# PRIMME: PReconditioned Iterative MultiMethod Eigensolver¶

PRIMME, pronounced as *prime*, computes
a few eigenvalues and their corresponding eigenvectors of a real symmetric or complex Hermitian matrix.
It can also compute singular values and vectors of a square or rectangular matrix.
It can find largest, smallest, or interior singular/eigenvalues and can use preconditioning to accelerate convergence.
It is especially optimized for large, difficult problems, and can be a useful tool for both non-experts and experts.
PRIMME is written in C99, but complete interfaces are provided for Fortran 77, MATLAB, Python, and R.

## Incompatibilities¶

From PRIMME 2.0 to 2.1:

- Added members
`monitorFun`

and`monitor`

to`primme_params`

. - Added members
`monitorFun`

and`monitor`

to`primme_svds_params`

. - Renamed
`PRIMME_SUBSPACE_ITERATION`

as`PRIMME_STEEPEST_DESCENT`

.

From PRIMME 1.x to 2.0:

- Prototype of callbacks has changed:
`matrixMatvec`

,`applyPreconditioner`

,`massMatrixMatvec`

and`globalSumReal`

. - The next parameters are
`PRIMME_INT`

:`n`

,`nLocal`

,`maxMatvecs`

,`iseed`

,`numOuterIterations`

,`numRestarts`

,`numMatvecs`

and`numMatvecs`

; use the macro`PRIMME_INT_P`

to print the values. - Rename the values of the enum
`primme_preset_method`

. - Rename
`primme_Free`

to`primme_free()`

. - Integer parameters in Fortran functions are of the same size as
`PRIMME_INT`

, which is`integer*8`

by default. - Extra parameter in many Fortran functions to return the error code.
- Removed
`primme_display_stats_f77`

.

## Changelog¶

Changes in PRIMME 2.1 (released on April 4, 2017):

- Improve robustness by broadcasting the result of critical LAPACK operations instead of replicating them on every process; this is useful when using a threaded BLAS/LAPACK or when some parallel processes may run on different architectures or libraries.
- New stopping criteria in QMR that improve performance for interior problems.
- MATLAB interface reimplementation with support for singular value
problems,
`primme_svds()`

, with double and single precision, and compatible with Octave. - R interface
- Proper reporting of convergence history for singular value solvers.

Changes in PRIMME 2.0 (released on September 19, 2016):

- Changed license to BSD 3-clause.
- New support for singular value problems; see
`dprimme_svds()`

. - New support for
`float`

and`complex float`

arithmetic. - Support for problem dimensions larger than 2^31, without requiring BLAS and LAPACK compiled with 64-bits integers.
- Improve robustness and performance for interior problems; implemented advanced refined and harmonic-Ritz extractions.
- Python interface compatible with NumPy and SciPy Library.
- Added parameter to indicate the leading dimension of the input/output matrices and to
return an error code in callbacks
`matrixMatvec`

,`applyPreconditioner`

,`massMatrixMatvec`

and`globalSumReal`

. - Changed to type
`PRIMME_INT`

the options`n`

,`nLocal`

,`maxMatvecs`

and`iseed`

, and the stats counters`numOuterIterations`

,`numRestarts`

,`numMatvecs`

,`numPreconds`

. Also changed`realWorkSize`

to`size_t`

. Fortran interface functions will expect an`interger`

of size compatible with`PRIMME_INT`

for all parameters with integer type:`int`

,`PRIMME_INT`

and`size_t`

; see also parameter`value`

in functions`primmetop_set_member_f77()`

,`primmetop_get_member_f77()`

,`primme_set_member_f77()`

and`primme_get_member_f77()`

. - Added parameter to return an error code in Fortran interface functions:
`primmetop_set_member_f77()`

,`primmetop_get_member_f77()`

,`primme_set_member_f77()`

and`primme_get_member_f77()`

. - Added leading dimension for
`evecs`

`ldevecs`

and preferred leading dimension for the operators`ldOPs`

, such as`matrixMatvec`

. - Optional user-defined convergence function,
`convTestFun`

. - Prefixed methods with
`PRIMME_`

. Rename Fortran constants from`PRIMMEF77_`

to`PRIMME_`

. - Removed
`primme_display_stats_f77`

.

Changes in PRIMME 1.2.2 (released on October 13, 2015):

- Fixed wrong symbols in
`libdprimme.a`

and`libzprimme.a`

. `primme_set_method()`

sets`PRIMME_JDQMR`

instead of`PRIMME_JDQMR_ETol`

for preset methods`PRIMME_DEFAULT_MIN_TIME`

and`PRIMME_DYNAMIC`

when seeking interior values.- Fixed compilation of driver with a PETSc installation without HYPRE.
- Included the content of the environment variable
`INCLUDE`

for compiling the driver.

Changes in PRIMME 1.2.1 (released on September 7, 2015):

- Added MATLAB interface to full PRIMME functionality.
- Support for BLAS/LAPACK with 64bits integers (
`-DPRIMME_BLASINT_SIZE=64`

). - Simplified configuration of Make_flags and Make_links (removed
`TOP`

variable and replaced defines`NUM_SUM`

and`NUM_IBM`

by`F77UNDERSCORE`

). - Replaced directories
`DTEST`

and`ZTEST`

by`TEST`

, that has: - Fixed a few minor bugs and improved documentation (especially the F77 interface).
- Using Sphinx to manage documentation.

Changes in PRIMME 1.2 (released on December 21, 2014):

A Fortran compiler is no longer required for building the PRIMME library. Fortran programs can still be linked to PRIMME’s F77 interface.

Fixed some uncommon issues with the F77 interface.

PRIMME can be called now multiple times from the same program.

Performance improvements in the QMR inner solver, especially for complex arithmetic.

Fixed a couple of bugs with the locking functionality.

- In certain extreme cases where all eigenvalues of a matrix were needed.
- The order of selecting interior eigenvalues.

The above fixes have improved robustness and performance.

PRIMME now assigns unique random seeds per parallel process for up to 4096^3 (140 trillion) processes.

For the

`PRIMME_DYNAMIC`

method, fixed issues with initialization and synchronization decisions across multiple processes.Fixed uncommon library interface bugs, coordinated better setting the method and the user setting of parameters, and improved the interface in the sample programs and makefiles.

Other performance and documentation improvements.

## License Information¶

PRIMME is licensed under the 3-clause license BSD.
Python and MATLAB interfaces have BSD-compatible licenses.
Source code under `tests`

is compatible with LGPLv3.
Details can be taken from `COPYING.txt`

:

```
Copyright (c) 2017, College of William & Mary
All rights reserved.
```

## Citing the code¶

Please cite (BibTeX):

[r1] | A. Stathopoulos and J. R. McCombs PRIMME: PReconditioned Iterative
MultiMethod Eigensolver: Methods and software description, ACM
Transaction on Mathematical Software Vol. 37, No. 2, (2010),
21:1-21:30. |

[r6] | L. Wu, E. Romero and A. Stathopoulos, PRIMME_SVDS: A High-Performance
Preconditioned SVD Solver for Accurate Large-Scale Computations,
arXiv:1607.01404 |

More information on the algorithms and research that led to this software can be found in the rest of the papers. The work has been supported by a number of grants from the National Science Foundation.

[r2] | A. Stathopoulos, Nearly optimal preconditioned methods for Hermitian
eigenproblems under limited memory. Part I: Seeking one eigenvalue, SIAM
J. Sci. Comput., Vol. 29, No. 2, (2007), 481–514. |

[r3] | A. Stathopoulos and J. R. McCombs, Nearly optimal preconditioned
methods for Hermitian eigenproblems under limited memory. Part II:
Seeking many eigenvalues, SIAM J. Sci. Comput., Vol. 29, No. 5, (2007),
2162-2188. |

[r4] | J. R. McCombs and A. Stathopoulos, Iterative Validation of
Eigensolvers: A Scheme for Improving the Reliability of Hermitian
Eigenvalue Solvers, SIAM J. Sci. Comput., Vol. 28, No. 6, (2006),
2337-2358. |

[r5] | A. Stathopoulos, Locking issues for finding a large number of eigenvectors
of Hermitian matrices, Tech Report: WM-CS-2005-03, July, 2005. |

[r7] | L. Wu and A. Stathopoulos, A Preconditioned Hybrid SVD Method for Computing
Accurately Singular Triplets of Large Matrices, SIAM J. Sci. Comput. 37-5(2015),
pp. S365-S388. |

## Contact Information¶

For reporting bugs or questions about functionality contact Andreas Stathopoulos by email, andreas at cs.wm.edu. See further information in the webpage http://www.cs.wm.edu/~andreas/software and on github.

## Directory Structure¶

The next directories and files should be available:

`COPYING.txt`

, license;`Make_flags`

, flags to be used by makefiles to compile library and tests;`Link_flags`

, flags needed in making and linking the test programs;`include/`

, directory with headers files;`src/`

, directory with the source code for`libprimme`

:`include/`

, common headers;`eigs/`

, eigenvalue interface and implementation;`svds/`

, singular value interface and implementation;`tools/`

, tools used to generated some headers;

`Matlab/`

, MATLAB interface;`Python/`

, Python interface;`examples/`

, sample programs in C, C++ and F77, both sequential and parallel;`tests/`

, drivers for testing purpose and test cases;`lib/libprimme.a`

, the PRIMME library (to be made);`makefile`

main make file;`readme.txt`

text version of the documentation;`doc/`

directory with the HTML and PDF versions of the documentation.

## Making and Linking¶

`Make_flags`

has the flags and compilers used to make `libprimme.a`

:

- CC, compiler program such as
`gcc`

,`clang`

or`icc`

. - CFLAGS, compiler options such as
`-g`

or`-O3`

and macro definitions - like the ones described next.

- CFLAGS, compiler options such as

Compiler flags for the BLAS and LAPACK libraries:

`-DF77UNDERSCORE`

, if Fortran appends an underscore to function names (usually it does).`-DPRIMME_BLASINT_SIZE=64`

, if the library integers are 64-bit integer (`kind=8`

) type, aka ILP64 interface; usually integers are 32-bits even in 64-bit architectures (aka LP64 interface).

By default PRIMME sets the integer type for matrix dimensions and counters (`PRIMME_INT`

)
to 64 bits integer `int64_t`

. This can be changed by setting the macro `PRIMME_INT_SIZE`

to one of the following values:

`0`

: use the regular`int`

of your compiler.`32`

: use C99`int32_t`

.`64`

: use C99`int64_t`

.

Note

When `-DPRIMME_BLASINT_SIZE=64`

is set the code uses the type `int64_t`

supported by the C99 standard. In case the compiler doesn’t honor the
standard, you can set the corresponding type name supported, for instance
`-DPRIMME_BLASINT_SIZE=__int64`

.

After customizing `Make_flags`

, type this to generate `libprimme.a`

:

```
make lib
```

Making can be also done at the command line:

```
make lib CC=clang CFLAGS='-O3'
```

`Link_flags`

has the flags for linking with external libraries and making the executables
located in `examples`

and `tests`

:

- LDFLAGS, linker flags such as
`-framework Accelerate`

. - LIBS, flags to link with libraries (BLAS and LAPACK are required), such as
`-lprimme -llapack -lblas -lgfortran -lm`

.

After that, type this to compile and execute a simple test:

```
$ make test
...
Test passed!
...
Test passed!
```

In case of linking problems check flags in LDFLAGS and LIBS and consider
to add/remove `-DF77UNDERSCORE`

from CFLAGS. If the execution fails consider
to add/remove `-DPRIMME_BLASINT_SIZE=64`

from CFLAGS.

Full description of actions that make can take:

- make lib, builds the static library
`libprimme.a`

. - make solib, builds the shared library
`libprimme.so`

. - make matlab, builds libprimme.a compatible with MATLAB and the MATLAB module.
- make octave, builds libprimme.a and the Octave module.
- make python, builds libprimme.a and the Python module.
- make python_install, install the Python module.
- make R_install, builds and installs the R package.
- make test, build and execute simple examples.
- make clean, removes all
`*.o`

,`a.out`

, and core files from`src`

.

### Considerations using an IDE¶

PRIMME can be built in other environments such as Anjuta, Eclipse, KDevelop, Qt Creator, Visual Studio and XCode. To build the PRIMME library do the following:

- Create a new project and include the source files under the directory
`src`

. - Add the directories
`include`

and`src/include`

as include directories.

To build an example code using PRIMME make sure:

## Tested Systems¶

PRIMME is primary developed with GNU gcc, g++ and gfortran (versions 4.8 and later). Many users have reported builds on several other platforms/compilers:

- SUSE 13.1 & 13.2
- CentOS 6.6
- Ubuntu 14.04
- MacOS X 10.9 & 10.10
- Cygwin & MinGW
- Cray XC30
- SunOS 5.9, quad processor Sun-Fire-280R, and several other UltraSparcs
- AIX 5.2 IBM SP POWER 3+, 16-way SMP, 375 MHz nodes (seaborg at nersc.gov)

## Main Contributors¶

- James R. McCombs
- Eloy Romero Alcalde
- Andreas Stathopoulos
- Lingfei Wu