NAG Library, Mark 27

NLMI627DGL - Licence Managed

macOS, 64-bit, GNU C or GNU Fortran, 32-bit integers

Users' Note


1. Introduction

This document is essential reading for every user of the NAG Library implementation specified in the title. It provides implementation-specific detail that augments the information provided in the NAG Mark 27 Library Manual (which we will refer to as the Library Manual). Wherever that manual refers to the "Users' Note for your implementation", you should consult this note.

In addition, NAG recommends that before calling any Library routine you should read the following reference material from the Library Manual (see Section 5):

(a) How to Use the NAG Library
(b) Chapter Introduction
(c) Routine Document

2. Supplementary Information

Please check the following URL:

for details of any new information related to the applicability or usage of this implementation.

3. General Information

This implementation of the NAG Library provides static and shareable libraries that use Apple vecLib (Accelerate Framework), a third-party vendor performance library, to provide Basic Linear Algebra Subprograms (BLAS) and Linear Algebra PACKage (LAPACK) routines (except for any routines listed in Section 4). It also provides static and shareable libraries that use the NAG versions of these routines (referred to as the self-contained libraries). This implementation has been tested with version 3.11 of vecLib. Please see the Apple web site for further information about vecLib ( For best performance, we recommend that you use one of the variants of the NAG Library which is based on vecLib, i.e. libnag_vl.a or libnag_vl.dylib, in preference to using one of the self-contained NAG libraries, libnag_nag.a or libnag_nag.dylib.

The NAG AD Library is not included in this implementation.

Note that the NAG Library is carefully designed so that any memory used can be reclaimed – either by the Library itself or, for C routines, by the user invoking calls of NAG_FREE(). However, the Library does itself depend on the use of compiler run-time and other libraries which may sometimes leak memory, and memory tracing tools used on programs linked to the NAG Library may report this. The amount of memory leaked will vary from application to application, but should not be excessive and should never increase without limit as more calls are made to the NAG Library.

If you intend to use the NAG library within a multithreaded application please refer to the document CL Interface Multithreading or FL Interface Multithreading (as appropriate) for more information.

The libraries supplied with this implementation do not contain OpenMP or any other threading mechanisms. However, vecLib may employ multithreading to enhance performance.

3.1. Accessing the Library

In this section we assume that the Library has been installed in the directory [INSTALL_DIR]. By default [INSTALL_DIR] (see Installer's Note (in.html)) is $HOME/NAG/nlmi627dgl; however it could have been changed by the person who did the installation, in which case you should consult that person. Note that the environment variable DYLD_LIBRARY_PATH must be set correctly at link time and run time to point to the appropriate library locations underneath [INSTALL_DIR]. See below for how to do this.

The NAG Library is a combined replacement for users of both the NAG C Library and the NAG Fortran Library (including the C wrappers to the Fortran Library interfaces). To help with calling the different libraries included with this implementation the scripts and nagvars.csh are included to set NAG-specific environment variables to assist with compiling and linking applications that call any of the NAG routines. They also amend the standard environment variables PATH and DYLD_LIBRARY_PATH so that NAG executable programs and libraries can be found at compile, link and run time.

The nagvars scripts are designed to be used as follows:

   . [INSTALL_DIR]/scripts/ [-help] [-unset] [-quiet] {int32} \
     {vendor,nag} {static,dynamic}
   source [INSTALL_DIR]/scripts/nagvars.csh [-help] [-unset] [-quiet] {int32} \
          {vendor,nag} {static,dynamic}
where: No default values are used, so all three options must be set. The order of specifying them is not important. The optional nagvars script arguments are as follows: Thus, an example of using the nagvars script to set the environment in Bash is:
   source [INSTALL_DIR]/scripts/ int32 vendor dynamic
The NAG-specific environment variables set are: To use the NAG Library and the Apple vecLib (Accelerate Framework) libraries (if desired), you may then link in the following manner:

For C programs:

or for C++ programs:
or for Fortran programs:

You will also need to edit the variable _nag_compiler_runtimedir in the script files nagvars.csh and so that it points to the location of the appropriate lib directory. If you are not using the scripts, then set DYLD_LIBRARY_PATH to point to the compiler run-time libraries.

3.2. Fortran Interface Blocks

The NAG Library interface blocks define the type and arguments of each user-callable NAG Library Fortran routine. While they are not essential to calling the NAG Library from Fortran programs, their use is highly recommended, and they are essential if the supplied examples are used.

Their purpose is to allow the Fortran compiler to check that NAG Library routines are called correctly. The interface blocks enable the compiler to check that:

(a) subroutines are called as such;
(b) functions are declared with the right type;
(c) the correct number of arguments are passed; and
(d) all arguments match in type and structure.

The NAG Library interface block files are organised by Library chapter. They are aggregated into one module named

The modules are supplied in compiled form (.mod files) for the GNU gfortran compiler. They can be accessed by specifying the -Ipathname option on each compiler invocation, where pathname ([INSTALL_DIR]/lp64/nag_interface_blocks) is the path of the directory containing the compiled interface blocks.

The .mod module files were compiled with the Fortran compiler shown in Section 2.2 of the Installer's Note. Such module files are compiler-dependent, so if you wish to use the NAG example programs, or use the interface blocks in your own programs, when using a compiler that is incompatible with these modules, you will first need to recompile the interface blocks with your own compiler version. A recompiled set of interface blocks can be created in a separate directory (e.g. nag_interface_blocks_alt) using the supplied script command

  [INSTALL_DIR]/scripts/nag_recompile_mods {int32} nag_interface_blocks_alt
from the [INSTALL_DIR] directory. This script uses the version of the GNU Fortran compiler from your PATH environment; to specify an alternative version it is safest to first run any GNU Fortran compiler environment scripts for that version prior to running [INSTALL_DIR]/scripts/nag_recompile_mods.

To make the new set of compiled modules the default set, you may use the following commands (with [INSTALL_DIR] substituted by the actual directory path as appropriate):

  mv [INSTALL_DIR]/lp64/nag_interface_blocks [INSTALL_DIR]/lp64/nag_interface_blocks_original
  mv [INSTALL_DIR]/lp64/nag_interface_blocks_alt [INSTALL_DIR]/lp64/nag_interface_blocks

You should now be able to use the newly compiled module files in the usual way.

3.3. Example Programs

The example results distributed were generated at Mark 27 using the software described in Section 2.2 of the Installer's Note. These example results may not be exactly reproducible if the example programs are run in a slightly different environment (for example, a different C or Fortran compiler, a different compiler runtime library, or a different set of BLAS or LAPACK routines). The results which are most sensitive to such differences are: eigenvectors (which may differ by a scalar multiple, often -1, but sometimes complex); numbers of iterations and function evaluations; and residuals and other "small" quantities of the same order as the machine precision.

The distributed example results are those obtained with the static library libnag_vl.a (i.e. using the vecLib BLAS and LAPACK routines). Running the examples with NAG BLAS or LAPACK may give slightly different results.

Note that the example material has been adapted, if necessary, from that published in the Library Manual, so that programs are suitable for execution with this implementation with no further changes. The distributed example programs should be used in preference to the versions in the Library Manual wherever possible. The example programs are most easily accessed by using the script nag_example, which is located in the directory [INSTALL_DIR]/scripts.

This script will provide you with a copy of an example program (and its data and options file, if any), compile the program and link it with the appropriate libraries. Finally, the executable program will be run (with appropriate arguments specifying data, options and results files as needed), with the results being sent to a file and to the command window. By default nag_example selects to use static linking to the self-contained libnag_nag.a library. These choices can be changed by optional switches -shared and -vendor respectively. The -quiet option may be specified to minimize printing of both comments and the commands being executed at each of the stages described above. Run

nag_example -help
to display a list of the available options.

The nag_example script demonstrates use of the nagvars script discussed in Section 3.1, but note that it does not alter the environment in the calling shell.

The example program concerned is specified by the argument to the command, e.g. for a NAG C routine

nag_example e04ucc
will copy the example program and its data and options files (e04ucce.c, e04ucce.d and e04ucce.opt) into the current directory, compile and link the program and run it to produce the example program results in the file e04ucce.r.

Similarly, for a NAG Fortran routine

nag_example e04nrf
will copy the example program and its data and options files (e04nrfe.f90, e04nrfe.d and e04nrfe.opt) into the current directory, compile and link the program and run it to produce the example program results in the file e04nrfe.r.

You will need to set the variable _nag_compiler_runtimedir with the location of the appropriate compiler run-time libraries in the script as this is used by the nag_example script. See Section 3.1 for details.

3.4. Maintenance Level

The maintenance level of the Library can be determined by compiling and executing the example that calls a00aaf or a00aac, or you could call the nag_example script with the argument a00aaf or a00aac. See Section 3.3. This example prints out details of the implementation, including title and product code, compiler and precision used, mark and maintenance level.

3.5. C Data Types

In this implementation, the NAG C types Integer and Pointer are defined as follows:
 NAG Type   C Type   Size (bytes) 
 Integer   int     4 
 Pointer   void *   8 

The values for sizeof(Integer) and sizeof(Pointer) are also given by the a00aac example program. Information on other NAG data types is available in Section 3.1.1 of the NAG CL Interface Introduction component of the Library Manual (see Section 5 below).

3.6. Fortran Data Types and Interpretation of Bold Italicised Terms

This implementation of the NAG Library includes libraries for 32-bit integers only. The libraries are located in [INSTALL_DIR]/lp64/lib.

The NAG Library and documentation use parameterized types for floating-point variables. Thus, the type

appears in the documentation of all NAG Library routines, where nag_wp is a Fortran KIND parameter. The value of nag_wp will vary between implementations, and its value can be obtained by use of the nag_library module. We refer to the type nag_wp as the NAG Library "working precision" type, because most floating-point arguments and internal variables used in the Library are of this type.

In addition, a small number of routines use the type

where nag_rp stands for "reduced precision" type. Another type, not currently used in the Library, is
for "higher precision" type or "additional precision" type.

For correct use of these types, see almost any of the example programs distributed with the Library.

For this implementation, these types have the following meanings:

      REAL (kind=nag_rp)      means REAL (i.e. single precision)
      REAL (kind=nag_wp)      means DOUBLE PRECISION
      COMPLEX (kind=nag_rp)   means COMPLEX (i.e. single precision complex)
      COMPLEX (kind=nag_wp)   means double precision complex (e.g. COMPLEX*16)

In addition, the FL Interface section of the Manual has adopted a convention of using bold italics to distinguish some terms. See Section 2.5 of the FL Interface Introduction for details.

3.7. Calling NAG Fortran Routines from C or C++

With care, the Fortran routines in the NAG Library may be used from within a C, C++ or compatible environment. Using the Fortran routines in this manner may be preferable to calling the C routines, either to access a legacy Fortran routine for which a C routine equivalent is not available, or to have a lower level C interface, using only elementary C data types, which may be more convenient for use from other languages. To assist the user make the mapping between Fortran and C types, a description of the Fortran interface from a C perspective (the C Header Interface) is included in each Fortran routine document. A C/C++ header file ([INSTALL_DIR]/lp64/include/nag.h) is also provided. It is recommended that users wishing to use a NAG Fortran routine in this manner #include this header file in their application.

A document, alt_c_interfaces.html, giving advice on calling the Fortran routines in the NAG Library from C and C++ is also available. (In previous Marks of the NAG Library, this document was called techdoc.html.)

4. Routine-specific Information

Any further information which applies to one or more routines in this implementation is listed below, chapter by chapter.
  1. F06, F07, F08 and F16

    In Chapters F06, F07, F08 and F16, alternative routine names are available for BLAS and LAPACK derived routines. For details of the alternative routine names please refer to the relevant Chapter Introduction. Note that applications should reference routines by their BLAS/LAPACK names, rather than their NAG-style names, for optimum performance.

    Many LAPACK routines have a "workspace query" mechanism which allows a caller to interrogate the routine to determine how much workspace to supply. Note that LAPACK routines from the vecLib library may require a different amount of workspace from the equivalent NAG versions of these routines. Care should be taken when using the workspace query mechanism.

    In this implementation, calls to BLAS and LAPACK routines in the non-self-contained NAG libraries are implemented by calls to vecLib, except for the following routines:

    blas_damax_val  blas_damin_val  blas_daxpby     blas_ddot       blas_dmax_val
    blas_dmin_val   blas_dsum       blas_dwaxpby    blas_zamax_val  blas_zamin_val
    blas_zaxpby     blas_zsum       blas_zwaxpby
    daxpyi  dbdsvdx ddoti   dgeesx  dgehrd  dgejsv  dgemqrt dgemv   dgeqrt  dgesvdx
    dgesvj  dgetrs  dgges   dgges3  dggev3  dggevx  dgghd3  dggsvd3 dggsvp3 dgthr
    dgthrz  dhgeqz  dorcsd  drot    droti   dsctr   dsgesv  dtpmqrt dtpqrt  dtrsen
    dtrsm   dznrm2  sasum   scasum  sdot    zaxpyi  zcgesv  zcopy   zdotc   zdotci
    zdotu   zdotui  zgeesx  zgejsv  zgelsy  zgemqrt zgeqp3  zgeqrt  zgesvdx zgesvj
    zgges3  zggev3  zgghd3  zggsvd3 zggsvp3 zgthr   zgthrz  zsctr   ztgsen  ztpmqrt
    ztpqrt  zuncsd  zunmrz
  2. S07 - S21

    The behaviour of functions in these Chapters may depend on implementation-specific values.

    General details are given in the Library Manual, but the specific values used in this implementation are as follows:

    s07aa[f] (nag[f]_specfun_tan)
        F_1 = 1.0e+13
        F_2 = 1.0e-14
    s10aa[fc] (nag[f]_specfun_tanh)
        E_1 = 1.8715e+1
    s10ab[fc] (nag[f]_specfun_sinh)
        E_1 = 7.080e+2
    s10ac[fc] (nag[f]_specfun_cosh)
        E_1 = 7.080e+2
    s13aa[fc] (nag[f]_specfun_integral_exp)
        x_hi = 7.083e+2
    s13ac[fc] (nag[f]_specfun_integral_cos)
        x_hi = 1.0e+16
    s13ad[fc] (nag[f]_specfun_integral_sin)
        x_hi = 1.0e+17
    s14aa[fc] (nag[f]_specfun_gamma)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.70e+2
        ifail = 2 (NE_REAL_ARG_LT) if x < -1.70e+2
        ifail = 3 (NE_REAL_ARG_TOO_SMALL) if abs(x) < 2.23e-308
    s14ab[fc] (nag[f]_specfun_gamma_log_real)
        ifail = 2 (NE_REAL_ARG_GT) if x > x_big = 2.55e+305
    s15ad[fc] (nag[f]_specfun_erfc_real)
        x_hi = 2.65e+1
    s15ae[fc] (nag[f]_specfun_erf_real)
        x_hi = 2.65e+1
    s15ag[fc] (nag[f]_specfun_erfcx_real)
        ifail = 1 (NW_HI) if x >= 2.53e+307
        ifail = 2 (NW_REAL) if 4.74e+7 <= x < 2.53e+307
        ifail = 3 (NW_NEG) if x < -2.66e+1
    s17ac[fc] (nag[f]_specfun_bessel_y0_real)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.0e+16
    s17ad[fc] (nag[f]_specfun_bessel_y1_real)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.0e+16
        ifail = 3 (NE_REAL_ARG_TOO_SMALL) if 0 < x <= 2.23e-308
    s17ae[fc] (nag[f]_specfun_bessel_j0_real)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) > 1.0e+16
    s17af[fc] (nag[f]_specfun_bessel_j1_real)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) > 1.0e+16
    s17ag[fc] (nag[f]_specfun_airy_ai_real)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.038e+2
        ifail = 2 (NE_REAL_ARG_LT) if x < -5.7e+10
    s17ah[fc] (nag[f]_specfun_airy_bi_real)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.041e+2
        ifail = 2 (NE_REAL_ARG_LT) if x < -5.7e+10
    s17aj[fc] (nag[f]_specfun_airy_ai_deriv)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.041e+2
        ifail = 2 (NE_REAL_ARG_LT) if x < -1.9e+9
    s17ak[fc] (nag[f]_specfun_airy_bi_deriv)
        ifail = 1 (NE_REAL_ARG_GT) if x > 1.041e+2
        ifail = 2 (NE_REAL_ARG_LT) if x < -1.9e+9
    s17dc[fc] (nag[f]_specfun_bessel_y_complex)
        ifail = 2 (NE_OVERFLOW_LIKELY) if abs(z) < 3.92223e-305
        ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n-1 > 3.27679e+4
        ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n-1 > 1.07374e+9
    s17de[fc] (nag[f]_specfun_bessel_j_complex)
        ifail = 2 (NE_OVERFLOW_LIKELY) if AIMAG(z) > 7.00921e+2
        ifail = 3 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n-1 > 3.27679e+4
        ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n-1 > 1.07374e+9
    s17dg[fc] (nag[f]_specfun_airy_ai_complex)
        ifail = 3 (NW_SOME_PRECISION_LOSS) if abs(z) > 1.02399e+3
        ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) > 1.04857e+6
    s17dh[fc] (nag[f]_specfun_airy_bi_complex)
        ifail = 3 (NW_SOME_PRECISION_LOSS) if abs(z) > 1.02399e+3
        ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) > 1.04857e+6
    s17dl[fc] (nag[f]_specfun_hankel_complex)
        ifail = 2 (NE_OVERFLOW_LIKELY) if abs(z) < 3.92223e-305
        ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n-1 > 3.27679e+4
        ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n-1 > 1.07374e+9
    s18ad[fc] (nag[f]_specfun_bessel_k1_real)
        ifail = 2 (NE_REAL_ARG_TOO_SMALL) if 0 < x <= 2.23e-308
    s18ae[fc] (nag[f]_specfun_bessel_i0_real)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) > 7.116e+2
    s18af[fc] (nag[f]_specfun_bessel_i1_real)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) > 7.116e+2
    s18dc[fc] (nag[f]_specfun_bessel_k_complex)
        ifail = 2 (NE_OVERFLOW_LIKELY) if abs(z) < 3.92223e-305
        ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n-1 > 3.27679e+4
        ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n-1 > 1.07374e+9
    s18de[fc] (nag[f]_specfun_bessel_i_complex)
        ifail = 2 (NE_OVERFLOW_LIKELY) if REAL(z) > 7.00921e+2
        ifail = 3 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n-1 > 3.27679e+4
        ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n-1 > 1.07374e+9
    s19aa[fc] (nag[f]_specfun_kelvin_ber)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) >= 5.04818e+1
    s19ab[fc] (nag[f]_specfun_kelvin_bei)
        ifail = 1 (NE_REAL_ARG_GT) if abs(x) >= 5.04818e+1
    s19ac[fc] (nag[f]_specfun_kelvin_ker)
        ifail = 1 (NE_REAL_ARG_GT) if x > 9.9726e+2
    s19ad[fc] (nag[f]_specfun_kelvin_kei)
        ifail = 1 (NE_REAL_ARG_GT) if x > 9.9726e+2
    s21bc[fc] (nag[f]_specfun_ellipint_symm_2)
        ifail = 3 (NE_REAL_ARG_LT) if an argument < 1.583e-205
        ifail = 4 (NE_REAL_ARG_GE) if an argument >= 3.765e+202
    s21bd[fc] (nag[f]_specfun_ellipint_symm_3)
        ifail = 3 (NE_REAL_ARG_LT) if an argument < 2.813e-103
        ifail = 4 (NE_REAL_ARG_GT) if an argument >= 1.407e+102
  3. X01

    The values of the mathematical constants are:

    x01aa[fc] (nag[f]_math_pi)
         = 3.1415926535897932
    x01ab[fc] (nag[f]_math_euler)
         = 0.5772156649015328
  4. X02

    The values of the machine constants are:

    The basic parameters of the model

    x02bh[fc] (nag[f]_machine_model_base)
         = 2
    x02bj[fc] (nag[f]_machine_model_digits)
         = 53
    x02bk[fc] (nag[f]_machine_model_minexp)
         = -1021
    x02bl[fc] (nag[f]_machine_model_maxexp)
         = 1024

    Derived parameters of the floating-point arithmetic

    x02aj[fc] (nag[f]_machine_precision)
         = 1.11022302462516e-16
    x02ak[fc] (nag[f]_machine_real_smallest)
         = 2.22507385850721e-308
    x02al[fc] (nag[f]_machine_real_largest)
         = 1.79769313486231e+308
    x02am[fc] (nag[f]_machine_real_safe)
         = 2.22507385850721e-308
    x02an[fc] (nag[f]_machine_complex_safe)
         = 4.78687214269110e-168

    Parameters of other aspects of the computing environment

    x02ah[fc] (nag[f]_machine_sinarg_max)
         = 1.42724769270596e+45
    x02bb[fc] (nag[f]_machine_integer_max)
         = 2147483647
    x02be[fc] (nag[f]_machine_decimal_digits)
         = 15
  5. X04

    Fortran routines: The default output units for error and advisory messages for those routines which can produce explicit output are both Fortran Unit 6.
  6. X06

    Chapter X06 routines do not change the behaviour of vecLib threading in this implementation of the Library.

5. Documentation

The Library Manual is available as a separate installation, via download from the NAG website. The most up-to-date version of the documentation is accessible via the NAG website at

The Library Manual is supplied in HTML5, a fully linked version of the manual using HTML and MathML. These documents can be accessed using your web browser.

The following master index file has been provided:

Use your web browser to navigate from here.

Advice on viewing and navigating the documentation can be found in

In addition the following are provided:

6. Support from NAG

Please see

for information about the NAG Technical Support Service, including details of the NAG Technical Support Service contact points. We would also be delighted to receive your feedback on NAG's products and services.

7. Contact Addresses

Please see

for worldwide contact details for the Numerical Algorithms Group.