NAG Fortran Library, Mark 26

FLMI626D9L - License Managed

Intel-based Mac, Mac OS X 64, NAG Fortran, Double Precision, 32-bit integers

Users' Note


1. Introduction

This document is essential reading for every user of the NAG Fortran Library implementation specified in the title. It provides implementation-specific detail that augments the information provided in the NAG Mark 26 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 and its Documentation
(b) Chapter Introduction
(c) Routine Document

Please note that the long names for Library routines mentioned in Section 3.1.1 of How to Use the NAG Library and its Documentation are not available in this implementation.

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 Fortran Library provides static and shareable libraries that use the Apple ® vecLib Library (VL), 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(a)). 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 3.6 of VL. For best performance, we recommend that you use one of the variants of the NAG Fortran Library which is based on VL, 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.

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/flmi626d9l; however it could have been changed by the person who did the installation, in which case you should consult that person.

To use the NAG Fortran Library and the VL libraries, you may link in the following manner:

  nagfor -thread_safe -I[INSTALL_DIR]/nag_interface_blocks -ieee=full \
         driver.f90 [INSTALL_DIR]/lib/libnag_vl.a \
         -framework Accelerate -lc++
         -framework IOKit -framework CoreFoundation
where driver.f90 is your application program;


  nagfor -thread_safe -I[INSTALL_DIR]/nag_interface_blocks -ieee=full \
         driver.f90 [INSTALL_DIR]/lib/libnag_vl.dylib \
         -framework Accelerate \
         -framework IOKit -framework CoreFoundation
if the shareable library is required.

However, if you prefer to link to a version of the NAG Fortran Library which does not require the use of VL you may wish to use the self-contained libraries as follows:

  nagfor -thread_safe -I[INSTALL_DIR]/nag_interface_blocks -ieee=full \
         driver.f90 [INSTALL_DIR]/lib/libnag_nag.a \
        -framework IOKit -framework CoreFoundation \
  nagfor -thread_safe -I[INSTALL_DIR]/nag_interface_blocks -ieee=full \
         driver.f90 [INSTALL_DIR]/lib/libnag_nag.dylib \
        -framework IOKit -framework CoreFoundation
if the shareable library is required.

If your application has been linked with the shareable NAG and VL libraries then the environment variable DYLD_LIBRARY_PATH must be set (or extended) to allow run-time linkage.

In the C shell type:

to extend DYLD_LIBRARY_PATH if you already have it set.

In the Bourne shell, type:

to extend DYLD_LIBRARY_PATH if you already have it set.

Note that you may also need to set DYLD_LIBRARY_PATH to point at other items such as compiler run-time libraries, for example if you are using a newer version of the compiler.

3.1.1. Calling the Library from C or C++

With care, the NAG Fortran Library may be used from within a C or C++ environment. To assist the user make the mapping between Fortran and C types, a C/C++ header file ([INSTALL_DIR]/c_headers/nagmk26.h) is provided. It is recommended that users wishing to use a Fortran Library routine either copy and paste the relevant section of the file into their C or C++ application (making sure that the relevant #defines etc. are also copied from the top of the file) or simply include the header file with their application.

A document, techdoc.html, giving advice on calling the NAG Fortran Library from C and C++ is also available in [INSTALL_DIR]/c_headers.

3.2. Interface Blocks

The NAG Fortran Library interface blocks define the type and arguments of each user callable NAG Fortran Library routine. These are not essential to calling the NAG Fortran Library from Fortran programs. However, they are required if the supplied examples are used. Their purpose is to allow the Fortran compiler to check that NAG Fortran 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 Fortran Library interface block files are organised by Library chapter. They are aggregated into one module named


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

The .mod module files were compiled with the 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 nag_interface_blocks_alt
from the [INSTALL_DIR] directory. This script uses the version of the NAG Fortran compiler from your PATH environment.

To make the new set of compiled modules the default set, move the directory [INSTALL_DIR]/nag_interface_blocks to [INSTALL_DIR]/nag_interface_blocks_original, and then move the directory containing the new set of modules [INSTALL_DIR]/nag_interface_blocks_alt to [INSTALL_DIR]/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 26, 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 Fortran compiler, a different compiler 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 VL 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 one of the following scripts, which are located in the directory [INSTALL_DIR]/scripts.

Each command 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 (showing you the compile command so that you can recompile your own version of the program). 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.

The example program concerned is specified by the argument to the command, e.g.

  nag_example_vl 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.

3.4. Fortran Types and Interpretation of Bold Italicised Terms

This implementation of the NAG Fortran Library uses 32-bit integers.

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

appears in documentation of all NAG Fortran 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 Manual has adopted a convention of using bold italics to distinguish some terms. See Section 4.4 of How to Use the NAG Library and its Documentation for details.

3.5. Maintenance Level

The maintenance level of the Library can be determined by compiling and executing the example that calls A00AAF, or you could call one of the nag_example* scripts with the argument a00aaf. 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.

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

    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 VL may require a different amount of workspace from the equivalent NAG versions of these routines.

    In this implementation calls to the NAG version of the following BLAS and LAPACK routines may be included in the libraries libnag_vl.a and libnag_vl.dylib to avoid problems with the vendor version:

      daxpyi dbdsdc dbdsqr dbdsvdx ddoti dgebal dgeesx dgeevx
      dgejsv dgelsd dgelsy dgemqrt dgemv dgeqlf dgeqp3 dgeqrf
      dgeqrt dgerqf dgesdd dgesv dgesvd dgesvdx dgesvj dgesvx
      dgetrf dgges dgges3 dggev dggev3 dgghd3 dgglse dggqrf
      dggrqf dggsvd3 dggsvp3 dgthr dgthrz dhgeqz dhseqr dnrm2
      dorcsd dormhr dormql dormrz dposvx dpotrf dpteqr droti
      dsbgvd dsctr dsgesv dstegr dstevr dsyevr dsymv dtpmqrt
      dtpqrt dtrsen dtrsv dtzrzf dznrm2 sasum scasum sdot
      zaxpyi zbdsqr zcgesv zcopy zdotc zdotci zdotu zdotui
      zgeesx zgejsv zgelsd zgelsy zgemqrt zgeqp3 zgeqrt zgesdd
      zgesvd zgesvdx zgesvj zgesvx zgetrf zgges3 zggev3 zgghd3
      zggqrf zggrqf zggsvd3 zggsvp3 zgthr zgthrz zhbgvd zheevr
      zher2k zhsein zposvx zpotrf zpteqr zsctr zstegr ztgsen
      ztpmqrt ztpqrt zuncsd zunmhr 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:

    S07AAF  F_1 = 1.0E+13
            F_2 = 1.0E-14
    S10AAF  E_1 = 1.8715E+1
    S10ABF  E_1 = 7.080E+2
    S10ACF  E_1 = 7.080E+2
    S13AAF  x_hi = 7.083E+2
    S13ACF  x_hi = 1.0E+16
    S13ADF  x_hi = 1.0E+17
    S14AAF  IFAIL = 1 if X > 1.70E+2
            IFAIL = 2 if X < -1.70E+2
            IFAIL = 3 if abs(X) < 2.23E-308
    S14ABF  IFAIL = 2 if X > x_big = 2.55E+305
    S15ADF  x_hi = 2.65E+1
    S15AEF  x_hi = 2.65E+1
    S15AGF  IFAIL = 1 if X >= 2.53E+307
            IFAIL = 2 if 4.74E+7 <= X < 2.53E+307
            IFAIL = 3 if X < -2.66E+1
    S17ACF  IFAIL = 1 if X > 1.0E+16
    S17ADF  IFAIL = 1 if X > 1.0E+16
            IFAIL = 3 if 0 < X <= 2.23E-308
    S17AEF  IFAIL = 1 if abs(X) > 1.0E+16
    S17AFF  IFAIL = 1 if abs(X) > 1.0E+16
    S17AGF  IFAIL = 1 if X > 1.038E+2
            IFAIL = 2 if X < -5.7E+10
    S17AHF  IFAIL = 1 if X > 1.041E+2
            IFAIL = 2 if X < -5.7E+10
    S17AJF  IFAIL = 1 if X > 1.041E+2
            IFAIL = 2 if X < -1.9E+9
    S17AKF  IFAIL = 1 if X > 1.041E+2
            IFAIL = 2 if X < -1.9E+9
    S17DCF  IFAIL = 2 if abs(Z) < 3.92223E-305
            IFAIL = 4 if abs(Z) or FNU+N-1 > 3.27679E+4
            IFAIL = 5 if abs(Z) or FNU+N-1 > 1.07374E+9
    S17DEF  IFAIL = 2 if AIMAG(Z) > 7.00921E+2
            IFAIL = 3 if abs(Z) or FNU+N-1 > 3.27679E+4
            IFAIL = 4 if abs(Z) or FNU+N-1 > 1.07374E+9
    S17DGF  IFAIL = 3 if abs(Z) > 1.02399E+3
            IFAIL = 4 if abs(Z) > 1.04857E+6
    S17DHF  IFAIL = 3 if abs(Z) > 1.02399E+3
            IFAIL = 4 if abs(Z) > 1.04857E+6
    S17DLF  IFAIL = 2 if abs(Z) < 3.92223E-305
            IFAIL = 4 if abs(Z) or FNU+N-1 > 3.27679E+4
            IFAIL = 5 if abs(Z) or FNU+N-1 > 1.07374E+9
    S18ADF  IFAIL = 2 if 0 < X <= 2.23E-308
    S18AEF  IFAIL = 1 if abs(X) > 7.116E+2
    S18AFF  IFAIL = 1 if abs(X) > 7.116E+2
    S18DCF  IFAIL = 2 if abs(Z) < 3.92223E-305
            IFAIL = 4 if abs(Z) or FNU+N-1 > 3.27679E+4
            IFAIL = 5 if abs(Z) or FNU+N-1 > 1.07374E+9
    S18DEF  IFAIL = 2 if REAL(Z) > 7.00921E+2
            IFAIL = 3 if abs(Z) or FNU+N-1 > 3.27679E+4
            IFAIL = 4 if abs(Z) or FNU+N-1 > 1.07374E+9
    S19AAF  IFAIL = 1 if abs(X) >= 5.04818E+1
    S19ABF  IFAIL = 1 if abs(X) >= 5.04818E+1
    S19ACF  IFAIL = 1 if X > 9.9726E+2
    S19ADF  IFAIL = 1 if X > 9.9726E+2
    S21BCF  IFAIL = 3 if an argument < 1.583E-205
            IFAIL = 4 if an argument >= 3.765E+202
    S21BDF  IFAIL = 3 if an argument < 2.813E-103
            IFAIL = 4 if an argument >= 1.407E+102
  3. X01

    The values of the mathematical constants are:

    X01AAF (pi) = 3.1415926535897932
    X01ABF (gamma) = 0.5772156649015328
  4. X02

    The values of the machine constants are:

    The basic parameters of the model

    X02BHF   = 2
    X02BJF   = 53
    X02BKF   = -1021
    X02BLF   = 1024

    Derived parameters of the floating-point arithmetic

    X02AJF   = 1.11022302462516E-16
    X02AKF   = 2.22507385850721E-308
    X02ALF   = 1.79769313486231E+308
    X02AMF   = 2.22507385850721E-308
    X02ANF   = 2.22507385850721E-308

    Parameters of other aspects of the computing environment

    X02AHF   = 1.42724769270596E+45
    X02BBF   = 2147483647
    X02BEF   = 15
  5. X04

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

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 the following formats:

The following main index files have been provided for these formats:

Use your web browser to navigate from here. For convenience, a master index file containing links to the above files has been provided at

Advice on viewing and navigating the formats available 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.