DQFUN: A double-quad precision package with special functions Revision date: 18 Mar 2023 AUTHOR: David H. Bailey Lawrence Berkeley National Lab (retired) and University of California, Davis Email: dhbailey@lbl.gov COPYRIGHT AND DISCLAIMER All software in this package (c) 2023 David H. Bailey. By downloading or using this software you agree to the copyright, disclaimer and license agreement in the accompanying file DISCLAIMER.txt. INDEX OF THIS README FILE: I. PURPOSE OF PACKAGE II. INSTALLING CODING ENVIRONMENT (FOR MAC OS X SYSTEMS) III. INSTALLING FORTRAN COMPILER (IF NEEDED) IV. DOWNLOADING AND COMPILING DQFUN V. BRIEF SUMMARY OF CODING INSTRUCTIONS AND USAGE VI. SAMPLE APPLICATION PROGRAMS AND TESTS VII. RECENT UPDATES VIII. APPENDIX: TRANSCENDENTAL, SPECIAL AND MISCELLANEOUS FUNCTIONS +++++ I. PURPOSE OF PACKAGE This package permits one to perform floating-point computations (real and complex) to double-quad precision (approximately 65 digits), by making only relatively minor changes to existing Fortran programs. All basic arithmetic operations and transcendental functions are supported, together with numerous special functions. The package should run correctly on any Unix-based system supporting a Fortran-2008 compiler and IEEE 128-bit floating-point arithmetic (in hardware or software). Note however that results are NOT guaranteed to the last bit. In addition to fast execution times, one key feature of this package is a 100% THREAD-SAFE design, which means that user-level applications can be easily converted for parallel execution, say using a threaded parallel environment such as OpenMP. Two related software packages by the same author are DDFUN (double-double real and complex) and MPFUN-2020 (arbitrary precision real and complex). They are available from the same site as this package: http://www.davidhbailey.com/dhbsoftware/ II. INSTALLING CODING ENVIRONMENT (FOR MAC OS X SYSTEMS) For Apple Mac OS X systems (highly recommended for DQFUN), first install the latest supported version of Xcode, which is available for free from the Apple Developer website: https://developer.apple.com/ Here click on Account, then enter your Apple ID and password, then go to https://developer.apple.com/download/more/?=xcode From this list, select Xcode. As of the above date, the latest version of Xcode is 13.3. Download this package on your system, double click to decompress, and place the resulting Xcode app somewhere on your system (typically in the Applications folder). Double-click on the app to run Xcode, allowing it install additional components, then quit Xcode. Open a terminal window, using the Terminal application in the Utilities folder, and type xcode-select --install which installs various command-line tools. The entire process of downloading Xcode and installing command-line tools takes roughly 30 minutes. When this is completed, you should be ready to continue with the installation. III. INSTALLING FORTRAN COMPILER (IF NEEDED) Running DQFUN is relatively straightforward, provided that one has a Unix-based system, such as Linux or Apple OS X, and a Fortran-2008 compliant compiler that supports IEEE 128-bit arithmetic. These requirements are met by the GNU gfortran compiler, the Intel ifort compiler, and others. The gfortran compiler (highly recommended for DQFUN) is available free for a variety of systems at this website: https://gcc.gnu.org/wiki/GFortranBinaries For Apple Mac OS X systems, download the installer file here: https://github.com/fxcoudert/gfortran-for-macOS/releases The gfortran compiler is normally placed in /usr/local/lib and /usr/local/bin. Thus before one uses gfortran, one must insert a line in one's shell initialization file (if the Z shell is used, as on most Apple OS X systems, the shell initialization file is ~/.zshrc). The line to be included is: PATH=/usr/local/lib:/usr/local/bin:$PATH The following line is also recommended for gfortran compiler users: GFORTRAN_UNBUFFERED_ALL=yes; export GFORTRAN_UNBUFFERED_ALL The following line is recommended for inclusion in the shell initialization file, no matter what compiler is used (it prevents stack overflow system errors): ulimit -s unlimited On most Unix systems (including Apple Mac OS X systems), the shell initialization file must be manually executed upon initiating a terminal shell, typically by typing "source .zshrc". IV. DOWNLOADING AND COMPILING DQFUN From the website https://www.davidhbailey.com/dhbsoftware, download the file "dqfun-vnn.tar.gz" (replace "vnn" by whatever is the current version on the website, such as "v03"). If the file is not decompressed by your browser, use gunzip at the shell level to do this. Some browsers (such as the Apple Safari browser) do not drop the ".gz" suffix after decompression; if so, remove this suffix manually at the shell level. Then type tar xfv dqfun-vnn.tar (where again "vnn" is replaced by the downloaded version). This should create the directory and unpack all files. Compiling the library (in directory "fortran") is straightforward -- just compile the four library code files: dqfuna.f90, dqfune.f90, dqmodule.f90 and second.f90. Compile/link scripts are available for the GNU gfortran compiler. To compile the library with this script, type ./gnu-complib-dq.scr Then to compile and link the application program tpslqm1dq.f90, using the GNU gfortran compiler, producing the executable file tpslqm1dq, type ./gnu-complink-dq.scr tpslqm1dq To execute the program, with output to tpslqm1dq.txt, type ./tpslqm1dq > tpslqm1dq.txt These scripts assume that the user program is in the same directory as the library files; this can easily be changed by editing the script files. Several sample test programs, together with reference output files, are included in fortran directory -- see Section VI below. V. BRIEF SUMMARY OF CODING INSTRUCTIONS AND USAGE First place the following line in every subprogram of the user's application code that contains a double-quad variable or array, at the beginning of the declaration section, before any implicit or type statements: use dqmodule To designate a variable or array as double-quad real (DQR) in your application code, use the Fortran-90 type statement with the type "dq_real", as in this example: type (dq_real) a, b(m), c(m,n) Similarly, to designate a variable or array as double-quad complex (DQC), use a type statement with "dq_complex", as in type (dq_complex) x, y(m), z(m,n) Thereafter when one of these variables or arrays appears in code, e.g., d = a + b(i) * sqrt(3.q0 - c(i,j)) the proper underlying routines from are DQFUN package are automatically called by the Fortran compiler. Most common mixed-mode combinations (arithmetic operations, comparisons and assignments) involving DQR, DQC, quad-real (QP) and quad-complex (QC) variables are supported. However, most mixed-mode operations involving DQR or DQC with double precision or double complex are typically NOT supported. Some conversion functions are provided between double precision (DP), QP and DQR; and between double complex (DC), QC and DQC -- see table below. However, there are some hazards when combining say quad precision and double-quad entities. For example, the code r1 = 3.14159q0, where r1 is DQR, does NOT produce the true double-quad equivalent of 3.14159, unless the numerical value is a modest-sized whole number or exact binary fraction. To obtain the full double-quad converted value, write this as r1 = '3.14159' or r1 = '3.14150d0' instead. Similarly, the code r2 = r1 + 3.q0 * sqrt (2.q0), where r1 and r2 are DQR, does NOT produce the true double-quad value, since the expression 3.q0 * sqrt (2.q0) will be performed in quad precision (according to standard Fortran-90 precedence rules). To obtain the fully accurate DQR result, write this as r2 = r1 + 3.q0 * sqrt (dqreal (2.q0)). As noted above, quad precision constants should be written with a q exponent, such as 3.25q0. Technically speaking, the q exponent, while almost universally supported by compilers that support quad precision, is nonstandard; some fully standard-conforming options are 3.25_qxknd or 3.25e0_qxknd or real(3.25d0,qxknd). Here "qxknd" is the kind parameter for IEEE quad precision, which is set automatically in qxfune.f90. Input and output of DQR and DQC data are performed using the subroutines dqread and dqwrite. For example, to output the variable r1 in E format to Fortran unit 6 (standard output), to 65-digit accuracy, in a field of width 80 characters, use the line of code call dqwrite (6, 80, 65, r1) The second argument (80 in the above example) must be at least 10 larger than the third argument (65 in the above example). To read the variable r1 from Fortran unit 5 (standard input), use the line of code call dqread (5, r1) Most Fortran-2008 intrinsic functions are supported with DQR and DQC arguments, as appropriate, and numerous special functions are also supported. A complete list of supported functions and subroutines is summarized in the Appendix below (section VIII). VI. SAMPLE APPLICATION PROGRAMS AND TESTS Four application programs are included, together with corresponding reference output files (e.g., tpphixdq.ref.txt) for comparison with user results: testdqfun.f90 Tests most arithmetic and transcendental functions. tpslqm1dq.f90 Performs the 1-level multipair PSLQ integer relation algorithm. tpphixdq.f90 Performs a Poisson polynomial application, using 1-level multipair PSLQ. tquaddq.f90 Evaluates definite integrals, using tanh-sinh, exp-sinh and sinh-sinh algorithms. In addition, the package includes a test script that compiles the library and run each of the above sample programs above. To run this script, type: ./gnu-dqfun-tests.scr For each test program, the script outputs either TEST PASSED or TEST FAILED. If all tests pass, then one can be fairly confident that the DQFUN software is working properly. VII. RECENT UPDATES 6 Jan 2023: This release (version v02) fixed problem with dqeformat and dqfformat; corrected several other minor bugs; added the acosh, asinh, atanh, dqeform and dqfform routines; changed the calling sequence for dqwrite to include width and field parameters; declared the subroutine names in dqfun.f90 as private in dqmodule.f90; inserted intent statements for all routines; changed all parameter statements to the object oriented style; changed all data statements to parameter statements; added several conversion functions; inserted test messages in the test programs and included a test script. 27 Feb 2023: Included a large set of special functions, by means of a special translator program to convert the file mpfune.f90 from MPFUN20-Fort. VIII. APPENDIX: TRANSCENDENTAL, SPECIAL AND MISCELLANEOUS FUNCTIONS In these tables, "F" denotes function, "S" denotes subroutine, "DQR" denotes double quad real, "DQC" denotes double quad complex, "QP" denotes quad precision, "QC" denotes quad complex, "DP" denotes double precision, "DC" denotes double complex, and "Int" denotes integer. The variable names r1,r2,r3 are DQR; z1 is DQC; d1 is DP, dc1 is DC, q1 is QP; qc1 is QC; i1,i2,i3,n,nb,np,nq are integers; s1 is character(1); sn is character(n); rb is an DQR vector of length nb; ss is an DQR vector of length n; aa is an DQR vector of length np; bb is an DQR vector of length nq. 1. Standard Fortran-2008 transcendental functions: Type Name Description DQR abs(r1) Absolute value DQR abs(z1) Absolute value of complex arg DQR acos(r1) Inverse cosine DQR acosh(r1) Inverse hyperbolic cosine DQR aimag(z1) Imaginary part of complex arg DQR aint(r1) Truncates to integer DQR anint(r1) Rounds to closest integer DQR asin(r1) Inverse sine DQR asinh(r1) Inverse hyperbolic sine DQR atan(r1) Inverse tangent DQR atan2(r1,r2) Arctangent with two args DQR atanh(r1) Inverse hyperbolic tangent DQR bessel_j0(r1) Bessel function of the first kind, order 0 DQR bessel_j1(r1) Bessel function of the first kind, order 1 DQR bessel_jn(n,r1) Besel function of the first kind, order n DQR bessel_y0(r1) Bessel function of the second kind, order 0 DQR bessel_y1(r1) Bessel function of the second kind, order 1 DQR bessel_yn(n,r1) Besel function of the second kind, order n DQC conjg(z1) Complex conjugate DQR cos(r1) Cosine of real arg DQC cos(z1) Cosine of complex arg DQR cosh(r1) Hyperbolic cosine DP dble(r1) Converts DQR argument to DP QC dcmplx(z1) Converts DQC argument to DC DQR erf(r1) Error function DQR erfc(r1) Complementary error function DQR exp(r1) Exponential function of real arg DQC exp(z1) Exponential function of complex arg DQR gamma(r1) Gamma function DQR hypot(r1,r2) Hypotenuse of two args DQR log(r1) Natural logarithm of real arg DQC log(z1) Natural logarithm of complex arg DQR log10(r1) Base-10 logarithm of real arg DQR max(r1,r2) Maximum of two (or three) args DQR min(r1,r2) Minimum of two (or three) args DQR mod(r1,r2) Mod function = r1 - r2*aint(r1/r2) QP qreal(r1) Converts DQR to QP. QC qcmplx(z1) Converts DQC to QC. DQR sign(r1,r2) Transfers sign from r2 to r1 DQR sin(r1) Sine function of real arg DQC sin(z1) Sine function of complex arg DQR sinh(r1) Hyperbolic sine DQR sqrt(r1) Square root of real arg DQC sqrt(z1) Square root of complex arg DQR tan(r1) Tangent function DQR tanh(r1) Hyperbolic tangent function 2. Special functions: Type Name Description hline F:DQR agm(r1,r2) Arithmetic-geometric mean of r1 and r2 S dqberne(nb,rb) Initialize array rb of length nb with first nb even Bernoulli numbers [3] F:DQR bessel_i(r1,r2) BesselI function, order r1, of r2 F:DQR bessel_in(n,r1) BesselI function, order n, of r1 F:DQR bessel_j(r1,r2) BesselJ function, order r1, of r2 F:DQR bessel_jn(n,r1) BesselJ function, order n, of r1 F:DQR bessel_i(r1,r2) BesselK function, order r1, of r2 F:DQR bessel_kn(n,r1) BesselK function, order n, of r1 F:DQR bessel_y(r1,r2) BesselY function, order r1, of r2 F:DQR bessel_yn(n,r1) BesselY function, order n, of r1 F:DQR digamma_be(nb,rb,r1) Digamma function of r1, using nb even Bernoulli numbers in rb [3] F:DQR erf(r1) Error function F:DQR erfc(r1) Complementary error function F:DQR expint(r1) Exponential integral function F:DQR gamma(r1) Gamma function F:DQR hurwitz_zetan(k,r1) Hurwitz zeta function, order n >= 2, of r1 [4] F:DQR hurwitz_zetan_be (nb,rb,n,r1) Hurwitz zeta function, order n >= 2, of r1, using nb even Bernoulli numbers in rb [3] F:DQR hypergeom_pfq(np,nq,aa,bb,r1) Hypergeometric pFq function of aa, bb and r1; dimensions are aa(np) and bb(nq) [5] F:DQR incgamma(r1,r2) Incomplete gamma function [6] F:DQR polygamma(k,r1) Polygamma function, order k >= 1, of r1 [4] F:DQR polygamma(nb,rb,k,r1) Polygamma function, order k >= 1, of r1, using nb even Bernoulli numbers in rb [3] S polylog_ini(n,ss) Initialize array ss, of size |n|, for computing polylogarithms of order n when n < 0 [6] F:DQR polylog_neg(n,ss,r1) Polylogarithm function of r1, for n < 0, using precomputed data in ss [7] F:DQR polylog_pos(n,r1) Polylogarithm function, order n >= 0, of r1 [8] F:DQR struve_hn(n,r1) StruveH function, order n >= 0, of r1 [9] F:DQR zeta(r1) Zeta function of r1 F:DQR zeta_be(nb,rb,r1) Zeta function of r1, using nb even Bernoulli numbers in rb [3] F:DQR zeta_int(n) Zeta function of integer argument n Notes: [3]: For most applications, set nb >= 140; see dqberne above. [4]: For hurwitz_zetan and polygamma, the argument r1 is limited to the range (0, 1). [5]: For hypergeom_pfq, the integers np and nq must not exceed 10. [6]: For incgamma, r1 must not be zero, and must not be negative unless r2 = 0. [7]: For polylog_ini and polylog_neg, the integer n is limited to the range [-1000, -1]. [8]: For polylog_pos, the argument r1 is limited to the range (-1,1). [9]: For struve_hn, the argument r1 is limited to the range [-1000, 1000]. 3. Miscellaneous conversion, I/O and transcendental functions: Type Name Description F:DP dble(r1) Converts DQR argument to DP F:QC dcmplx(z1) Converts DQC argument to DC F:DQC dqcmplx(r1,r2) Converts (r1,r2) to DQC F:DQC dqcmplx(dc1) Converts DC arg to DQC F:DQC dqcmplx(qc1) Converts QC arg to DQC F:DQC dqcmplx(z1) Converts DQC arg to DQC S dqcssh(r1,r2,r3) Returns both cosh and sinh of r1, in the same time as calling just cosh or just sinh S dqcssn(r1,r2,r3) Returns both cos and sin of r1, in the same time as calling just cos or just sin S dqdecmd(r1,q1,i1) Converts r1 to the form q1*10^i1 S dqeform(r1,i1,i2,s1) Converts r1 to char(1) string in Ei1.i2 suitable for output S dqfform(r1,i1,i2,s1) Converts r1 to char(1) string in Fi1.i2 suitable for output F:DQR dqegamma() Returns Euler's gamma constant S dqinit() Initializes for extra-high precision F:DQR dqlog2() Returns log(2) F:DQR dqnrt(r1,i1) Returns the i1-th root of r1 F:DQR dqpi() Returns pi F:DQR dqrand(r1) Returns pseudorandom number, based on r1 Start with an irrational, say r1 = dqlog2() Typical iterated usage: r1 = dqrand(r1) S dqread(i1,r1) Inputs r1 from Fortran unit i1; up to five DQR args may be listed S dqread(i1,z1) Inputs z1 from Fortran unit i1; up to five DQC args may be listed F:DQR dqreal(i1) Converts Int arg to DQR F:DQR dqreal(d1) Converts DP arg to DQR F:DQR dqreal(q1) Converts QP arg to DQR F:DQR dqreal(z1) Converts DQC arg to DQR F:DQR dqreal(s1,i1) Converts character(1) string of length i1 to DQR F:DQR dqreal(sn) Converts character(n) string to DQR F:Int dqwprec(r1) Returns precision in words assigned to r1 F:Int dqwprec(z1) Returns precision in words assigned to z1 S dqwrite(i1,i2,i3,r1) Outputs r1 in Ei2.i3 format to unit i1; up to five DQR args may be listed S dqwrite(i1,i2,i3,z1) Outputs z1 in Ei2.i3 format to unit i1; up to five DQC args may be listed