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1 MBLAS MLAPACK; BLAS/LAPACK February 23, 2009

2 MPACK(MBLAS/MLAPACK)

3 ( ) (2007 ) ( )

4 BLAS/LAPACK BLAS (Basic Linear Algebra Subprograms) MBLAS LAPACK (Linear Algebra PACKage) MLAPACK :MBLAS 76 MLAPACK 40 (700 ) LU

5 ; ; ; ;

6 [Journal of Chemical Physics, 114, (2001)]

7 SDPA/SDPARA 7-8

8 brute force solution: : double :

9 SDPA-GMP:GMP SDPA SDPA GMP BLAS/LAPACK GMP(C++) SDPA-GMP Journal of Chemical Physics 128, 16, (2008). :GNU General Public License

10 MPACK(MBLAS/MLAPACK) SDPA-GMP SDPA-GMP

11 MPACK (MBLAS/MLAPACK) FORTRAN C++ double GMP(GNU Multiple Precision Arithmetic Library) GNU Lesser General Public License

12 GMP GMP(GNU Multiple Precision Arithmetic Library) the fastest bignum library on the planet! ( ) C++ (double) IEEE754 ( )

13 BLAS? The BLAS (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. The Level 1 BLAS perform scalar, vector and vector-vector operations, the Level 2 BLAS perform matrix-vector operations, and the Level 3 BLAS perform matrix-matrix operations. [1] Level 1 : t x x, t x y, etc. Level 2 : Ax = b, t Ax = b, etc. Level 3 : αab + βc etc. [1]

14 MBLAS Multiple precision arithmetic BLAS; BLAS C++ GMP BLAS double mpf class

15 MBLAS Prefix float, double R eal, complex, double complex C complex. saxpy, daxpy Raxpy caxpy, zaxpy Caxpy dgemm, sgemm Rgemm cgemm, zgemm Cgemm

16 MBLAS SDPA-GMP sdpa linear.cpp if (scalar==null) { scalar = &MONE; // scalar is local variable } // The Point is the first argument is "Transpose". Rgemm("Transpose","NoTranspose",retMat.nRow,retMat.nCol,aMat.nCol, *scalar,amat.de_ele,amat.ncol,bmat.de_ele,bmat.nrow, 0.0,retMat.de_ele,retMat.nRow); break; case DenseMatrix::COMPLETION: rerror("no support for COMPLETION"); break; } return _SUCCESS;

17 MBLAS Netlib ( ; GO TO 2007 FORTRAN77 ; MLAPACK

18 MBLAS MBLAS BLAS for (int k = MIN_K; k < MAX_K; k++) { for (int n = MIN_N; n < MAX_N; n++) { for (int m = MIN_M; m < MAX_M; m++) {... for (int lda = minlda; lda < MAX_LDA; lda++) { for (int ldb = minldb; ldb < MAX_LDB; ldb++) { for (int ldc = max(1, m); ldc < MAX_LDC; ldc++) {... Rgemm(transa, transb, m, n, k, alpha, A, lda, B, ldb, beta, C, ldc); dgemm_f77(transa, transb, &m, &n, &k, &alphad, Ad, &lda, Bd, &ldb, &betad, Cd, &ldc); diff = vec_diff(c, Cd, MAT_A(ldc, n), 1); if (fabs(diff) > EPSILON) { printf("#error %lf!!\n", diff); errorflag = TRUE; }

19 LAPACK? LAPACK provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. The associated matrix factorizations (LU, Cholesky, QR, SVD, Schur, generalized Schur) are also provided, as are related computations such as reordering of the Schur factorizations and estimating condition numbers. Dense and banded matrices are handled, but not general sparse matrices. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision. ( )

20 MLAPACK 700 Netlib dsyev.f f2c FORTRAN C C sed BLAS : dsyev Rsyev /day 1 /day emacs

21 MLAPACK IEEE754 LAPACK 3.2 IEEE754 LAPACK 3.0

22 MLAPACK ;37 SDPA-GMP7.1.2 MLAPACK/MBLAS Rsyev.cpp, Rsterf.cpp: ok Rtrtri.cpp: ok Rpotrf.cpp: ok LAPACK ilaenv FORTRAN

23 / SDPA-GMP QD (C++/Fortran-90 double-double and quad-double package)

24 QD (C++/Fortran-90 double-double and quad-double package) BLAS/LAPACK BLAS/LAPACK

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