTINUATION TO THE LEFT HALF PLANE !----------------------------------------------------------------------} NN := N - NZ; IF NN = 0 Then goto RETURN; For I:=1 to NN do begin STR := CYR[I]*CSGNR - CYI[I]*CSGNI; CYI[I] := CYR[I]*CSGNI + CYI[I]*CSGNR; CYR[I] := STR; CSGNR := -CSGNR; CSGNI := -CSGNI end; goto RETURN; 120: IF NZ = -2 Then GOTO 130; NZ := 0; IERR:=2; goto RETURN; 130: NZ:=0; IERR:=5; goto RETURN; 260: NZ:=0; IERR:=4; Return: End; {ZBESI} {main program} Begin n:=5; zr:=1.0; zi:=2.0; ZBESI(zr,zi,0,1,n,cyr,cyi,nz,ierr); writeln; for i:=1 to n do begin writeln(' zr(', i-1, ') = ', cyr[i]:10:6); writeln(' zi(', i-1, ') = ', cyi[i]:10:6) end; writeln(' NZ = ', NZ); writeln(' Error code: ', ierr); writeln; ReadKey; DoneWinCrt END. {end of file tzbesi.pas}