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
for details of any new information related to the applicability or usage of this implementation.
This implementation of the NAG Library provides static and shared libraries that use a third-party vendor performance library to provide Basic Linear Algebra Subprograms (BLAS) and Linear Algebra PACKage (LAPACK) routines (see below). It also provides static and shared libraries that use the NAG versions of these routines (referred to as the self-contained libraries).
This implementation of the NAG Library has been tested with version 2019.0.2 of the Intel ® Math Kernel Library for Windows (MKL) which is supplied as a part of this product. Please see the Intel website for further information about MKL (https://software.intel.com/en-us/mkl).
For best performance, we recommend that you use one of the variants of the NAG Library which is based on the supplied MKL, i.e. nag_mkl_MT.lib, nag_mkl_MD.lib or NLW6I27DE_mkl.lib/NLW6I27DE_mkl.dll. These libraries do not contain the NAG versions of the BLAS and LAPACK routines (except for any routines listed in Section 4).
The self-contained libraries, nag_nag_MT.lib, nag_nag_MD.lib and NLW6I27DE_nag.lib/NLW6I27DE_nag.dll, are also supplied. These contain the NAG versions of the BLAS and LAPACK routines.
Which static variant of the NAG Library you should use will also depend on how you wish to link to the Microsoft run-time libraries. For example, if you are linking with the multithreaded static run-time libraries, you should use nag_mkl_MT.lib or nag_nag_MT.lib, whereas if you are linking with the multithreaded dynamic link run-time libraries, you should use nag_mkl_MD.lib or nag_nag_MD.lib. Alternatively, if you wish to call a dynamic link library (DLL) variant of the NAG Library, you should link with the import library NLW6I27DE_mkl.lib or NLW6I27DE_nag.lib (and, at run time, make sure that the corresponding DLL, NLW6I27DE_mkl.dll or NLW6I27DE_nag.dll, is on your path). For more details, see Section 3.1.1.
The NAG AD Library is also included as the variant add-on libraries: nag_nag_ad_MT.lib and nag_nag_ad_MD.lib.
Note that the NAG Library is carefully designed so that any memory used can be reclaimed – either by the Library itself or 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. Further information about using the supplied Intel MKL libraries with threaded applications is available at https://software.intel.com/en-us/articles/intel-math-kernel-library-intel-mkl-using-intel-mkl-with-threaded-applications.
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 3.1.0 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/en-us/mkl-windows-developer-guide-reproducibility-conditions), 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.
In this section we assume that the Library has been installed in the default folder, namely
C:\Program Files\NAG\NL27\nlw6i27delThe actual name of the "Program Files" folder may appear differently, depending on your locale. If the above folder does not exist, please consult the system manager (or the person who did the installation). In some of the following subsections, this folder is referred to as install_dir.
We also assume that the shortcut for the Library command prompt is in the NAG Library (NLW6I27DEL) section of the Start Menu or All apps under:
NAG NLW6I27DEL Command PromptIf this shortcut does not exist, please consult the system manager (or the person who did the installation). (Other shortcuts created as part of the Library installation procedure are also assumed to be in this location.)
If you are using a DLL form of the Library (see Section 3.1.1), you need to ensure that the NAG DLL (NLW6I27DE_mkl.dll or NLW6I27DE_nag.dll) is accessible at run time; therefore the install_dir\bin folder must be on the path. The install_dir\rtl\bin folder must be on the path too (unless you have the appropriate Intel run-time libraries on your path already). If an MKL-based version of the Library is to be used, the install_dir\mkl\bin folder must also be on the path, but should appear later in the path than the install_dir\bin folder, since the NAG versions of a few BLAS / LAPACK routines may be included in the NAG Libraries to avoid problems with the vendor versions. (See Section 4 for details.)
To check the accessibility of the NAG DLLs, run the program NAG_Library_DLL_info.exe which is available from the Start Menu or All apps shortcut
Check NAG NLW6I27DEL DLL AccessibilitySee Section 4.2.2 of the Installer's Note for details of this utility.
set OMP_NUM_THREADS=Nwhere N is the number of threads required. Environment variables can also be set in the usual way via the Windows Control Panel. The environment variable OMP_NUM_THREADS may be re-set 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 comma-separated list to specify the number of threads desired at each level. e.g.
set OMP_NUM_THREADS=N,PThis 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 performance-critical 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/en-us/articles/intel-math-kernel-library-intel-mkl-intel-mkl-100-threading. 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.
NAG NLW6I27DEL Command Promptmay be used to start a command prompt window with the correct settings for the INCLUDE, LIB and PATH environment variables for the Library and the supplied MKL. The environment variable NAG_NLW6I27DEL, which is needed by the nag_example_*.bat batch files is also set.
If the shortcut is not used, you can set the environment variables by running the batch file envvars.bat for this implementation. The default location for this file is:
C:\Program Files\NAG\NL27\nlw6i27del\batch\envvars.batIf this file is not in the default location, you can locate it by searching for the file envvars.bat containing nlw6i27del.
You may then compile and link to the NAG Library on the command line using one of the following commands:
cl /MD driver.c NLW6I27DE_mkl.lib ifort /MD driver.f90 NLW6I27DE_mkl.lib cl /MD driver.c NLW6I27DE_nag.lib ifort /MD driver.f90 NLW6I27DE_nag.lib cl /MT driver.c nag_mkl_MT.lib mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib /link /nodefaultlib:ifconsol.lib /nodefaultlib:ifmodintr.lib /nodefaultlib:libifcoremt.libwhere driver.c or driver.f90 is your application program. (Note – we assume above use of the Microsoft C compiler cl. You may also use the Intel C compiler icl. Options for both compilers are the same.)
/nodefaultlib:libifport.lib /nodefaultlib:ifwin.lib ifort /MT driver.f90 nag_mkl_MT.lib mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib cl /MT driver.c nag_nag_MT.lib user32.lib /link /nodefaultlib:ifconsol.lib /nodefaultlib:ifmodintr.lib /nodefaultlib:libifcoremt.lib
/nodefaultlib:libifport.lib /nodefaultlib:ifwin.lib ifort /MT driver.f90 nag_nag_MT.lib user32.lib cl /MD driver.c nag_mkl_MD.lib mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib /link /nodefaultlib:ifconsol.lib /nodefaultlib:ifmodintr.lib /nodefaultlib:libifcoremd.lib
/nodefaultlib:libifportmd.lib /nodefaultlib:ifwin.lib ifort /MD driver.f90 nag_mkl_MD.lib mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib cl /MD driver.c nag_nag_MD.lib user32.lib /link /nodefaultlib:ifconsol.lib /nodefaultlib:ifmodintr.lib /nodefaultlib:libifcoremd.lib
/nodefaultlib:libifportmd.lib /nodefaultlib:ifwin.lib ifort /MD driver.f90 nag_nag_MD.lib user32.lib
The instructions above show how to use the NAG Library without the NAG AD Libraries. To add the NAG AD Libraries you must add the appropriate variants of those libraries after the named NAG Library by using just one of these library and linker option sets:
The compiler/linker options:
NLW6I27DE_mkl.lib is a DLL import library that makes use of MKL for BLAS/LAPACK routines. NLW6I27DE_nag.lib is a DLL import library that includes NAG BLAS/LAPACK. Both libraries have been compiled with the /MD option. This option must be used when compiling applications to be linked with such libraries to ensure linking to the correct compiler run-time libraries.
nag_mkl_MT.lib is a static library that does not include BLAS/LAPACK and should be linked to the MKL static libraries. nag_nag_MT.lib is a static library that includes NAG BLAS/LAPACK. Both libraries have been compiled with the /MT option. This option must be used when compiling applications to be linked with such libraries to ensure linking to the correct compiler run-time libraries.
nag_mkl_MD.lib is a static library that does not include BLAS/LAPACK and should be linked to the MKL static libraries. nag_nag_MD.lib is a static library that includes NAG BLAS/LAPACK. Both libraries have been compiled with the /MD option. This option must be used when compiling applications to be linked with such libraries to ensure linking to the correct compiler run-time libraries.
If it is planned to use Microsoft Visual Studio to build programs that use the NAG Library, each user should set the appropriate options.
Start Visual Studio and create your project in the usual way. We assume that your project is going to make use of the NAG Library.
The library is intended to be run in fully optimized mode, so to avoid any warning messages, you might decide to set the active configuration to Release. Once Visual Studio has been opened, you can do this from the Toolbar or alternatively via the Build|Configuration Manager menus. Note that if you work in Debug mode, you may receive a warning message about conflicting run-time libraries.
Make sure the Platform is set to x64 (to ensure compatibility with this 64-bit implementation of the NAG Library). This can be changed via the Configuration Manager... button on the Property Pages.
The following steps show how to add the NAG Library to the project:
The default folders are as follows:
Include Directories C:\Program Files\NAG\NL27\nlw6i27del\include Library Directories C:\Program Files\NAG\NL27\nlw6i27del\lib C:\Program Files\NAG\NL27\nlw6i27del\rtl\lib C:\Program Files\NAG\NL27\nlw6i27del\mkl\libClick on the Apply button to accept the changes.
Click on the OK button to accept the changes and close the form.
If your project is a Microsoft C or C++ project:
First add your source file, e.g. a NAG example program, to the project, using Add Existing Item... from the Project menu. (If you don't have a C or C++ file in the project, the C++ options may not be visible.)
Open the Property Pages again (as detailed above) and click/expand Configuration Properties (if required) and then C/C++, then click on Code Generation in the left hand panel. Then, from the right hand panel, select Runtime Library and change this to the appropriate version, for example /MD or /MT. This must match the version of the NAG Library that you link to.
Click on the OK button to accept the changes and close the form.
If your project is an Intel Fortran project:
From the Properties form, click/expand Fortran in the leftmost panel and then choose Libraries. The right hand panel will now have a Runtime Library entry, and you need to select Multithreaded if your project uses one of the two libraries nag_nag_MT.lib or nag_mkl_MT.lib. If your project uses any of the other NAG libraries you need to select Multithread DLL. After you select the correct run-time library click on the OK button to accept the changes and close the form.
|NAG Library||MKL and other Libraries||Run-time Libraries|
|NLW6I27DE_mkl.lib||user32.lib (only needed if also linking to AD library)||Multi-threaded DLL (/MD)|
|NLW6I27DE_nag.lib||user32.lib (only needed if also linking to AD library)||Multi-threaded DLL (/MD)|
|nag_mkl_MT.lib||mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib||Multi-threaded (/MT)|
|nag_mkl_MD.lib||mkl_intel_lp64.lib mkl_intel_thread.lib mkl_core.lib libiomp5md.lib user32.lib||Multi-threaded DLL (/MD)|
|nag_nag_MD.lib||user32.lib||Multi-threaded DLL (/MD)|
Click on the OK button to accept the changes and close the form.
The project should now compile and link using the appropriate choice from the Build menu.
To run a program from within the Microsoft Development Environment, the program may be executed via the Debug menu (by selecting Start Without Debugging (Ctrl+F5), for example). Note that the PATH environment variable must be set appropriately, as detailed in Section 3.1.1 above.
If a data file needs to be attached to the standard input or the output of a program needs to be redirected to the standard output, this can be achieved by selecting the Debugging section on the Properties form and inserting the appropriate commands in the Command Arguments field, e.g.
< input_file > output_fileIf the input and output files are not in the application's working directory, full or relative paths may need to be specified. For NAG examples that use an .opt file, this should be placed in the working directory. This directory may be set via the Working Directory field, which is also on the Debugging page of the Properties form.
Note that if you build your project in Debug mode (the default), it is not possible to use the Undefined variables option which is accessible on the Fortran Compiler / Runtime Check tab of Project Settings. This is because the NAG Library was not compiled with this option. Trying to use it will cause a compile-time error in Fortran Builder, showing an "Incompatible option setting" when using the NAG interface blocks.
Again, it is important to note that you must link to the DLL itself, not the associated import library.
From a Windows Command Prompt, first make sure that the PATH environment variable is correctly set, as described in Section 3.1.1.
You may then compile and link to the NAG Library on the command line using one of the following commands:
nagfor -ieee=full -I"install_dir\nag_interface_blocks_nagfor" driver.f90 "install_dir\bin\NLW6I27DE_mkl.dll" -o driver.exe nagfor -ieee=full -I"install_dir\nag_interface_blocks_nagfor" driver.f90 "install_dir\bin\NLW6I27DE_nag.dll" -o driver.exedepending on whether you wish to link to the MKL-supported version of the library or the all-NAG version.
The full pathname of the NLW6I27DE_mkl or NLW6I27DE_nag library file must be specified and must be enclosed within quotes if it contains spaces.
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 pre-compiled form (.mod files) for use by the Intel Fortran compiler, ifort.
If you use the Library command prompt shortcut or set the environment variables by running the batch file envvars.bat for this implementation (see Section 3.1.1), and the Intel ifort compiler, you can use any of the commands described in Section 3.1.1 to access these modules since the environment variable INCLUDE will be set.
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 create your own module files, as described here.
Create a folder named nag_interface_blocks_original in a location of your choice (the exact folder name is not important), and copy the contents of nag_interface_blocks to nag_interface_blocks_original, thus saving the original set of interface blocks.
Then in folder nag_interface_blocks recompile all the .f90 files into objects using your compiler. Because the interface blocks contain some inter-dependencies, the order of compilation is important, but the following compilation order should work, where FCOMP is the name of your Fortran compiler:
FCOMP -c nag_precisions.f90 FCOMP -c nag_a_ib.f90 FCOMP -c nag_blast_ib.f90 FCOMP -c nag_blas_consts.f90 FCOMP -c nag_blas_ib.f90 FCOMP -c nag_c_ib.f90 FCOMP -c nag_d_ib.f90 FCOMP -c nag_e_ib.f90 FCOMP -c nag_f_ib.f90 FCOMP -c nag_g_ib.f90 FCOMP -c nag_h_ib.f90 FCOMP -c nag_lapack_ib.f90 FCOMP -c nag_m_ib.f90 FCOMP -c nag_s_ib.f90 FCOMP -c nag_x_ib.f90 FCOMP -c nag_long_names.f90 FCOMP -c nag_library.f90The object files generated by the compilation may be discarded – only the module files are needed.
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).
The distributed example results are those obtained with the static library nag_mkl_MD.lib (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 static libraries nag_mkl_MD.lib and nag_nag_ad_MD.lib.
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 using the batch files nag_example_DLL.bat, nag_example_static_MT.bat and nag_example_static_MD.bat, which can be found in the install_dir\batch folder.
These batch files require that the environment variables for your C/C++ or Fortran compiler and the NAG Library are set. In particular, the environment variable NAG_NLW6I27DEL needs to be set to the location of the NAG Library. Please see Section 3.1.1 for details of how to do this.
Each of the nag_example_*.bat batch files mentioned above 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.
Example programs in both C and Fortran are provided.
The example program concerned is specified by the argument to the command, e.g.
nag_example_DLL e04ucc nag_example_DLL e04ucfwill copy the example program and its data and options files (e04ucce.c, e04ucce.d and e04ucce.opt for C, or e04ucfe.f90 and e04ucfe.d for Fortran) into the current folder, compile and link the program and run it to produce the example program results in the file e04ucce.r.
nag_example_DLL.bat links to the DLL version of the NAG Library using NAG BLAS/LAPACK.
To link with the MKL version of the DLL, use the -mkl option, e.g.
nag_example_DLL -mkl e04ucc nag_example_DLL -mkl e04ucf
The nag_example_static_MD.bat batch file is used in the same way and links to the static NAG library compiled with /MD.
nag_example_static_MD e04ucc nag_example_static_MD e04ucfAgain, it is possible to link the MKL BLAS/LAPACK by using the -mkl option
nag_example_static_MD -mkl e04ucc nag_example_static_MD -mkl e04ucf
The nag_example_static_MT.bat batch file links to the static library compiled with /MT, e.g.
nag_example_static_MT e04ucc nag_example_static_MT e04ucf nag_example_static_MT -mkl e04ucc nag_example_static_MT -mkl e04ucf
For any of the above bat files, adding the -ad switch will run a NAG AD library example program instead, e.g.
nag_example_static_MT -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, their location should be added to the INCLUDE environment variable. Alternatively, the files may be copied to the install_dir\include folder.
Alternatively, run the diagnostic program NAG_Library_DLL_info.exe which itself calls a00aac and a00aaf (see Installer's Note, Section 4.2.2).
|NAG Type||C Type||Size (bytes)|
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).
This implementation of the NAG Library includes libraries for 32-bit integers only. The libraries are located in install_dir\lib.
The NAG Library and documentation use parameterized types for floating-point variables. Thus, the type
REAL(KIND=nag_wp)appears in 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
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 FL Interface Introduction for details.
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.)
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 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 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 dgejsv dgesvdx dgesvj zgejsv zgesvdx zgesvj zhegvx
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
The values of the mathematical constants are:
x01aa[fc] (nag[f]_math_pi) = 3.1415926535897932 x01ab[fc] (nag[f]_math_euler) = 0.5772156649015328
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) = 2.22507385850721e-308
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
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