NAG Fortran Library, Mark 24

FLL6I24DCL - License Managed

Linux 64 (Intel 64 / AMD64), Intel Fortran, Double Precision

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 24 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 (see Section 5):

(a) Essential Introduction
(b) Chapter Introduction
(c) Routine Document

The libraries supplied with this implementation have been compiled in a manner that facilitates the use of multiple threads.

2. Post Release Information

Please check the following URL:

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

3. General Information

For best performance, we recommend that you use one of the variants of the NAG Fortran Library which is based on the supplied Intel ® Math Kernel Library (MKL) i.e. libnag_mkl.a or However if you use a version of the MKL different from the version used in creating this implementation and you have problems when calling a NAG routine, we suggest that you use one of the above libraries with the supplied MKL, or one of the self-contained libraries libnag_nag.a or

For repeatability of results, we recommend that you use one of the self-contained variants of the NAG Fortran Library, i.e. libnag_nag.a or These have been compiled using the -fp-model precise option and so are optimized for repeatability rather than speed.

If your machine has more than one processor or a multicore chip, then it is recommended that you set the environment variable OMP_NUM_THREADS to the number of available threads, e.g.

  setenv OMP_NUM_THREADS 4
in the C shell, or
in the Bourne shell. This will enable the Intel MKL BLAS to make use of the extra processor(s) / core(s) and will thus speed up the computation of many of the Library procedures.

With MKL version 10.0 or newer, this is the default behaviour. In this case, if you do not want MKL to make use of multiple cores, OMP_NUM_THREADS should be set to 1.

If you are running on an Intel processor and using an MKL-based variant of the NAG Library, performance may be enhanced by using the Conditional Numerical Reproducibility settings introduced in MKL 11.0. To get the best performance from the MKL routines, set the environment variable MKL_CBWR appropriately for your processor. See for the various settings available. Alternatively, call the mkl_set_cbwr_branch function from your code prior to calling any NAG Library routines.

Please note that this implementation is not compatible with versions of MKL earlier than 10.3.

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 /opt/NAG/fll6i24dcl or /usr/local/NAG/fll6i24dcl depending on your system; however it could have been changed by the person who did the installation. To identify [INSTALL_DIR] for this installation:

To use the NAG Fortran Library and the supplied MKL libraries, you may link in the following manner:
  ifort -I[INSTALL_DIR]/nag_interface_blocks [INSTALL_DIR]/lib/libnag_mkl.a driver.f90 \
                    -Wl,--start-group [INSTALL_DIR]/mkl_intel64/libmkl_intel_lp64.a \
                                      [INSTALL_DIR]/mkl_intel64/libmkl_intel_thread.a \
                                      [INSTALL_DIR]/mkl_intel64/libmkl_core.a -Wl,--end-group \
                   -liomp5 -lpthread
where driver.f90 is your application program;


  ifort -I[INSTALL_DIR]/nag_interface_blocks driver.f90 \
      [INSTALL_DIR]/lib/ -L[INSTALL_DIR]/mkl_intel64 \
if the shareable library is required. Please note that the shareable library is not fully resolved unless linked against the run-time library explicitly; this requires the environment variable LD_LIBRARY_PATH to be set correctly at link time (see below).

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

  ifort -I[INSTALL_DIR]/nag_interface_blocks driver.f90 \
  ifort -I[INSTALL_DIR]/nag_interface_blocks driver.f90 \
if the shareable library is required.

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

In the C shell type:

  setenv LD_LIBRARY_PATH [INSTALL_DIR]/lib:[INSTALL_DIR]/mkl_intel64
to set LD_LIBRARY_PATH, or
  setenv LD_LIBRARY_PATH \
to extend LD_LIBRARY_PATH if you already have it set.

In the Bourne shell, type:

to set LD_LIBRARY_PATH, or
to extend LD_LIBRARY_PATH if you already have it set.

Note that you may also need to set LD_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.

If you are using a different compiler, you may need to link against the Intel ifort run-time libraries provided in [INSTALL_DIR]/rtl.

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/nagmk24.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) and 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 2.1 of the Installer's Note. Such module files are compiler-dependent. If you wish to use NAG modules with a version of the Intel Fortran compiler other than the one used to build this NAG Library, you may 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 Intel Fortran compiler from your PATH environment; to specify an alternative version it is safest to first run the Intel 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, 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.

See the Post Release Information page

where more information may be available, or contact NAG for further help.

3.3. Example Programs

The example results distributed were generated at Mark 24, 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 Basic Linear Algebra Subprograms (BLAS) or Linear Algebra PACKage (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_mkl.a, (i.e. using the MKL BLAS and LAPACK routines).

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 directory [INSTALL_DIR]/scripts contains four scripts: nag_example_mkl, nag_example_shar_mkl, nag_example and nag_example_shar.

The example programs are most easily accessed by one of the commands

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.

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

nag_example_mkl e04nrf
will copy the example program and its data and options files (e04nrfe.f90, e04nrfe.d and e04nrfe.opt) into the current directory, compile 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.

One important bold italicised term is machine precision, which denotes the relative precision to which DOUBLE PRECISION floating-point numbers are stored in the computer, e.g. in an implementation with approximately 16 decimal digits of precision, machine precision has a value of approximately 1.0D-16.

The precise value of machine precision is given by the routine X02AJF. Other routines in Chapter X02 return the values of other implementation-dependent constants, such as the overflow threshold, or the largest representable integer. Refer to the X02 Chapter Introduction for more details.

The bold italicised term block size is used only in Chapters F07 and F08. It denotes the block size used by block algorithms in these chapters. You only need to be aware of its value when it affects the amount of workspace to be supplied – see the parameters WORK and LWORK of the relevant routine documents and the Chapter Introduction.

3.5. Explicit Output from NAG Routines

Certain routines produce explicit error messages and advisory messages via output units which have default values that can be reset by using X04AAF for error messages and X04ABF for advisory messages. (The default values are given in Section 4.)

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, alternate routine names are available for BLAS and LAPACK derived routines. For details of the alternate 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 MKL 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 the NAG version of the following BLAS and LAPACK routines are included in the libraries libnag_mkl.a and to avoid problems with the vendor version:

  2. G02

    The value of ACC, the machine-dependent constant mentioned in several documents in the chapter, is 1.0D-13.
  3. 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
  4. X01

    The values of the mathematical constants are:

    X01AAF (pi) = 3.1415926535897932
    X01ABF (gamma) = 0.5772156649015328
  5. 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
  6. 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 part of the installation or 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 the Online Documentation document.

In addition the following are provided:

6. Support from NAG

(a) Contact with NAG

Queries concerning this document or the implementation generally should be directed to NAG at one of the addresses given in the Appendix. Users subscribing to the support service are encouraged to contact one of the NAG Response Centres (see below).

(b) NAG Response Centres

The NAG Response Centres are available for general enquiries from all users and also for technical queries from sites with an annually licensed product or support service.

The Response Centres are open during office hours, but contact is possible by fax, email and phone (answering machine) at all times.

When contacting a Response Centre, it helps us deal with your enquiry quickly if you can quote your NAG site reference or account number and NAG product code (in this case FLL6I24DCL).

(c) NAG Websites

The NAG websites provide information about implementation availability, descriptions of products, downloadable software, product documentation and technical reports. The NAG websites can be accessed at the following URLs:,, or

(d) NAG Electronic Newsletter

If you would like to be kept up to date with news from NAG then please register to receive our free electronic newsletter, which will alert you to announcements about new products or product/service enhancements, technical tips, customer stories and NAG's event diary. You can register via one of our websites, or by contacting us at

(e) Product Registration

To ensure that you receive information on updates and other relevant announcements, please register this product with us. For NAG Library products this may be accomplished by filling in the online registration form at

7. User Feedback

Many factors influence the way that NAG's products and services evolve, and your ideas are invaluable in helping us to ensure that we meet your needs. If you would like to contribute to this process, we would be delighted to receive your comments. Please contact any of the NAG Response Centres (shown below).

Appendix - Contact Addresses

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Tel: +44 (0)1865 511245                 Tel: +44 (0)1865 311744
Fax: +44 (0)1865 310139                 Fax: +44 (0)1865 310139

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USA                                     email:

Tel: +1 630 971 2337                    Tel: +1 630 971 2337
Fax: +1 630 971 2706                    Fax: +1 630 971 2706

Nihon NAG KK
Hatchobori Frontier Building 2F
Tokyo 104-0032                          Nihon NAG Response Centre
Japan                                   email:

Tel: +81 3 5542 6311                    Tel: +81 3 5542 6311
Fax: +81 3 5542 6312                    Fax: +81 3 5542 6312

NAG Taiwan Branch Office
5F.-5, No.36, Sec.3
Minsheng E. Rd.
Taipei City 10480                       NAG Taiwan Response Centre
Taiwan                                  email:

Tel: +886 2 25093288                    Tel: +886 2 25093288
Fax: +886 2 25091798                    Fax: +886 2 25091798