CEPHES MATHEMATICAL FUNCTION LIBRARY This computer software library is a collection of more than 400 high quality mathematical routines for scientific and engineering applications. All are written entirely in C language. Many of the functions are supplied in six different arithmetic precisions: 32 bit single (24-bit significand), 64 bit IEEE double (53-bit), 64 bit DEC (56-bit), 80 or 96 bit IEEE long double (64-bit), and extended precision formats having 144-bit and 336-bit significands. The extended precision arithmetic is included with the function library. The library treats about 180 different mathematical functions. In addition to the elementary arithmetic and transcendental routines, the library includes a substantial collection of probability integrals, Bessel functions, and higher transcendental functions. There are complex variable routines covering complex arithmetic, complex logarithm and exponential, and complex trigonometric functions. Each function subroutine has been tested by comparing at a large number of points against high precision check routines. The test programs use floating point arithmetic having 144 bit (43 decimal) precision. Thus the actual accuracy of each program is reported, not merely the result of a consistency test. Test results are given with the description of each routine. The routines have been characterized and tested in IEEE Std 754 double precision arithmetic (both Intel and Motorola formats), used on IBM PC and a growing number of other computers, and also in the popular DEC/IBM double precision format. For DEC and IEEE arithmetic, numerical constants and approximation coefficients are supplied as integer arrays in order to eliminate conversion errors that might be introduced by the language compiler. All coefficients are also supplied in the normal decimal scientific notation so that the routines can be compiled and used on other machines that do not support either of the above numeric formats. A single, common error handling routine is supplied. Error conditions produce a display of the function name and error type. The user may easily insert modifications to implement any desired action on specified types of error. The following table summarizes the current contents of the double precision library. See also the corresponding documentation for the single and long double precision libraries. Accuracies reported for DEC and IEEE arithmetic are with arithmetic rounding precision limited to 56 and 53 bits, respectively. Higher precision may be realized if an arithmetic unit such as the 8087 or 68881 is used in conjunction with an optimizing compiler. The accuracy figures are experimentally measured; they are not guaranteed maximum errors. Documentation is included on the distribution media as Unix-style manual pages that describe the functions and their invocation. The primary documentation for the library functions is the book by Moshier, Methods and Programs for Mathematical Functions, Prentice-Hall, 1989. Function Name Accuracy -------- ---- DEC IEEE ---- ---- Arithmetic and Algebraic Square root sqrt 2e-17 2e-16 Long integer square root lsqrt 1 1 Cube root cbrt 2e-17 2e-16 Evaluate polynomial polevl Evaluate Chebyshev series chbevl Round to nearest integer value round Truncate upward to integer ceil Truncate downward to integer floor Extract exponent frexp Add integer to exponent ldexp Absolute value fabs Rational arithmetic euclid Roots of a polynomial polrt Reversion of power series revers IEEE 854 arithmetic ieee Polynomial arithmetic (polyn.c): Add polynomials poladd Subtract polynomials polsub Multiply polynomials polmul Divide polynomials poldiv Substitute polynomial variable polsbt Evaluate polynomial poleva Set all coefficients to zero polclr Copy coefficients polmov Display coefficients polprt Note, polyr.c contains routines corresponding to the above for polynomials with rational coefficients. Power series manipulations (polmisc.c): Square root of a polynomial polsqt Arctangent polatn Sine polsin Reversion of power series revers Exponential and Trigonometric Arc cosine acos 3e-17 3e-16 Arc hyperbolic cosine acosh 4e-17 5e-16 Arc hyperbolic sine asinh 5e-17 4e-16 Arc hyperbolic tangent atanh 3e-17 2e-16 Arcsine asin 6e-17 5e-16 Arctangent atan 4e-17 3e-16 Quadrant correct arctangent atan2 4e-17 4e-16 Cosine cos 3e-17 2e-16 Cosine of arg in degrees cosdg 4e-17 2e-16 Exponential, base e exp 3e-17 2e-16 Exponential, base 2 exp2 2e-17 2e-16 Exponential, base 10 exp10 3e-17 2e-16 Hyperbolic cosine cosh 3e-17 3e-16 Hyperbolic sine sinh 4e-17 3e-16 Hyperbolic tangent tanh 3e-17 3e-16 Logarithm, base e log 2e-17 2e-16 Logarithm, base 2 log2 2e-16 Logarithm, base 10 log10 3e-17 2e-16 Power pow 1e-15 2e-14 Integer Power powi 9e-14 Sine sin 3e-17 2e-16 Sine of arg in degrees sindg 4e-17 2e-16 Tangent tan 4e-17 3e-16 Tangent of arg in degrees tandg 3e-17 3e-16 Exponential integral Exponential integral expn 2e-16 2e-15 Hyperbolic cosine integral shichi 9e-17 8e-16 Hyperbolic sine integral shichi 9e-17 7e-16 Cosine integral sici 8e-17A 7e-16 Sine integral sici 4e-17A 4e-16 Gamma Beta beta 8e-15 8e-14 Factorial fac 2e-17 2e-15 Gamma gamma 1e-16 1e-15 Logarithm of gamma function lgam 5e-17 5e-16 Incomplete beta integral incbet 4e-14 4e-13 Inverse beta integral incbi 3e-13 8e-13 Incomplete gamma integral igam 5e-15 4e-14 Complemented gamma integral igamc 3e-15 1e-12 Inverse gamma integral igami 9e-16 1e-14 Psi (digamma) function psi 2e-16 1e-15 Reciprocal Gamma rgamma 1e-16 1e-15 Error function Error function erf 5e-17 4e-16 Complemented error function erfc 5e-16 6e-14 Dawson's integral dawsn 7e-16 7e-16 Fresnel integral (C) fresnl 2e-16 2e-15 Fresnel integral (S) fresnl 2e-16 2e-15 Bessel Airy (Ai) airy 6e-16A 2e-15A Airy (Ai') airy 6e-16A 5e-15A Airy (Bi) airy 6e-16A 4e-15A Airy (Bi') airy 6e-16A 5e-15A Bessel, order 0 j0 4e-17A 4e-16A Bessel, order 1 j1 4e-17A 3e-16A Bessel, order n jn 7e-17A 2e-15A Bessel, noninteger order jv 5e-15A Bessel, second kind, order 0 y0 7e-17A 1e-15A Bessel, second kind, order 1 y1 9e-17A 1e-15A Bessel, second kind, order n yn 3e-16A 3e-15A Bessel, noninteger order yv see struve.c Modified Bessel, order 0 i0 8e-17 6e-16 Exponentially scaled i0 i0e 8e-17 5e-16 Modified Bessel, order 1 i1 1e-16 2e-15 Exponentially scaled i1 i1e 1e-16 2e-15 Modified Bessel, nonint. order iv 3e-15 2e-14 Mod. Bessel, 3rd kind, order 0 k0 1e-16 1e-15 Exponentially scaled k0 k0e 1e-16 1e-15 Mod. Bessel, 3rd kind, order 1 k1 9e-17 1e-15 Exponentially scaled k1 k1e 9e-17 8e-16 Mod. Bessel, 3rd kind, order n kn 1e-9 2e-8 Hypergeometric Confluent hypergeometric hyperg 1e-15 2e-14 Gauss hypergeometric function hyp2f1 4e-11 9e-8 2F0 hyp2f0f see hyperg.c 1F2 onef2f see struve.c 3F0 threef0f see struve.c Elliptic Complete elliptic integral (E) ellpe 3e-17 2e-16 Incomplete elliptic integral (E) ellie 2e-16 2e-15 Complete elliptic integral (K) ellpk 4e-17 3e-16 Incomplete elliptic integral (K) ellik 9e-17 6e-16 Jacobian elliptic function (sn) ellpj 5e-16A 4e-15A Jacobian elliptic function (cn) ellpj 4e-15A Jacobian elliptic function (dn) ellpj 1e-12A Jacobian elliptic function (phi) ellpj 9e-16 Probability Binomial distribution bdtr 4e-14 4e-13 Complemented binomial bdtrc 4e-14 4e-13 Inverse binomial bdtri 3e-13 8e-13 Chi square distribution chdtr 5e-15 3e-14 Complemented Chi square chdtrc 3e-15 2e-14 Inverse Chi square chdtri 9e-16 6e-15 F distribution fdtr 4e-14 4e-13 Complemented F fdtrc 4e-14 4e-13 Inverse F distribution fdtri 3e-13 8e-13 Gamma distribution gdtr 5e-15 3e-14 Complemented gamma gdtrc 3e-15 2e-14 Negative binomial distribution nbdtr 4e-14 4e-13 Complemented negative binomial nbdtrc 4e-14 4e-13 Normal distribution ndtr 2e-15 3e-14 Inverse normal distribution ndtri 1e-16 7e-16 Poisson distribution pdtr 3e-15 2e-14 Complemented Poisson pdtrc 5e-15 3e-14 Inverse Poisson distribution pdtri 3e-15 5e-14 Student's t distribution stdtr 2e-15 2e-14 Miscellaneous Dilogarithm spence 3e-16 4e-15 Riemann Zeta function zetac 1e-16 1e-15 Two argument zeta function zeta Struve function struve Matrix Fast Fourier transform fftr Simultaneous linear equations simq Simultaneous linear equations gels (symmetric coefficient matrix) Matrix inversion minv Matrix multiply mmmpy Matrix times vector mvmpy Matrix transpose mtransp Eigenvectors (symmetric matrix) eigens Levenberg-Marquardt nonlinear equations lmdif Numerical Integration Simpson's rule simpsn Runge-Kutta runge - see de118 Adams-Bashforth adams - see de118 Complex Arithmetic Complex addition cadd 1e-17 1e-16 Subtraction csub 1e-17 1e-16 Multiplication cmul 2e-17 2e-16 Division cdiv 5e-17 4e-16 Absolute value cabs 3e-17 3e-16 Square root csqrt 3e-17 3e-16 Complex Exponential and Trigonometric Exponential cexp 4e-17 3e-16 Logarithm clog 9e-17 5e-16A Cosine ccos 5e-17 4e-16 Arc cosine cacos 2e-15 2e-14 Sine csin 5e-17 4e-16 Arc sine casin 2e-15 2e-14 Tangent ctan 7e-17 7e-16 Arc tangent catan 1e-16 2e-15 Cotangent ccot 7e-17 9e-16 Applications Minimax rational approximations to functions remes Digital elliptic filters ellf Numerical integration of the Moon and planets de118 IEEE compliance test for printf(), scanf() ieetst Long Double Precision Functions Function Name Accuracy -------- ---- -------- Arc hyperbolic cosine acoshl 2e-19 Arc cosine acosl 1e-19 Arc hyperbolic sine asinhl 2e-19 Arcsine asinl 3e-19 Arc hyperbolic tangent atanhl 1e-19 Arctangent atanl 1e-19 Quadrant correct arctangent atan2l 2e-19 Cube root cbrtl 7e-20 Truncate upward to integer ceill Hyperbolic cosine coshl 1e-19 Cosine cosl 1e-19 Cotangent cotl 2e-19 Exponential, base e expl 1e-19 Exponential, base 2 exp2l 9e-20 Exponential, base 10 exp10l 1e-19 Absolute value fabsl Truncate downward to integer floorl Extract exponent frexpl Add integer to exponent ldexpl Logarithm, base e logl 9e-20 Logarithm, base 2 log2l 1e-19 Logarithm, base 10 log10l 9e-20 Integer Power powil 4e-17 Power powl 3e-18 Hyperbolic sine sinhl 2e-19 Sine sinl 1e-19 Square root sqrtl 8e-20 Hyperbolic tangent tanhl 1e-19 Tangent tanl 2e-19 Single Precision Routines Function Name Accuracy -------- ---- -------- Arithmetic Truncate upward to integer ceilf Truncate downward to integer floorf Extract exponent frexpf Add integer to exponent ldexpf Absolute value fabsf Square root sqrtf 9e-8 Cube root cbrtf 8e-8 Polynomials and Power Series Polynomial arithmetic (polynf.c): Add polynomials poladdf Subtract polynomials polsubf Multiply polynomials polmulf Divide polynomials poldivf Substitute polynomial variable polsbtf Evaluate polynomial polevaf Set all coefficients to zero polclrf Copy coefficients polmovf Display coefficients polprtf Note, polyr.c contains routines corresponding to the above for polynomials with rational coefficients. Evaluate polynomial polevlf (coefficients in reverse order) Evaluate Chebyshev series chbevlf (coefficients in reverse order) Exponential and Trigonometric Arc cosine acosf 1e-7 Arc hyperbolic cosine acoshf 2e-7 Arc hyperbolic sine asinhf 2e-7 Arc hyperbolic tangent atanhf 1e-7 Arcsine asinf 3e-7 Arctangent atanf 2e-7 Quadrant correct arctangent atan2f 2e-7 Cosine cosf 1e-7 Cosine of arg in degrees cosdgf 1e-7 Cotangent cotf 3e-7 Cotangent of arg in degrees cotdgf 2e-7 Exponential, base e expf 2e-7 Exponential, base 2 exp2f 2e-7 Exponential, base 10 exp10f 1e-7 Hyperbolic cosine coshf 2e-7 Hyperbolic sine sinhf 1e-7 Hyperbolic tangent tanhf 1e-7 Logarithm, base e logf 8e-8 Logarithm, base 2 log2f 1e-7 Logarithm, base 10 log10f 1e-7 Power powf 1e-6 Integer Power powif 1e-6 Sine sinf 1e-7 Sine of arg in degrees sindgf 1e-7 Tangent tanf 3e-7 Tangent of arg in degrees tandgf 2e-7 Exponential integral Exponential integral expnf 6e-7 Hyperbolic cosine integral shichif 4e-7A Hyperbolic sine integral shichif 4e-7 Cosine integral sicif 2e-7A Sine integral sicif 4e-7A Gamma Beta betaf 4e-5 Factorial facf 6e-8 Gamma gammaf 6e-7 Logarithm of gamma function lgamf 7e-7(A) Incomplete beta integral incbetf 2e-4 Inverse beta integral incbif 3e-4 Incomplete gamma integral igamf 8e-6 Complemented gamma integral igamcf 8e-6 Inverse gamma integral igamif 1e-5 Psi (digamma) function psif 8e-7 Reciprocal Gamma rgammaf 9e-7 Error function Error function erff 2e-7 Complemented error function erfcf 4e-6 Dawson's integral dawsnf 4e-7 Fresnel integral (C) fresnlf 1e-6 Fresnel integral (S) fresnlf 1e-6 Bessel Airy (Ai) airyf 1e-5A Airy (Ai') airyf 9e-6A Airy (Bi) airyf 2e-6A Airy (Bi') airyf 2e-6A Bessel, order 0 j0f 2e-7A Bessel, order 1 j1f 2e-7A Bessel, order n jnf 4e-7A Bessel, noninteger order jvf 2e-6A Bessel, second kind, order 0 y0f 2e-7A Bessel, second kind, order 1 y1f 2e-7A Bessel, second kind, order n ynf 2e-6A Bessel, second kind, order v yvf see struvef.c Modified Bessel, order 0 i0f 4e-7 Exponentially scaled i0 i0ef 4e-7 Modified Bessel, order 1 i1f 2e-6 Exponentially scaled i1 i1ef 2e-6 Modified Bessel, nonint. order ivf 9e-6 Mod. Bessel, 3rd kind, order 0 k0f 8e-7 Exponentially scaled k0 k0ef 8e-7 Mod. Bessel, 3rd kind, order 1 k1f 5e-7 Exponentially scaled k1 k1ef 5e-7 Mod. Bessel, 3rd kind, order n knf 2e-4A Hypergeometric Confluent hypergeometric 1F1 hypergf 1e-5 Gauss hypergeometric function hyp2f1f 2e-3 2F0 hyp2f0f see hypergf.c 1F2 onef2f see struvef.c 3F0 threef0f see struvef.c Elliptic Complete elliptic integral (E) ellpef 1e-7 Incomplete elliptic integral (E) ellief 5e-7 Complete elliptic integral (K) ellpkf 1e-7 Incomplete elliptic integral (K) ellikf 3e-7 Jacobian elliptic function (sn) ellpjf 2e-6A Jacobian elliptic function (cn) ellpjf 2e-6A Jacobian elliptic function (dn) ellpjf 1e-3A Jacobian elliptic function (phi) ellpjf 4e-7 Probability Binomial distribution bdtrf 7e-5 Complemented binomial bdtrcf 6e-5 Inverse binomial bdtrif 4e-5 Chi square distribution chdtrf 3e-5 Complemented Chi square chdtrcf 3e-5 Inverse Chi square chdtrif 2e-5 F distribution fdtrf 2e-5 Complemented F fdtrcf 7e-5 Inverse F distribution fdtrif 4e-5A Gamma distribution gdtrf 6e-5 Complemented gamma gdtrcf 9e-5 Negative binomial distribution nbdtrf 2e-4 Complemented negative binomial nbdtrcf 1e-4 Normal distribution ndtrf 2e-5 Inverse normal distribution ndtrif 4e-7 Poisson distribution pdtrf 7e-5 Complemented Poisson pdtrcf 8e-5 Inverse Poisson distribution pdtrif 9e-6 Student's t distribution stdtrf 2e-5 Miscellaneous Dilogarithm spencef 4e-7 Riemann Zeta function zetacf 6e-7 Two argument zeta function zetaf 7e-7 Struve function struvef 9e-5 Complex Arithmetic Complex addition caddf 6e-8 Subtraction csubf 6e-8 Multiplication cmulf 1e-7 Division cdivf 2e-7 Absolute value cabsf 1e-7 Square root csqrtf 2e-7 Complex Exponential and Trigonometric Exponential cexpf 1e-7 Logarithm clogf 3e-7A Cosine ccosf 2e-7 Arc cosine cacosf 9e-6 Sine csinf 2e-7 Arc sine casinf 1e-5 Tangent ctanf 3e-7 Arc tangent catanf 2e-6 Cotangent ccotf 4e-7 QLIB Extended Precision Mathematical Library q100asm.bat Create 100-decimal Q type library (for IBM PC MSDOS) q100asm.rsp qlibasm.bat 43-decimal Q type library (for IBM PC MSDOS) qlibasm.rsp qlib.lib Q type library, 43 decimal qlib100.lib Q type library, 100 decimal qlib120.lib Q type library, 120 decimal Function calling arguments: NQ is the number of 16-bit short integers in a number (see qhead.h) short x[NQ], x1[NQ], ... are inputs short y[NQ], y1[NQ], ... are outputs mconf.h Machine configuration file mtherr.c Common error handling routine qacosh.c Arc hyperbolic cosine qacosh( x, y ); qairy.c Airy functions qairy( x, Ai, Ai', Bi, Bi' ); Also see source program for auxiliary functions. qasin.c Arc sine qasin( x, y ); qasinh.c Arc hyperbolic sine qasinh( x, y ); qatanh.c Arc hyperbolic tangent qatanh( x, y ); qatn.c Arc tangent qatn( x, y ); qatn2( x1, x2, y ); y = radian angle whose tangent is x2/x1 qbeta.c Beta function qbeta( x, y ); qcbrt.c Cube root qcbrt( x, y ); qcmplx.c Complex variable functions: qcabs absolute value qcabs( y ); qcadd add qcsub subtract qcsub( a, b, y ); y = b - a qcmul multiply qcdiv divide qcdiv( d, n, y ); y = n/d qcmov move qcneg negate qcneg( y ); qcexp exponential function qclog logarithm qcsin sine qccos cosine qcasin arcsine qcacos arccosine qcsqrt square root qctan tangent qccot cotangent qcatan arctangent qcos.c Cosine qcosm1( x, y ); y = cos(x) - 1 qcosh.c Hyperbolic cosine qctst1.c Universal function test program for complex variables qdawsn.c Dawson's integral qellie.c Incomplete elliptic integral (E) qellik.c Incomplete elliptic integral (K) qellpe.c Complete elliptic integral (E) qellpj.c Jacobian elliptic functions sn, cn, dn, phi qellpj( u, m, sn, cn, dn, phi ); sn = sn(u|m), etc. qellpk.c Complete elliptic integral (K) qerf.c Error integral qerfc.c Complementary error integral qeuclid.c Q type rational arithmetic: qradd add fractions qrsub subtract fractions qrmul multiply fractions qrdiv divide fractions qreuclid reduce to lowest terms qexp.c Exponential function qexp10.c Base 10 exponential function qexp2.c Base 2 exponential function qexp21.c 2**x - 1 qexpn.c Exponential integral qf68k.a Q type arithmetic for 68000 OS-9 qf68k.asm Q type arithmetic for 68000 (Definicon assembler) qf68k.s Q type arithmetic for 68000 (System V Unix) qfac.c Factorial qfresf.c Fresnel integral S(x) Fresnel integral C(x) qgamma.c Gamma function log Gamma function qhead.asm Q type configuration file for assembly language qhead.h Q type configuration file for C language qhy2f1.c Gauss hypergeometric function qhyp.c Confluent hypergeometric function qigam.c Incomplete gamma integral qigami.c Functional inverse of incomplete gamma integral qin.c Bessel function In qincb.c Incomplete beta integral qincbi.c Functional inverse of incomplete beta integral qine.c Exponentially weighted In qjn.c Bessel function Jv (noninteger order) qhank Hankel's asymptotic expansion qjypn.c Auxiliary Bessel functions qjyqn.c qkn.c modified Bessel function Kn qkne.c Exponentially weighted Kn qlog.c Natural logarithm qlog1.c log(1+x) qlog10.c Common logarithm qndtr.c Gaussian distribution function qndtri.c Functional inverse of Gaussian distribution function qpolyr.c Q type polynomial arithmetic, rational coefficients: poleva Evaluate polynomial a(t) at t = x. polprt Print the coefficients of a to D digits. polclr Set a identically equal to zero, up to a[na]. polmov Set b = a. poladd c = b + a, nc = max(na,nb) polsub c = b - a, nc = max(na,nb) polmul c = b * a, nc = na+nb poldiv c = b / a, nc = MAXPOL qpow.c Power function, also qpowi raise to integer power qprob.c Various probability integrals: qbdtr binomial distribution qbdtrc complemented binomial distribution qbdtri inverse of binomial distribution qchdtr chi-square distribution qchdti inverse of chi-square distribution qfdtr F distribution qfdtrc complemented F distribution qfdtri inverse of F distribution qgdtr gamma distribution qgdtrc complemented gamma distribution qnbdtr negative binomial distribution qnbdtc complemented negative binomial qpdtr Poisson distribution qpdtrc complemented Poisson distribution qpdtri inverse of Poisson distribution qpsi.c psi function qshici.c hyperbolic sine integral hyperbolic cosine integral qsici.c sine integral cosine integral qsimq.c solve simultaneous equations qsin.c sine qsinmx3(x,y); y = sin(x) - x qsindg.c sine of arg in degrees qsinh.obj hyperbolic sine qspenc.c Spence's integral (dilogarithm) qsqrt.c square root qsqrta.c strictly rounded square root qstudt.c Student's t distribution function qtan.c tangent qtanh.c hyperbolic tangent qtst1.c Universal function test program qyn.c Bessel function Yn (integer order), also qyaux0 auxiliary functions qyaux1 qymod modulus qyphase phase qzetac.c Riemann zeta function Arithmetic routines qflt.c Main Q type arithmetic package: asctoq decimal ASCII string to Q type dtoq DEC double precision to Q type etoq IEEE double precision to Q type ltoq long integer to Q type qabs absolute value qadd add qclear set to zero qcmp compare qdiv divide qifrac long integer part plus q type fraction qinfin set to infinity, leaving its sign alone qmov b = a qmul multiply qmuli multiply by small integer qneg negate qnrmlz adjust exponent and mantissa qsub subtract qtoasc Q type to decimal ASCII string qtod convert Q type to DEC double precision qtoe convert Q type to IEEE double precision qflta.c Q type arithmetic, C language loops, strict rounding qfltb.c Q type arithmetic, C language faster loops mulr.asm Q type multiply, IBM PC assembly language divn.asm Q type IBM PC divide routine subm.asm Q type assembly language add, subtract for MSDOS qfltd.asm Q type arithmetic for 68020 (Definicon assembler) qconst.c Q type common constants qc120.c 120 decimal version of qconst.c mul128.a Fast multiply algorithm (for OS-9 68000) mul128ts.c Test program for above mul32.a mul64.a qfloor.c Q type floor(), also qround() round to integer Applications calc100.doc Documentation for 100 digit calculator program qcalc.c Command interpreter for calculator program qcalc.h Include file for command interpreter qcalc120.exe 120 decimal calculator program qcalcasm.bat Make calculator program qcalclin.bat qccalc.mak Make complex variable calculator program qparanoi.c Paranoia arithmetic test for Q type arithmetic notes Paranoia documentation qparanoi.mak Paranoia makefile etst.c Arithmetic demo program etstasm.bat etstlink.bat dentst.c frexp(), ldexp() tester qstirling.c Find coefficients for Stirling's formula qbernum.c Generates Bernoulli numbers qbernum.lst qbernuma.bat Calculator programs for qcalc euler.tak Euler's constant gamcof.tak Bernoulli numbers for gamma function gamma.tak Gamma function lgamnum.doc Stirling's formula lgamnum.tak zeta.tak zeta function ctest.tak exercise complex variable calculator A: absolute error; others are relative error (i.e., % of reading) Copyright 1984 - 1992 by Stephen L. Moshier Release 1.0: July, 1984 Release 1.1: March, 1985 Release 1.2: May, 1986 Release 2.0: April, 1987 Release 2.1: March, 1989 Release 2.2: July, 1992