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
https://www.nag.co.uk/doc/inun/nl27/mi6dbl/supplementary.html
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 macOS (MKL), a third-party vendor performance library, to provide Basic Linear Algebra Subprograms (BLAS) and Linear Algebra PACKage (LAPACK) routines (except for any routines listed in Section 4). It also provides static and shareable libraries that use the NAG versions of these routines (referred to as the self-contained libraries). This implementation has been tested with version 2019.0.3 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/intel-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. libnag_mkl.a or libnag_mkl.dylib, in preference to using one of the self-contained NAG libraries, libnag_nag.a or libnag_nag.dylib. This implementation includes libraries (and associated files) for use with both 32-bit integers (denoted by lp64) and 64-bit integers (denoted by ilp64).
The NAG AD Library is not included in this implementation.
Note that the NAG Library is carefully designed so that any memory used can be reclaimed – either by the Library itself or, for C routines, by the user invoking calls of NAG_FREE(). However, the Library does itself depend on the use of compiler run-time and other libraries which may sometimes leak memory, and memory tracing tools used on programs linked to the NAG Library may report this. The amount of memory leaked will vary from application to application, but should not be excessive and should never increase without limit as more calls are made to the NAG Library.
If you intend to use the NAG library within a multithreaded application please refer to the document CL Interface Multithreading or FL Interface Multithreading (as appropriate) for more information. 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.
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/en-us/mkl-macos-developer-guide-obtaining-numerically-reproducible-results), 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 directory [INSTALL_DIR]. By default [INSTALL_DIR] (see Installer's Note (in.html)) is $HOME/NAG/nlmi627dbl; however it could have been changed by the person who did the installation, in which case you should consult that person. Note that the environment variable DYLD_LIBRARY_PATH must be set correctly at link time and run time to point to the appropriate library locations underneath [INSTALL_DIR]. See below for how to do this.
The NAG Library is a combined replacement for users of both the NAG C Library and the NAG Fortran Library (including the C wrappers to the Fortran Library interfaces). To help with calling the different libraries included with this implementation the scripts nagvars.sh and nagvars.csh are included to set NAG-specific environment variables to assist with compiling and linking applications that call any of the NAG routines. They also amend the standard environment variables PATH and DYLD_LIBRARY_PATH so that NAG executable programs and libraries can be found at compile, link and run time.
The nagvars scripts are designed to be used as follows:
. [INSTALL_DIR]/scripts/nagvars.sh [-help] [-unset] [-quiet] {int32,int64} \ {vendor,nag} {static,dynamic}or
source [INSTALL_DIR]/scripts/nagvars.csh [-help] [-unset] [-quiet] {int32,int64} \ {vendor,nag} {static,dynamic}where:
source [INSTALL_DIR]/scripts/nagvars.sh int64 vendor dynamicThe NAG-specific environment variables set are:
For C programs:
${NAGLIB_CC} ${NAGLIB_CFLAGS} ${NAGLIB_INCLUDE} program.c ${NAGLIB_LINK}or for C++ programs:
${NAGLIB_CXX} ${NAGLIB_CXXFLAGS} ${NAGLIB_INCLUDE} program.cpp ${NAGLIB_LINK}or for Fortran programs:
${NAGLIB_F77} ${NAGLIB_FFLAGS} ${NAGLIB_INCLUDE} program.f90 ${NAGLIB_LINK}
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.
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 run-time libraries provided in [INSTALL_DIR]/rtl/lib. This may be facilitated by adding
[INSTALL_DIR]/rtl/libto DYLD_LIBRARY_PATH.
In the C shell, type:
setenv OMP_NUM_THREADS NIn the Bourne shell, type:
OMP_NUM_THREADS=N export OMP_NUM_THREADSwhere N is the number of threads required. 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.
In the C shell, type:
setenv OMP_NUM_THREADS N,PIn the Bourne shell, type:
OMP_NUM_THREADS=N,P export OMP_NUM_THREADSThis 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.
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_libraryThe modules are supplied in compiled form (.mod files) for the Intel Fortran compiler. They can be accessed by specifying the
The .mod module files were compiled with the Fortran compiler shown in Section 2.2 of the Installer's Note. Such module files are compiler-dependent, so if you wish to use the NAG example programs, or use the interface blocks in your own programs, when using a compiler that is incompatible with these modules, you will first need to recompile the interface blocks with your own compiler version. A recompiled set of interface blocks can be created in a separate directory (e.g. nag_interface_blocks_alt) using the supplied script command
[INSTALL_DIR]/scripts/nag_recompile_mods {int32,int64} nag_interface_blocks_altfrom 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.
The distributed example results are those obtained with the 32-bit 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.
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 32-bit integers and static linking to the self-contained libnag_nag.a library. These options 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 -helpto 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 e04uccwill 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 e04nrfwill 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.
For the 32-bit 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 64-bit 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).
This implementation of the NAG Library includes libraries for both 32-bit integers (located in directory [INSTALL_DIR]/lp64/lib) and 64-bit integers (located in directory [INSTALL_DIR]/ilp64/lib).
The NAG Library and documentation use parameterized types for floating-point 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 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 DGEESX DGEHRD DGEJSV DGELSS DGER DGESV DGESVDX DGESVJ DSBGVD DSPOSV DTRSV ZGEJSV ZGESVDX ZGESVJ ZHBGVD
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 (for 32-bit integer libraries)
= 9223372036854775807 (for 64-bit integer libraries) x02be[fc] (nag[f]_machine_decimal_digits) = 15
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 https://www.nag.co.uk/numeric/nl/nagdoc_27/.
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/index.htmlUse your web browser to navigate from here.
Advice on viewing and navigating the documentation can be found in https://www.nag.co.uk/numeric/nl/nagdoc_27/nlhtml/genint/naglibdoc.html.
In addition the following are provided:
https://www.nag.co.uk/content/nag-technical-support-service
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.
https://www.nag.co.uk/content/worldwide-contact-information
for worldwide contact details for the Numerical Algorithms Group.