NLL6I272NL  Licence Managed
NAG Library, Mark 27.2
NLL6I272NL  Licence Managed
Linux, 64bit, C++98, Intel C/C++ or Intel Fortran
Users' Note
NLL6I272NL  Licence Managed
1
Introduction
This document is essential reading for every user of the NAG Library
implementation specified in the title. It provides implementationspecific
detail that augments the information provided in the NAG Mark 27.2 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.
This implementation of the NAG Library provides static and shareable libraries that use
the Intel ® Math Kernel Library for Linux (MKL), a thirdparty 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 selfcontained NAG reference versions of these routines
(referred to as the selfcontained libraries). This implementation has been tested with
version 2020.0.4 of MKL, which is supplied as a part of this product.
Please see the Intel website for further information about MKL (
https://software.intel.com/content/www/us/en/develop/tools/mathkernellibrary.html).
For best performance, we recommend that you use one of the variants of the NAG Library
which is based on the supplied MKL vendor Library, i.e.
libnag_mkl.a or
libnag_mkl.so, in preference to
using one of the selfcontained NAG libraries,
libnag_nag.a
or
libnag_nag.so.
This implementation includes libraries (and associated files) for use with
both 32bit integers (denoted by lp64) and 64bit
integers (denoted by ilp64).
The NAG AD Library is also included as the variant addon libraries:
libnag_nag_ad.a, libnag_nag_ad.so,
libnag_mkl_ad.a and libnag_mkl_ad.so.
If using routines from the NAG AD Library then one of these libraries,
along with a dcof interface layer library
libnag_dcof.a,
must be
used in addition to the libraries above.
The dcof interface layer library
libnag_dcof.a
is supplied for two tape memory models: blob and chunk.
For more details about the memorymodel feature, see the user guide for
NAG dco/c++.
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 runtime 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.
Please note that the NAG AD Library is not currently threadsafe.
Calls to NAG AD Library routines cannot be performed in parallel.
The libraries supplied with this implementation do not contain OpenMP
or any other threading mechanisms.
However, the MKL vendor Library is OpenMP threaded. See
Section 3.1.1 for more information on how to control this threading.
Intel have introduced a conditional bitwise reproducibility (BWR) option in MKL. Provided a user's code adheres to certain conditions (see
https://software.intel.com/enus/mkllinuxdeveloperguidereproducibilityconditions), BWR can be forced by setting the
MKL_CBWR environment variable. See the MKL documentation for further details. It should be noted, however, that many NAG routines do not adhere to these conditions. This means that for a given NAG library built on top of MKL, it may not be possible to ensure BWR for all NAG routines across different CPU architectures by setting
MKL_CBWR. See
Section 8.1
of How to Use the NAG Library for more general information on bitwise reproducibility.
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
$HOME/NAG/nll6i272nl;
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
LD_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 nagvars.sh and nagvars.csh are
included to set NAGspecific 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 LD_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/nagvars.sh [help] [unset] [quiet] [gnu] \
{int32,int64} {vendor,nag} {static,dynamic}
or
source [INSTALL_DIR]/scripts/nagvars.csh [help] [unset] [quiet] [gnu] \
{int32,int64} {vendor,nag} {static,dynamic}
where:
 •nagvars.sh should be used for Bourne, Bash or equivalent shells
(note: not Dash,
which is available on some Debianderived distributions of Linux such as Ubuntu)
and
nagvars.csh should be used for Csh, Tcsh or equivalent shells.
 •{int32,int64} specifies the default size for integer arguments and
variables within NAG routines.
 •{vendor,nag} specifies whether to use the set of NAG libraries that depend
upon the supplied MKL Library for
BLAS and LAPACK (option vendor), or to use the selfcontained NAG
libraries (option nag). For best performance, option vendor
is recommended.
 •{static,dynamic} specifies whether to link to static or dynamic (shared)
versions of the NAG libraries.
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:
 •help prints information about the script.
 •unset clears the NAGspecific environment variables and tries to
remove all references to material from this NAG implementation only from
the standard environment variables PATH and LD_LIBRARY_PATH.
 •quiet suppresses output to stdout.
 •gnu is explained at the end of this section.
Thus, an example of using the
nagvars script to set the environment
in Bash is:
source [INSTALL_DIR]/scripts/nagvars.sh int64 vendor dynamic
The NAGspecific environment variables set are:
 •NAGLIB_CC  The C compiler used in the creation of this NAG Library.
 •NAGLIB_CXX  The C++ compiler used in the creation of this NAG Library.
 •NAGLIB_F77  The Fortran compiler used in the creation of this NAG Library.
 •NAGLIB_CFLAGS  Necessary or recommended C compiler flags.
 •NAGLIB_CXXFLAGS  Necessary or recommended C++ compiler flags.
 •NAGLIB_FFLAGS  Necessary or recommended Fortran compiler flags.
 •NAGLIB_INCLUDE  Include paths to NAG C header and/or Fortran module files.
 •NAGLIB_CLINK  Libraries needed for linking to regular NAG (and optionally vendor BLAS and LAPACK) routines using the C compiler.
 •NAGLIB_CXXLINK  Libraries needed for linking to regular NAG (and optionally vendor BLAS and LAPACK) routines using the C++ compiler.
 •NAGLIB_FLINK  Libraries needed for linking to regular NAG (and optionally vendor BLAS and LAPACK) routines using the Fortran compiler.
 •NAGLIB_AD_LINK  Additional library needed for NAG AD routines, which should
be linked before NAGLIB_CLINK, NAGLIB_CXXLINK or NAGLIB_FLINK.
To use the NAG Library and the supplied Intel MKL
libraries (if desired), you may then link in the following manner:

For C programs:

or for C++ programs:

or for Fortran programs:
Note that you may also
need to set LD_LIBRARY_PATH to point
at other items such as compiler runtime libraries, for example if
you are using a newer version of the compiler.
If you are using a different compiler, or indeed a different
version of the Intel compiler, you may need to link against
the Intel compiler runtime libraries
provided in
[INSTALL_DIR]/rtl/lib/intel64.
This may be facilitated by adding
[INSTALL_DIR]/rtl/lib/intel64
to
LD_LIBRARY_PATH.
When NAG AD routines are required,
the nagvars scripts configure linking to be
against the blob version of the dcof interface layer library.
To switch to accessing the chunk version, set the environment variable
DCO_CHUNK_TAPE to 1
before using the nagvars or
nag_example scripts.
This version of the NAG Library can be called from the GNU gcc and g++ compilers,
using at least the versions of gcc mentioned in the
Supplementary Information page on our
website. The easiest way to do this is to use the
gnu option to
the
nagvars.sh and/or
nagvars.csh scripts. Using this option will modify the
NAGspecific environment variables to use GNU gcc and g++ (as appropriate).
Note that GNU gfortran is not supported, so the Fortranrelated NAG environment variables will continue to reference the Intel ifort compiler.
Also note that linking with MKL will continue to use the Intel compiler OpenMP runtime,
for consistency with any OpenMP routines present in the NAG Library.
3.1.1
Setting the Number of Threads to Use
MKL makes
use of OpenMP to implement threading in some of the Library routines. The number of threads
that will be used at run time can be controlled by setting the environment variable
OMP_NUM_THREADS to the appropriate value.
In the C shell, type:
setenv OMP_NUM_THREADS N
In the Bourne shell, type:
OMP_NUM_THREADS=N
export OMP_NUM_THREADS
where N is the number of threads required. The environment variable OMP_NUM_THREADS
may be reset between each execution of the program, as desired.
Multiple levels of OpenMP parallelism may be present in some
MKL routines,
and you may also call these multithreaded routines from within an OpenMP parallel region in your
own application. By default, OpenMP nested parallelism is disabled, so only the outermost parallel
region will actually be active, using N threads in the example above. The inner level(s)
will not be active, i.e. they will run on one thread. You can check
if OpenMP nested parallelism is enabled and choose to enable/disable it by
querying and setting the OMP_NESTED OpenMP environment variable.
If OpenMP nested parallelism is enabled, the above example will create N threads at each
parallel region for each thread at a higher level, thus N*N threads in total if there are two
levels of OpenMP parallelism, etc. To provide more detailed control of nested parallelism, the
environment variable OMP_NUM_THREADS can be set to be a commaseparated list to specify
the number of threads desired at each level.
In the C shell, type:
setenv OMP_NUM_THREADS N,P
In the Bourne shell, type:
OMP_NUM_THREADS=N,P
export OMP_NUM_THREADS
This will create N threads for the first level of parallelism, and then P threads
for each outer level thread when an inner level of parallelism is encountered.
Note: If the environment variable OMP_NUM_THREADS is not set, the default value can vary
from compiler to compiler, and for different vendor libraries, usually to either be 1 or else equal
to the maximum number of cores available on your system. The latter could be an issue if you are
sharing the system with other users, or are running a higher level of parallelism within your own
application. Thus it is recommended that you always set OMP_NUM_THREADS explicitly to your
desired value.
In general, the maximum number of threads you are recommended to use is the number
of physical cores on your shared memory system.
However, most Intel processors
support a facility known as Hyperthreading, which allows each physical core to support up to
two threads at the same time and thus appear to the operating system as two logical cores.
It may be beneficial to make use of this functionality, but this choice will depend
on the particular algorithms and problem size(s) used. You are advised to benchmark
performancecritical applications with and without making use of the additional logical cores,
to determine the best choice for you. This can normally be achieved simply by an appropriate choice
for the number of threads to use, via OMP_NUM_THREADS. Completely disabling Hyperthreading
normally requires setting the desired choice in the BIOS on your system at boot time.
The supplied Intel MKL libraries include additional environment variables to allow
greater control of the threading within MKL. These are discussed at
https://software.intel.com/enus/articles/intelmathkernellibraryintelmklintelmkl100threading.
Many NAG routines make calls to routines within MKL, thus the MKL environment variables
may indirectly affect the operation of the NAG Library as well. The default settings of the
MKL environment variables should be suitable for most purposes, thus it is recommended that
you do not explicitly set these variables.
Please contact NAG for further advice if required.
3.2
Fortran Interface Blocks
The NAG Library interface blocks define the type and arguments of each
usercallable 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
nag_library
The modules are supplied in compiled form (.mod files) for the
Intel Fortran compiler.
They can be accessed by specifying the
Ipathname
option on each compiler invocation, where
pathname ([INSTALL_DIR]/lp64/nag_interface_blocks or [INSTALL_DIR]/ilp64/nag_interface_blocks)
is the path of the directory containing the compiled interface blocks for the required integer size.
The
.mod module files were compiled with the Fortran compiler shown
in
Section 2.2 of the Installer's Note.
Such module files are compilerdependent, 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,int64} 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 any
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, it is first recommended that you generate a new set of modules for
both integer sizes using both options
{int32,int64} in separate calls to
[INSTALL_DIR]/scripts/nag_recompile_mods.
Then you may use the following commands (with [INSTALL_DIR] substituted by the actual directory path as appropriate) to
change both default sets:
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
mv [INSTALL_DIR]/ilp64/nag_interface_blocks [INSTALL_DIR]/ilp64/nag_interface_blocks_original
mv [INSTALL_DIR]/ilp64/nag_interface_blocks_alt [INSTALL_DIR]/ilp64/nag_interface_blocks
You should now be able to use the newly compiled module files in the
usual way.
Currently it is not possible for users to recompile the interface blocks for the AD routines. If this is required, please
contact NAG for support (see
Section 7 for contact details).
3.3
Example Programs
The example results distributed were generated at
Mark 27.2 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
32bit integer
static library
libnag_mkl.a
(i.e. using the MKL BLAS and LAPACK routines).
Running the examples with NAG BLAS or LAPACK may
give slightly different results.
The distributed NAG AD example results are those obtained with the
32bit integer
static
libraries libnag_mkl.a and libnag_mkl_ad.a.
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
32bit integers and
static linking
to the selfcontained
libnag_nag.a library. These
choices can be changed by optional switches
int64,
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.
To run one of the examples for the NAG AD routines, the
ad flag must be added, e.g.
nag_example ad s01ba_a1w_hcpp
Note that a few of the AD examples have a dependency on the
file
dco.hpp which is not shipped with this Library.
Please contact NAG if you are interested in using these examples
(see
Section 7 for contact details).
If you already have the dco.hpp file and other dco/c++ files
on your system, the environment variable DCODIR may be set to
point to their location; if this has been done when
the nagvars script is run, the DCODIR directory will
be added to the include paths used in the compile commands.
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, libraries for both 32bit and 64bit integers are included.
For the 32bit integer libraries (located in directory
[INSTALL_DIR]/lp64/lib)
the NAG C types
Integer and
Pointer are defined as follows:
NAG Type 
C Type 
Size (bytes) 
Integer 
int 
4 
Pointer 
void * 
8 
For the 64bit integer libraries (located in directory
[INSTALL_DIR]/ilp64/lib)
the NAG C types
Integer and
Pointer are defined as follows:
NAG Type 
C Type 
Size (bytes) 
Integer 
long 
8 
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 both 32bit integers
(located in directory [INSTALL_DIR]/lp64/lib)
and 64bit integers
(located in directory [INSTALL_DIR]/ilp64/lib).
The NAG Library and documentation use parameterized types for
floatingpoint variables.
Thus, the type
REAL(KIND=nag_wp)
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 floatingpoint
arguments and internal variables used in the Library are of this type.
In addition, a small number of routines use the type
REAL(KIND=nag_rp)
where
nag_rp stands for "reduced precision" type. Another
type, not currently
used in the Library, is
REAL(KIND=nag_hp)
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 NAG FL Interface Introduction for details.
3.7
Calling NAG Fortran Routines from C/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.
C/C++ header files ([INSTALL_DIR]/lp64/include/nag.h for 32bit integers
and [INSTALL_DIR]/ilp64/include/nag.h for 64bit integers) are
also provided. It is recommended that users wishing to use a NAG Fortran
routine in this manner #include
the appropriate 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.)
3.8
C declarations for LAPACK, BLAS, etc.
The NAG C/C++ header files include declarations for the
LAPACK,
BLAS and BLAS Technical Forum
(
BLAST) routines included
in the NAG Library. Users may prefer to get these definitions from the C include files
associated with other libraries, e.g. the supplied Intel MKL.
In these circumstances, to avoid clashes between the different C header declarations,
the NAG declarations of these routines may be disabled by adding the compile flags:
DNAG_OMIT_LAPACK_DECLARATION DNAG_OMIT_BLAS_DECLARATION DNAG_OMIT_BLAST_DECLARATION
to the C or C++ compile statements described in
Section 3.1.
Declarations for the alternative NAG F01, F06, F07 and F08 routine names will remain.
4
Routinespecific Information
Any further information which applies to one or more routines in this
implementation is listed below, chapter by chapter.

(a)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 NAGstyle 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 Library may require
a different amount of workspace from the equivalent NAG reference 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 nonselfcontained NAG libraries
are implemented by calls to MKL, 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
dbdsvdx dgesvdx dgesvj dsbgvd zgejsv zgesvdx zgesvj zhbgvd

(b)S07  S21
The behaviour of functions in these Chapters may depend on implementationspecific 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 = 1.0e14
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) < 2.23e308
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 <= 2.23e308
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) < 3.92223e305
ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n1 > 3.27679e+4
ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n1 > 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+n1 > 3.27679e+4
ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n1 > 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) < 3.92223e305
ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n1 > 3.27679e+4
ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n1 > 1.07374e+9
s18ad[fc] (nag[f]_specfun_bessel_k1_real)
ifail = 2 (NE_REAL_ARG_TOO_SMALL) if 0 < x <= 2.23e308
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) < 3.92223e305
ifail = 4 (NW_SOME_PRECISION_LOSS) if abs(z) or fnu+n1 > 3.27679e+4
ifail = 5 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n1 > 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+n1 > 3.27679e+4
ifail = 4 (NE_TOTAL_PRECISION_LOSS) if abs(z) or fnu+n1 > 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 < 1.583e205
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 < 2.813e103
ifail = 4 (NE_REAL_ARG_GT) if an argument >= 1.407e+102

(c)X01
The values of the mathematical constants are:
x01aa[fc] (nag[f]_math_pi)
= 3.1415926535897932
x01ab[fc] (nag[f]_math_euler)
= 0.5772156649015328

(d)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 floatingpoint arithmetic
x02aj[fc] (nag[f]_machine_precision)
= 1.11022302462516e16
x02ak[fc] (nag[f]_machine_real_smallest)
= 2.22507385850721e308
x02al[fc] (nag[f]_machine_real_largest)
= 1.79769313486231e+308
x02am[fc] (nag[f]_machine_real_safe)
= 2.22507385850721e308
x02an[fc] (nag[f]_machine_complex_safe)
= 2.22507385850721e308
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 (for 32bit integer libraries)
= 9223372036854775807 (for 64bit integer libraries)
x02be[fc] (nag[f]_machine_decimal_digits)
= 15

(e)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.

(f)X06
Chapter X06 routines do not change the behaviour of MKL 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 uptodate version of the documentation
is accessible via the NAG website at
NAG Library Manual, Mark 27.2.
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:
nagdoc_27.2/index.html
Use your web browser to navigate from here.
Advice on viewing and navigating the documentation can be found in
Guide to the NAG Library Documentation.
In addition the following are provided:
 in.html – Installer's Note
 un.html – Users' Note (this document)
 alt_c_interfaces.html – Advice on
calling the Fortran routines in the NAG Library from C and C++
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.
Please see
for worldwide contact details for the Numerical Algorithms Group.
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