Cross interpolation of high-dimensional arrays in tensor train format. This code implements the parallel version of the tensor cross interpolation algorithm, see

• Dmitry Savostyanov, Quasioptimality of maximum-volume cross interpolation of tensors, Linear Algebra and its Applications, 2014.
• Sergey Dolgov and Dmitry Savostyanov, Parallel cross interpolation for high-precision calculation of high-dimensional integrals, arXiv: 1903.11554, 2019.

If you benefit from this code, please consider citing our papers in your work.

0. Read the articles.
1. Download the code.
2. Edit the Makefile to match your system. Note: this software makes use of BLAS and LAPACK libraries. Please set the BLASLIB variable in the Makefile accordingly.
3. Compile the code by make command.
4. Run appropriate ./test_* command to execute an experiment. Use mpirun -np NN ./test_* to run the experiment in parallel on NN processors. Use OMP system variables (e.g. OMP_NUM_THREADS) to control the number of OMP threads.

• test_mc_ising.exe KIND INDEX Q REP: calculate Ising susceptibility integrals using Monte Carlo algorithm
• test_qmc_ising.exe KIND INDEX Q REP: calculate Ising susceptibility integrals using quasi Monte Carlo algorithm
• test_crs_ising.exe KIND INDEX N RANK PIV : calculate Ising susceptibility integrals using tensor cross interpolation algorithm

• KIND: which integral to calculate, 'C', 'D' or 'E'
• INDEX: the sub-index of the integral, e.g. 6 to calculate C_6, etc.
• Q: the size of the MC sampling or QMC lattice, the number of evaluations will be 2^Q
• REP: the number of repeats for MC or QMC to estimate standard deviation (1<=REP)
• N: the one-dimensional quadrature size for the integration
• RANK: the maximum TT-rank in the cross interpolation method (1<=RANK)
• PIV: the number of rook pivoting searches in the cross interpolation method (0<=PIV)

Double precision calculations are accurate to approximately 15 decimal digits. To reach higher precision, you can compile the code enabling -fdefault-real-8 or -fdefault-real-16 in the compiler. Note that you also will need to obtain and link quadruple-precision BLAS/LAPACK libraries. Use test_crs_ising.exe with the same parameters, adjusting the quadrature size N to the desired accuracy.

For those who desires even higher precision, we provide a multiple-precision version of the code. It relies on the MPFUN2015 library written by David H. Bailey. The source code of MPFUN2015 is included in the mpfun-mpfr-v08 directory. The code uses GNU MPFR Library https://www.mpfr.org/. Please install it on your system and set the MPFLIB variable in the Makefile accordingly. Compile the multi-precision version with make test_mpf_ising.exe, noting the following:

• No special BLAS/LAPACK libraries are needed for this experiment --- all multiple-precision BLAS/LAPACK subroutines are provided in the mpblas.f90 file. You will still need a standard BLAS/LAPACK library to link and run the code.
• The -fdefault-real-XX flag has to be disabled.
• The precision is adjusted via the parameter mpipl in ./mpfun-mpfr-v08/mpfunf.f90. After compiling, run the test_mpf_ising.exe to calculate Ising susceptibility integrals using tensor cross interpolation algorithm in multiple precision. Use the same parameters as in test_crs_ising.exe, adjusting the quadrature size N to reach the desired accuracy.