*DECK CBESK SUBROUTINE CBESK (Z, FNU, KODE, N, CY, NZ, IERR) C***BEGIN PROLOGUE CBESK C***PURPOSE Compute a sequence of the Bessel functions K(a,z) for C complex argument z and real nonnegative orders a=b,b+1, C b+2,... where b>0. A scaling option is available to C help avoid overflow. C***LIBRARY SLATEC C***CATEGORY C10B4 C***TYPE COMPLEX (CBESK-C, ZBESK-C) C***KEYWORDS BESSEL FUNCTIONS OF COMPLEX ARGUMENT, K BESSEL FUNCTIONS, C MODIFIED BESSEL FUNCTIONS C***AUTHOR Amos, D. E., (SNL) C***DESCRIPTION C C On KODE=1, CBESK computes an N member sequence of complex C Bessel functions CY(L)=K(FNU+L-1,Z) for real nonnegative C orders FNU+L-1, L=1,...,N and complex Z.NE.0 in the cut C plane -piCBESJ returns the scaled C functions C C CY(L) = exp(Z)*K(FNU+L-1,Z), L=1,...,N C C which remove the exponential growth in both the left and C right half planes as Z goes to infinity. Definitions and C notation are found in the NBS Handbook of Mathematical C Functions (Ref. 1). C C Input C Z - Nonzero argument of type COMPLEX C FNU - Initial order of type REAL, FNU>=0 C KODE - A parameter to indicate the scaling option C KODE=1 returns C CY(L)=K(FNU+L-1,Z), L=1,...,N C =2 returns C CY(L)=K(FNU+L-1,Z)*EXP(Z), L=1,...,N C N - Number of terms in the sequence, N>=1 C C Output C CY - Result vector of type COMPLEX C NZ - Number of underflows set to zero C NZ=0 Normal return C NZ>0 CY(L)=0 for NZ values of L (if Re(Z)>0 C then CY(L)=0 for L=1,...,NZ; in the C complementary half plane the underflows C may not be in an uninterrupted sequence) C IERR - Error flag C IERR=0 Normal return - COMPUTATION COMPLETED C IERR=1 Input error - NO COMPUTATION C IERR=2 Overflow - NO COMPUTATION C (abs(Z) too small and/or FNU+N-1 C too large) C IERR=3 Precision warning - COMPUTATION COMPLETED C (Result has half precision or less C because abs(Z) or FNU+N-1 is large) C IERR=4 Precision error - NO COMPUTATION C (Result has no precision because C abs(Z) or FNU+N-1 is too large) C IERR=5 Algorithmic error - NO COMPUTATION C (Termination condition not met) C C *Long Description: C C Equations of the reference are implemented to compute K(a,z) C for small orders a and a+1 in the right half plane Re(z)>=0. C Forward recurrence generates higher orders. The formula C C K(a,z*exp((t)) = exp(-t)*K(a,z) - t*I(a,z), Re(z)>0 C t = i*pi or -i*pi C C continues K to the left half plane. C C For large orders, K(a,z) is computed by means of its uniform C asymptotic expansion. C C For negative orders, the formula C C K(-a,z) = K(a,z) C C can be used. C C CBESK assumes that a significant digit sinh function is C available. C C In most complex variable computation, one must evaluate ele- C mentary functions. When the magnitude of Z or FNU+N-1 is C large, losses of significance by argument reduction occur. C Consequently, if either one exceeds U1=SQRT(0.5/UR), then C losses exceeding half precision are likely and an error flag C IERR=3 is triggered where UR=R1MACH(4)=UNIT ROUNDOFF. Also, C if either is larger than U2=0.5/UR, then all significance is C lost and IERR=4. In order to use the INT function, arguments C must be further restricted not to exceed the largest machine C integer, U3=I1MACH(9). Thus, the magnitude of Z and FNU+N-1 C is restricted by MIN(U2,U3). In IEEE arithmetic, U1,U2, and C U3 approximate 2.0E+3, 4.2E+6, 2.1E+9 in single precision C and 4.7E+7, 2.3E+15 and 2.1E+9 in double precision. This C makes U2 limiting in single precision and U3 limiting in C double precision. This means that one can expect to retain, C in the worst cases on IEEE machines, no digits in single pre- C cision and only 6 digits in double precision. Similar con- C siderations hold for other machines. C C The approximate relative error in the magnitude of a complex C Bessel function can be expressed as P*10**S where P=MAX(UNIT C ROUNDOFF,1.0E-18) is the nominal precision and 10**S repre- C sents the increase in error due to argument reduction in the C elementary functions. Here, S=MAX(1,ABS(LOG10(ABS(Z))), C ABS(LOG10(FNU))) approximately (i.e., S=MAX(1,ABS(EXPONENT OF C ABS(Z),ABS(EXPONENT OF FNU)) ). However, the phase angle may C have only absolute accuracy. This is most likely to occur C when one component (in magnitude) is larger than the other by C several orders of magnitude. If one component is 10**K larger C than the other, then one can expect only MAX(ABS(LOG10(P))-K, C 0) significant digits; or, stated another way, when K exceeds C the exponent of P, no significant digits remain in the smaller C component. However, the phase angle retains absolute accuracy C because, in complex arithmetic with precision P, the smaller C component will not (as a rule) decrease below P times the C magnitude of the larger component. In these extreme cases, C the principal phase angle is on the order of +P, -P, PI/2-P, C or -PI/2+P. C C***REFERENCES 1. M. Abramowitz and I. A. Stegun, Handbook of Mathe- C matical Functions, National Bureau of Standards C Applied Mathematics Series 55, U. S. Department C of Commerce, Tenth Printing (1972) or later. C 2. D. E. Amos, Computation of Bessel Functions of C Complex Argument, Report SAND83-0086, Sandia National C Laboratories, Albuquerque, NM, May 1983. C 3. D. E. Amos, Computation of Bessel Functions of C Complex Argument and Large Order, Report SAND83-0643, C Sandia National Laboratories, Albuquerque, NM, May C 1983. C 4. D. E. Amos, A Subroutine Package for Bessel Functions C of a Complex Argument and Nonnegative Order, Report C SAND85-1018, Sandia National Laboratory, Albuquerque, C NM, May 1985. C 5. D. E. Amos, A portable package for Bessel functions C of a complex argument and nonnegative order, ACM C Transactions on Mathematical Software, 12 (September C 1986), pp. 265-273. C C***ROUTINES CALLED CACON, CBKNU, CBUNK, CUOIK, I1MACH, R1MACH C***REVISION HISTORY (YYMMDD) C 830501 DATE WRITTEN C 890801 REVISION DATE from Version 3.2 C 910415 Prologue converted to Version 4.0 format. (BAB) C 920128 Category corrected. (WRB) C 920811 Prologue revised. (DWL) C***END PROLOGUE CBESK