インテル(R) 数値演算ライブラリ(MKL) リファレンス・マニュアル · インテル® 数値演算ライブラリ－リファレンス・マニュアル－目次

1 ................................................................................... 1-1

BLAS ............................................................................................. 1-2 BLAS .............................................................................. 1-2 ......................................................................................... 1-2LAPACK ........................................................................................ 1-2VML ...................................................................................................... 1-3 ................................................................................................... 1-3 ........................................................................................................... 1-3 ................................................................................. 1-4

..................................................................................................... 1-4 ............................................................................................ 1-5 ................................................................................................... 1-5 ................................................................................................... 1-6 ............................................................................ 1-6 .............................................................................................. 1-6

....................................................................................................... 1-8

2 BLAS BLAS .............................................................................................. 2-2 ................................................................................... 2-3BLAS 1 ..................................................................... 2-4

?asum ..................................................................................................... 2-5

i

?axpy ....................................................................................................... 2-6?copy ....................................................................................................... 2-7

?dot ......................................................................................................... 2-8?dotc ....................................................................................................... 2-9?dotu ..................................................................................................... 2-10?nrm2 .................................................................................................... 2-11?rot ........................................................................................................ 2-12?rotg ...................................................................................................... 2-14?rotm ..................................................................................................... 2-15?rotmg ................................................................................................... 2-17?scal ...................................................................................................... 2-18?swap .................................................................................................... 2-20i?amax ................................................................................................... 2-21i?amin .................................................................................................... 2-22

BLAS 2 .............................................................................. 2-23?gbmv ................................................................................................... 2-24?gemv ................................................................................................... 2-27?ger ....................................................................................................... 2-30?gerc ..................................................................................................... 2-31?geru ..................................................................................................... 2-33?hbmv ................................................................................................... 2-35?hemv ................................................................................................... 2-38?her ....................................................................................................... 2-40?her2 ..................................................................................................... 2-42?hpmv ................................................................................................... 2-44?hpr ....................................................................................................... 2-47?hpr2 ..................................................................................................... 2-49?sbmv .................................................................................................... 2-51?spmv .................................................................................................... 2-54?spr ....................................................................................................... 2-56?spr2 ..................................................................................................... 2-58?symv .................................................................................................... 2-60?syr ....................................................................................................... 2-62?syr2 ..................................................................................................... 2-64

ii

?tbmv .................................................................................................... 2-66?tbsv ...................................................................................................... 2-69?tpmv .................................................................................................... 2-72

?tpsv ...................................................................................................... 2-75?trmv ..................................................................................................... 2-77?trsv ...................................................................................................... 2-79

BLAS 3 .............................................................................. 2-82MKL ..................... 2-82

?gemm .................................................................................................. 2-83?hemm .................................................................................................. 2-86?herk ..................................................................................................... 2-89?her2k ................................................................................................... 2-92?symm ................................................................................................... 2-96?syrk ................................................................................................... 2-100?syr2k .................................................................................................. 2-103?trmm .................................................................................................. 2-107?trsm ................................................................................................... 2-110

BLAS ................................................................ 2-114 BLAS .................................................... 2-114 BLAS ........................................................... 2-115 BLAS ......................................... 2-115 BLAS .................................. 2-116

?axpyi .................................................................................................. 2-116?doti .................................................................................................... 2-118?dotci ................................................................................................... 2-119?dotui .................................................................................................. 2-120?gthr .................................................................................................... 2-121?gthrz .................................................................................................. 2-122?roti ..................................................................................................... 2-123?sctr .................................................................................................... 2-124

3 1 FFT ......................................................................................................... 3-1 ............................................................................................ 3-2 ......................................................................................... 3-2

iii

1 FFT ............................................................... 3-3cfft1d/zfft1d .............................................................................................. 3-4cfft1dc/zfft1dc .......................................................................................... 3-5

1 FFT ................................................................... 3-6scfft1d/dzfft1d .......................................................................................... 3-8scfft1dc/dzfft1dc .................................................................................... 3-10

1 FFT ................................................................. 3-11csfft1d/zdfft1d ........................................................................................ 3-13csfft1dc/zdfft1dc .................................................................................... 3-15

2 FFT ....................................................................................................... 3-17 2 FFT ............................................................. 3-18

cfft2d/zfft2d ............................................................................................ 3-19cfft2dc/zfft2dc ........................................................................................ 3-20

2 FFT ................................................................. 3-21scfft2d/dzfft2d ........................................................................................ 3-22scfft2dc/dzfft2dc .................................................................................... 3-24

2 FFT ................................................................. 3-27csfft2d/zdfft2d ........................................................................................ 3-28csfft2dc/zdfft2dc .................................................................................... 3-29

4 LAPACK : 1 .......................................................................................... 4-2 ................................................................................... 4-3 ......................................................................................................... 4-3 ......................................................................................................... 4-4 ..................................................................................................... 4-5 ........................................................ 4-7

?getrf ....................................................................................................... 4-7?gbtrf ..................................................................................................... 4-10?gttrf ...................................................................................................... 4-12?potrf ..................................................................................................... 4-14?pptrf ..................................................................................................... 4-16?pbtrf ..................................................................................................... 4-18?pttrf ...................................................................................................... 4-20?sytrf ..................................................................................................... 4-22

iv

?hetrf ..................................................................................................... 4-25?sptrf ..................................................................................................... 4-28?hptrf ..................................................................................................... 4-31

1 ...................................................... 4-33?getrs .................................................................................................... 4-34?gbtrs .................................................................................................... 4-36?gttrs ..................................................................................................... 4-38?potrs .................................................................................................... 4-41?pptrs .................................................................................................... 4-43?pbtrs .................................................................................................... 4-45?pttrs ..................................................................................................... 4-47?sytrs ..................................................................................................... 4-49?hetrs .................................................................................................... 4-51?sptrs .................................................................................................... 4-53?hptrs .................................................................................................... 4-55?trtrs ...................................................................................................... 4-57?tptrs ..................................................................................................... 4-59?tbtrs ..................................................................................................... 4-61

............................................................. 4-63?gecon .................................................................................................. 4-63?gbcon .................................................................................................. 4-65?gtcon ................................................................................................... 4-67?pocon .................................................................................................. 4-70?ppcon .................................................................................................. 4-72?pbcon .................................................................................................. 4-74?ptcon ................................................................................................... 4-76?sycon ................................................................................................... 4-78?hecon .................................................................................................. 4-80?spcon ................................................................................................... 4-82?hpcon .................................................................................................. 4-84?trcon .................................................................................................... 4-86?tpcon ................................................................................................... 4-88?tbcon ................................................................................................... 4-90

..................................................................... 4-92?gerfs .................................................................................................... 4-92

v

?gbrfs .................................................................................................... 4-95?gtrfs ..................................................................................................... 4-98?porfs .................................................................................................. 4-101

?pprfs .................................................................................................. 4-104?pbrfs .................................................................................................. 4-107?ptrfs ................................................................................................... 4-110?syrfs ................................................................................................... 4-113?herfs .................................................................................................. 4-116?sprfs .................................................................................................. 4-119?hprfs .................................................................................................. 4-122?trrfs .................................................................................................... 4-124?tprfs ................................................................................................... 4-127?tbrfs ................................................................................................... 4-130

........................................................... 4-133?getri ................................................................................................... 4-133?potri ................................................................................................... 4-135?pptri ................................................................................................... 4-137?sytri .................................................................................................... 4-139?hetri ................................................................................................... 4-141?sptri ................................................................................................... 4-143?hptri ................................................................................................... 4-145?trtri ..................................................................................................... 4-147?tptri .................................................................................................... 4-148

............................................................................. 4-150?geequ ................................................................................................ 4-150?gbequ ................................................................................................ 4-153?poequ ................................................................................................ 4-155?ppequ ................................................................................................ 4-157?pbequ ................................................................................................ 4-159

...................................................................................... 4-161?gesv ................................................................................................... 4-162?gesvx ................................................................................................. 4-163?gbsv ................................................................................................... 4-170?gbsvx ................................................................................................. 4-172?gtsv .................................................................................................... 4-179

vi

?gtsvx .................................................................................................. 4-181?posv ................................................................................................... 4-186?posvx ................................................................................................. 4-188

?ppsv ................................................................................................... 4-193?ppsvx ................................................................................................. 4-195?pbsv ................................................................................................... 4-200?pbsvx ................................................................................................. 4-202?ptsv .................................................................................................... 4-207?ptsvx .................................................................................................. 4-209?sysv ................................................................................................... 4-212?sysvx ................................................................................................. 4-215?hesvx ................................................................................................. 4-220?hesv ................................................................................................... 4-224?spsv ................................................................................................... 4-227?spsvx ................................................................................................. 4-230?hpsvx ................................................................................................. 4-235?hpsv ................................................................................................... 4-239

5 LAPACK : .............................................................................................. 5-4 ................................................................................... 5-5 ............................................................................................................ 5-5 ..................................................................................................... 5-6 ................................................................................................ 5-6

?geqrf ...................................................................................................... 5-8?geqpf ................................................................................................... 5-11?geqp3 .................................................................................................. 5-14?orgqr .................................................................................................... 5-17?ormqr ................................................................................................... 5-19?ungqr ................................................................................................... 5-21?unmqr .................................................................................................. 5-23?gelqf .................................................................................................... 5-25?orglq .................................................................................................... 5-28?ormlq ................................................................................................... 5-30?unglq ................................................................................................... 5-32

vii

?unmlq .................................................................................................. 5-34?geqlf .................................................................................................... 5-36?orgql .................................................................................................... 5-38

?ungql ................................................................................................... 5-40?ormql ................................................................................................... 5-42?unmql .................................................................................................. 5-45?gerqf .................................................................................................... 5-48?orgrq .................................................................................................... 5-50?ungrq ................................................................................................... 5-52?ormrq ................................................................................................... 5-54?unmrq .................................................................................................. 5-57?tzrzf ..................................................................................................... 5-60?ormrz ................................................................................................... 5-62?unmrz .................................................................................................. 5-65?ggqrf .................................................................................................... 5-68?ggrqf .................................................................................................... 5-71

................................................................................................. 5-74?gebrd ................................................................................................... 5-76?gbbrd ................................................................................................... 5-79?orgbr .................................................................................................... 5-82?ormbr ................................................................................................... 5-85?ungbr ................................................................................................... 5-88?unmbr .................................................................................................. 5-91?bdsqr ................................................................................................... 5-94?bdsdc ................................................................................................... 5-98

........................................................................................ 5-101?sytrd .................................................................................................. 5-105?orgtr ................................................................................................... 5-107?ormtr .................................................................................................. 5-109?hetrd .................................................................................................. 5-111?ungtr .................................................................................................. 5-113?unmtr ................................................................................................. 5-115?sptrd .................................................................................................. 5-117?opgtr .................................................................................................. 5-119?opmtr ................................................................................................. 5-120

viii

?hptrd .................................................................................................. 5-122?upgtr .................................................................................................. 5-124?upmtr ................................................................................................. 5-125

?sbtrd .................................................................................................. 5-128?hbtrd .................................................................................................. 5-130?sterf ................................................................................................... 5-132?steqr .................................................................................................. 5-134?stedc .................................................................................................. 5-137?stegr .................................................................................................. 5-141?pteqr .................................................................................................. 5-146?stebz .................................................................................................. 5-149?stein ................................................................................................... 5-152?disna .................................................................................................. 5-154

................................................................................. 5-157?sygst .................................................................................................. 5-158?hegst ................................................................................................. 5-160?spgst .................................................................................................. 5-162?hpgst ................................................................................................. 5-164?sbgst .................................................................................................. 5-166?hbgst ................................................................................................. 5-169?pbstf .................................................................................................. 5-172

..................................................................................... 5-174?gehrd ................................................................................................. 5-178?orghr .................................................................................................. 5-180?ormhr ................................................................................................. 5-182?unghr ................................................................................................. 5-185?unmhr ................................................................................................ 5-187?gebal ................................................................................................. 5-190?gebak ................................................................................................ 5-193?hseqr ................................................................................................. 5-195?hsein .................................................................................................. 5-199?trevc .................................................................................................. 5-205?trsna .................................................................................................. 5-210?trexc .................................................................................................. 5-215?trsen .................................................................................................. 5-217

ix

?trsyl .................................................................................................... 5-222 ............................................................................. 5-225

?gghrd ................................................................................................. 5-226

?ggbal ................................................................................................. 5-230?ggbak ................................................................................................ 5-233?hgeqz ................................................................................................ 5-235?tgevc .................................................................................................. 5-242?tgexc .................................................................................................. 5-247?tgsen ................................................................................................. 5-250?tgsyl ................................................................................................... 5-256?tgsna ................................................................................................. 5-261

........................................................................................ 5-266?ggsvp ................................................................................................. 5-267?tgsja ................................................................................................... 5-271

...................................................................................... 5-278 (LLS) .......................................................................... 5-278

?gels .................................................................................................... 5-279?gelsy .................................................................................................. 5-282?gelss .................................................................................................. 5-286?gelsd .................................................................................................. 5-289

LLS ........................................................................................... 5-293?gglse .................................................................................................. 5-293?ggglm ................................................................................................ 5-296

........................................................................................ 5-298?syev ................................................................................................... 5-299?heev ................................................................................................... 5-301?syevd ................................................................................................. 5-303?heevd ................................................................................................. 5-306?syevx ................................................................................................. 5-309?heevx ................................................................................................. 5-313?syevr .................................................................................................. 5-317?heevr ................................................................................................. 5-322?spev ................................................................................................... 5-327?hpev ................................................................................................... 5-329?spevd ................................................................................................. 5-331

x

?hpevd ................................................................................................. 5-334?spevx ................................................................................................. 5-338?hpevx ................................................................................................. 5-342

?sbev ................................................................................................... 5-346?hbev .................................................................................................. 5-348?sbevd ................................................................................................. 5-350?hbevd ................................................................................................ 5-353?sbevx ................................................................................................. 5-357?hbevx ................................................................................................. 5-361?stev .................................................................................................... 5-365?stevd .................................................................................................. 5-367?stevx .................................................................................................. 5-370?stevr .................................................................................................. 5-374

..................................................................................... 5-379?gees .................................................................................................. 5-379?geesx ................................................................................................. 5-384?geev .................................................................................................. 5-390?geevx ................................................................................................. 5-394

............................................................................................... 5-400?gesvd ................................................................................................. 5-400?gesdd ................................................................................................ 5-405?ggsvd ................................................................................................. 5-409

................................................................................. 5-415?sygv ................................................................................................... 5-416?hegv .................................................................................................. 5-419?sygvd ................................................................................................. 5-422?hegvd ................................................................................................ 5-425?sygvx ................................................................................................. 5-429?hegvx ................................................................................................. 5-434?spgv ................................................................................................... 5-439?hpgv .................................................................................................. 5-442?spgvd ................................................................................................. 5-445?hpgvd ................................................................................................ 5-448?spgvx ................................................................................................. 5-452?hpgvx ................................................................................................. 5-456

xi

?sbgv ................................................................................................... 5-460?hbgv .................................................................................................. 5-463?sbgvd ................................................................................................. 5-466

?hbgvd ................................................................................................ 5-469?sbgvx ................................................................................................. 5-473?hbgvx ................................................................................................. 5-478

............................................................................. 5-483?gges .................................................................................................. 5-483?ggesx ................................................................................................. 5-490?ggev ................................................................................................... 5-498?ggevx ................................................................................................. 5-503

........................................................................................................ 5-510

6 ............................................................................ 6-2 ................................................................................................. 6-2 ................................................................................. 6-3

............................................................................ 6-6 ......................................................................................................... 6-6VML .................................................................................................... 6-8

Inv .......................................................................................................... 6-10Div ......................................................................................................... 6-11Sqrt ....................................................................................................... 6-12InvSqrt ................................................................................................... 6-13Cbrt ....................................................................................................... 6-15InvCbrt ................................................................................................... 6-16Pow ........................................................................................................ 6-17Exp ........................................................................................................ 6-18Ln .......................................................................................................... 6-20Log10 .................................................................................................... 6-21Cos ........................................................................................................ 6-22Sin ......................................................................................................... 6-23SinCos ................................................................................................... 6-24Tan ......................................................................................................... 6-25Acos ...................................................................................................... 6-26

xii

Asin ....................................................................................................... 6-27Atan ....................................................................................................... 6-28Atan2 ..................................................................................................... 6-29

Cosh ...................................................................................................... 6-30Sinh ....................................................................................................... 6-31Tanh ...................................................................................................... 6-33Acosh .................................................................................................... 6-34Asinh ..................................................................................................... 6-35Atanh ..................................................................................................... 6-36

VML Pack/Unpack ................................................................................... 6-37Pack ...................................................................................................... 6-37Unpack .................................................................................................. 6-39

VML ........................................................................................... 6-42SetMode ................................................................................................ 6-42GetMode ............................................................................................... 6-45SetErrStatus .......................................................................................... 6-46GetErrStatus ......................................................................................... 6-48ClearErrStatus ....................................................................................... 6-48SetErrorCallBack ................................................................................... 6-49GetErrorCallBack .................................................................................. 6-51ClearErrorCallBack ............................................................................... 6-52

A BLAS ....................................................................................... A-1VML ......................................................................................... A-3 .............................................................................................. A-4

B

C BLAS CBLASCBLAS ................................................................................................... C-1

..................................................................................................... C-2 1 CBLAS ............................................................................................ C-3 2 CBLAS ............................................................................................ C-6 3 CBLAS .......................................................................................... C-14

xiii

: 630813J-011

Web : http://www.intel.co.jp/jp/developer/http://www.intel.co.jp/jp/developer/design/develop.htm

-001 1994 11

-002 crotg zrotg?her2k?symm?syrk ?syr2k

1995 5

-003 Intel BLAS Library for the Pentium Processor Reference Manual?rotm ?rotmg C

1996 1

-004 2.0 1 2BLAS 3

1996 11

-005 2 FFT1 FFT 2 FFT C

1997 8

-006 2.1 2 BLAS

1998 1

-007 3.0LAPACK ( 4 5 ) CBLAS ( C)MKL 3.0 HTML

1999 1

-008 3.2FFT 4 5 LAPACK NAG

1999 6

-009 LAPACK 5 1999 11

-010 4.0VML 6

2000 6

-011 5.1LAPACK

2001 4

IntelIntel Pentium Intel Corporation

*

1994-2001 Intel Corporation. Chapters 4 and 5 include derivative work portions that have been copyrighted 1991, 1992, and 1998 by The Numerical Algorithms Group, Ltd.

1-1

1 (MKL) FortranFortran CMKL MKL

BLAS (Basic Linear Algebra Subprograms) - -

BLAS( ) (Fortran C ) LAPACK

LAPACK

VML (Vector Mathematical Library) (Fortran C )

1 BLAS

BLAS

BLAS 1

BLAS 2 - 1 2 -

BLAS 3 - k -

BLAS

BLAS ( )BLAS 1 BLAS

(FFT) (PDE)FFTBLAS

LAPACK

LAPACK

( 4 )

( 5 )

1-2

1 VML

VML ( 6 ) ( )

dgemm() -

dgemm

SIMD

MKL (SMP)SMP

1MKL

1-3

1 FFT BLAS 3BLAS 3( 2BLAS 2 )

1 LAPACK QR

BLAS 3 LAPACK

(MKL) FortranMKLFortran VML C

4 ( )

1

1-4

1

a REAL(saxpy )DOUBLE PRECISION(daxpy )

1

2 BLAS BLASBLAS BLAS

3 (FFT)

4 LAPACK : 1 ( ) LAPACK

5 LAPACK : LAPACK

1-5

1 6

VML

A BLAS VML ( )

B BLAS

C BLAS CBLASBLAS C

(cungqr/zungqr ?ungqr )

(?)

?swap -?swap4 sswapdswapcswap zswap

UPPERCASE COURIER FortranCHARACTER*1

lowercase courier a(k+i,j) = matrix(i,j) Cconst float*

1-6

1 lowercase courier mixed Cwith UpperCase courier vmlSetMode

lowercase courier italic incx

* Fortran

1-7

1

BLAS BLASLAPACK VML

BLAS 1C. Lawson, R. Hanson, D. Kincaid, and F. Krough, Basic Linear Algebra Subprograms for Fortran Usage, ACM Transactions on Mathematical Software, Vol.5, No.3 (September 1979) 308-325.

BLAS 2J. Dongarra, J. Du Croz, S. Hammarling, and R. Hanson, An Extended Set of Fortran Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, Vol.14, No.1 (March 1988) 1-32.

BLAS 3J. Dongarra, J. DuCroz, I. Duff, and S. Hammarling, A Set of Level 3 Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software (December 1989).

BLASD. Dodson, R. Grimes, and J. Lewis. Sparse Extensions to the FORTRAN Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software. Vol.17, No.2 (June 1991).

D. Dodson, R. Grimes, and J. Lewis. Algorithm 692: Model Implementation and Test Package for the Sparse Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software. Vol.17, No.2 (June 1991).

LAPACKE. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Userss Guide, Third Edition, Society for Industrial and Applied Mathematics (SIAM), 1999.

G. Golub and C. Van Loan, Matrix Computations, Johns Hopkins University Press, 1989.

VMLJ.M. Muller, Elementary functions: algorithms and implementation, Birkhauser Boston, 1997.

IEEE Standard for Binary Floating-Point Arithmetic. ANSI/IEEE Std 754-1985.

BLAS BLAS LAPACK (

1-8

)www.netlib.org

2-1

BLAS
2 BLAS
BLASBLASBLAS 4

BLAS 1 ( ) BLAS 2 ( ) BLAS 3 ( ) BLAS ?asum ?axpy (sdcz )

BLAS XERBLA XERBLAXERBLA

BLAS 1 i?amax i?aminiBLAS 1

BLAS BLAS 2

BLAS ( )

s c

d z

sc dz scasum

BLAS 1BLAS 1 ?dot?rot?swap

BLAS 2 3ge gb sy sp ( )sb he hp ( )hb tr tp ( )tb

( )BLAS 1

c u g Givens

BLAS 2 mv sv 1

2-2

r 1r2 2

BLAS BLAS 2 BLAS 3 mm sm 1rk kr2k 2k

BLAS

ddot

cdotc

scasum

cdotu

sgemv

ztrmm

BLASBLAS 1 BLAS

BLAS A 2 a aij a(i,j)

1

2

A

2-3

BLAS BLAS 2 BLAS 1

BLAS 12-1BLAS 1

2-1 BLAS 1

?asum s, d, sc, dz ( )

?axpy s, d, c, z ( )

?copy s, d, c, z ( )

?dot s, d ( )

?dotc c, z ( )

?dotu c, z ( )

?nrm2 s, d, sc, dz 2- ( )( )

?rot s, d, cs, zd ( )

?rotg s, d, c, z Givens ( )

?rotm s, d

?rotmg s, d Givens

?scal s, d, c, z, cs, zd ( )

?swap s, d, c, z ( )

i?amax s, d, c, z ( )i

i?amin s, d, c, z ( )i

2-4

BLAS BLAS 2

?asum

res = sasum ( n, x, incx )

res = scasum ( n, x, incx )

res = dasum ( n, x, incx )

res = dzasum ( n, x, incx )

?asum x

res = |Rex(1)| + |Imx(1)| + |Rex(2)| + |Imx(2)|+ ... + |Rex(n)| + |Imx(n)|x n

n INTEGER x

x REAL (sasum )DOUBLE PRECISION (dasum )COMPLEX (scasum )DOUBLE COMPLEX (dzasum )

DIMENSION(1+ (n-1)*abs(incx))

incx INTEGERx

res REAL (sasum )

DOUBLE PRECISION (dasum )REAL (scasum )DOUBLE PRECISION (dzasum )

2-5

BLAS BLAS 2

?axpy

call saxpy ( n, a, x, incx, y, incy )

call daxpy ( n, a, x, incx, y, incy )

call caxpy ( n, a, x, incx, y, incy )

call zaxpy ( n, a, x, incx, y, incy )

?axpyy := a*x + Y

a

x y n

n INTEGER x y

a REAL (saxpy )DOUBLE PRECISION (daxpy )COMPLEX (caxpy )DOUBLE COMPLEX (zaxpy )

a

x REAL (saxpy )DOUBLE PRECISION (daxpy )COMPLEX (caxpy )DOUBLE COMPLEX (zaxpy )

DIMENSION(1+ (n-1)*abs(incx))

incx INTEGERx

2-6

BLAS BLAS 2 y REAL (saxpy )

DOUBLE PRECISION (daxpy )COMPLEX (caxpy )DOUBLE COMPLEX (zaxpy )

DIMENSION(1+ (n-1)*abs(incy))

incy INTEGERy

y y

?copy

call scopy ( n, x, incx, y, incy )

call dcopy ( n, x, incx, y, incy )

call ccopy ( n, x, incx, y, incy )

call zcopy ( n, x, incx, y, incy )

?copyy = x

x y

n INTEGER x y

x REAL (scopy )DOUBLE PRECISION (dcopy )COMPLEX (ccopy )DOUBLE COMPLEX (zcopy )

2-7

DIMENSION (1+ (n-1)*abs(incx))

incx INTEGERx

BLAS BLAS 2 y REAL (scopy )

DOUBLE PRECISION (dcopy )COMPLEX (ccopy )DOUBLE COMPLEX (zcopy )

DIMENSION(1+ (n-1)*abs(incy))

incy INTEGERy

n x

?dot

res = sdot ( n, x, incx, y, incy )

res = ddot ( n, x, incx, y, incy )

?dot

x y

n INTEGER x y

x REAL (sdot )DOUBLE PRECISION (ddot )

DIMENSION (1+(n-1)*abs(incx))

incx INTEGERx

res xy( )=

2-8

BLAS BLAS 2 y REAL (sdot )

DOUBLE PRECISION (ddot )

DIMENSION (1+(n-1)*abs(incy))

incy INTEGERy

res REAL (sdot )

DOUBLE PRECISION (ddot )

nx yres 0

?dotc

res = cdotc ( n, x, incx, y, incy )

res = zdotc ( n, x, incx, y, incy )

?dotc

x y n

n INTEGER x y

x COMPLEX (cdotc )DOUBLE COMPLEX (zdotc )

DIMENSION (1 + (n-1)*abs(incx))

incx INTEGERx

res conjg x( )y( )=

2-9

BLAS BLAS 2 y COMPLEX (cdotc )

DOUBLE COMPLEX (zdotc )

DIMENSION(1 + (n-1)*abs(incy))

incy INTEGERy

res COMPLEX (cdotc )

DOUBLE COMPLEX (zdotc )

n x yres 0

?dotu

res = cdotu ( n, x, incx, y, incy )

res = zdotu ( n, x, incx, y, incy )

?dotu

x y n

n INTEGER x y

x COMPLEX (cdotu )DOUBLE COMPLEX (zdotu )


incx INTEGER x

res x*y( )=

2-10

BLAS BLAS 2 y COMPLEX (cdotu )

DOUBLE COMPLEX (zdotu )

DIMENSION (1 + (n-1)*abs(incy))incy INTEGERy

res COMPLEX (cdotu )

DOUBLE COMPLEX (zdotu )

nx yres 0

?nrm2

res = snrm2 ( n, x, incx )

res = dnrm2 ( n, x, incx )

res = scnrm2 ( n, x, incx )

res = dznrm2 ( n, x, incx )

?nrm2res = ||X||

x

resx

2-11

BLAS BLAS 2 n INTEGER x

x REAL (snrm2 )DOUBLE PRECISION (dnrm2 )COMPLEX (scnrm2 )DOUBLE COMPLEX (dznrm2 )

DIMENSION (1 + (n-1)*abs (incx))

incx INTEGERx

res REAL (snrm2 )

DOUBLE PRECISION (dnrm2 )REAL (scnrm2 )DOUBLE PRECISION (dznrm2 )

x

?rot

call srot ( n, x, incx, y, incy, c, s )

call drot ( n, x, incx, y, incy, c, s )

call csrot ( n, x, incx, y, incy, c, s )

call zdrot ( n, x, incx, y, incy, c, s )

2 x yx(i) = c*X(i) + s*y(i)

y(i) = c*Y(i) - s*y(i)

2-12

BLAS BLAS 2 n INTEGER x y

x REAL (srot )DOUBLE PRECISION (drot )COMPLEX (csrot )DOUBLE COMPLEX (zdrot )


incx INTEGERx

y REAL (srot )DOUBLE PRECISION (drot )COMPLEX (csrot )DOUBLE COMPLEX (zdrot )

DIMENSION (1 + (n-1)*abs(incy))

incy INTEGERy

c REAL (srot )DOUBLE PRECISION (drot )REAL (csrot )DOUBLE PRECISION (zdrot )

s REAL (srot )DOUBLE PRECISION (drot )REAL (csrot )DOUBLE PRECISION (zdrot )

x c*x + s*y

y c*y - s*x

2-13

BLAS BLAS 2

?rotg Givens

call srotg ( a, b, c, s )

call drotg ( a, b, c, s )

call crotg ( a, b, c, s )

call zrotg ( a, b, c, s )

p (a, b) y Givens abc s

a REAL (srotg )

DOUBLE PRECISION (drotg )COMPLEX (crotg )DOUBLE COMPLEX (zrotg )

p x

b REAL (srotg )DOUBLE PRECISION (drotg )COMPLEX (crotg )DOUBLE COMPLEX (zrotg )

p y

a Givens r

b Givens z

2-14

BLAS BLAS 2 c REAL (srotg )

DOUBLE PRECISION (drotg )REAL (crotg )DOUBLE PRECISION (zrotg )

Givens c

s REAL (srotg )DOUBLE PRECISION (drotg )COMPLEX (crotg )DOUBLE COMPLEX (zrotg )

Givens s

?rotm

call srotm ( n, x, incx, y, incy, param )

call drotm ( n, x, incx, y, incy, param )

2 x y

x(i) = H*x(i) + H*y(i)

y(i) = H*y(i) - H*x(i)

H Givens param(2) param(5)param

2-15

BLAS BLAS 2 n INTEGER x y

x REAL (srotm )DOUBLE PRECISION (drotm )DIMENSION (1 + (n-1)*abs(incx))

incx INTEGERx

y REAL (srotm )DOUBLE PRECISION (drotm )DIMENSION (1 + (n-1)*abs(incy))

incy INTEGERy

param REAL (srotm )DOUBLE PRECISION (drotm )DIMENSION 5

param

param(1)flagparam(2-5) H h11h21h12 h22

flagH

flag = -1.: H =

flag = 0.: H =

flag = 1.: H =

flag = -2.: H =

1.-1. 0flag

h11 h12

h21 h22

1. h12

h21 1.

h11 1.

1. h22

1. 0.

0. 1.

2-16

3 param

BLAS BLAS 2 x h11*x + h12*y

y h21*x + h22*y

H Givens

?rotmg Givens

call srotmg ( d1, d2, x1, y1, param )

call drotmg ( d1, d2, x1, y1, param )

(x1, y1) y Givens H

d1 REAL (srotmg )

DOUBLE PRECISION (drotmg ) (sqrt(d1)x1) x

d2 REAL (srotmg )DOUBLE PRECISION (drotmg ) (sqrt(d2)y1) y

x1 REAL (srotmg )DOUBLE PRECISION (drotmg ) x

x

0H x1

y1=

2-17

BLAS BLAS 2 y1 REAL (srotmg )

DOUBLE PRECISION (drotmg ) y

param REAL (srotmg )

DOUBLE PRECISION (drotmg )DIMENSION 5

param

param(1)flagparam(2-5) H h11h21h12 h22

flagH

flag = -1.: H =

flag = 0.: H =

flag = 1.: H =

flag = -2.: H =

1.-1. 0flag 3 param

?scal

h11 h12

h21 h22

1. h12

h21 1.

h11 1.

1. h22

1. 0.

0. 1.

2-18

call sscal ( n, a, x, incx )

BLAS BLAS 2 call dscal ( n, a, x, incx )

call cscal ( n, a, x, incx )

call zscal ( n, a, x, incx )

call csscal ( n, a, x, incx )

call zdscal ( n, a, x, incx )

?scalx = a*x

ax n

n INTEGER x

a REAL (sscalcsscal )DOUBLE PRECISION (dscalzdscal )REAL (cscal )DOUBLE PRECISION (zscal )

a

x REAL (sscal )DOUBLE PRECISION (dscal )COMPLEX (cscalcsscal )DOUBLE COMPLEX (zscalcsscal )


incx INTEGERx

x x

2-19

BLAS BLAS 2

?swap

call sswap ( n, x, incx, y, incy )

call dswap ( n, x, incx, y, incy )

call cswap ( n, x, incx, y, incy )

call zswap ( n, x, incx, y, incy )

2 x y?swap y x

n INTEGER x y

x REAL (sswap )DOUBLE PRECISION (dswap )COMPLEX (cswap )DOUBLE COMPLEX (zswap )

DIMENSION(1 + (n-1)*abs(incx))

incx INTEGERx

y REAL (sswap )DOUBLE PRECISION (dswap )COMPLEX (cswap )DOUBLE COMPLEX (zswap )

DIMENSION(1 + (n-1)*abs(incy))

incy INTEGERy

x x

y y

2-20

BLAS BLAS 2

i?amax

index = isamax ( n, x, incx )

index = idamax ( n, x, incx )

index = icamax ( n, x, incx )

index = izamax ( n, x, incx )

xi?amax x(i) |Re x(i)| + |Im x(i)|

n0

2

n INTEGER x

x REAL (isamax )DOUBLE PRECISION (idamax )COMPLEX (icamax )DOUBLE COMPLEX (izamax )


incx INTEGERx

index INTEGER x

2-21

BLAS BLAS 2

i?amin

index = isamin ( n, x, incx )

index = idamin ( n, x, incx )

index = icamin ( n, x, incx )

index = izamin ( n, x, incx )

xi?amin x(i) |Rex(i)| + |Imx(i)|

n , 0

2

n INTEGERn x

x REAL (isamin )DOUBLE PRECISION (idamin )COMPLEX (icamin )DOUBLE COMPLEX (izamin )


incx INTEGERx

index INTEGER x

2-22

BLAS BLAS 2BLAS 2

BLAS 2 2-2BLAS 2

2-2 BLAS 2

?gbmv s, d, c, z

?gemv s, d, c, z

?ger s, d 1

?gerc c, z 1

?geru c, z 1

?hbmv c, z

?hemv c, z

?her c, z 1

?her2 c, z 2

?hpmv c, z

?hpr c, z 1

?hpr2 c, z 2

?sbmv s, d

?spmv s, d

?spr s, d 1

?spr2 s, d 2

?symv s, d

?syr s, d 1

?syr2 s, d 2

( )

2-23

BLAS BLAS 2

?gbmv

CALL sgbmv ( trans, m, n, kl, ku, alpha, a, lda, x, inxc, beta, y, incy )

call dgbmv ( trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy )

call cgbmv ( trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy )

call zgbmv ( trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy )

?gbmvy := alpha*a*x + beta*y

y := alpha*a'*x + beta*y

?tbmv s, d, c, z

?tbsv s, d, c, z 1

?tpmv s, d, c, z

?tpsv s, d, c, z 1

?trmv s, d, c, z

?trsv s, d, c, z 1

2-2 BLAS 2 ( )

2-24

y := alpha*conjg(a')*x + beta*y

alpha beta

BLAS BLAS 2x y

akl ku m n

trans CHARACTER*1

m INTEGER am

n INTEGER an

kl INTEGER akl0 kl

ku INTEGER aku0 ku

alpha REAL (sgbmv )DOUBLE PRECISION (dgbmv )COMPLEX (cgbmv )DOUBLE COMPLEX (zgbmv )

alpha

a REAL (sgbmv )DOUBLE PRECISION (dgbmv )OMPLEX (cgbmv )DOUBLE COMPLEX (zgbmv )

DIMENSION (lda, n) a (kl + ku + 1) n (ku + 1) (ku + 1)

trans

N n y:= alpha*a*x + beta*y

T t y:= alpha*a'*x + beta*y

C c y:= alpha*conjg(a')*x +beta*y

2-25

2 (ku + 2) 1

BLAS BLAS 2 a ( ku ku )

do 20, j = 1, n k = ku + 1 - j

do 10, i = max(1, j-ku), min(m, j+kl) a(k+i, j) = matrix(i,j) 10 continue 20 continue

lda INTEGER ( )a 1lda (kl + ku + 1)

x REAL (sgbmv )DOUBLE PRECISION (dgbmv )COMPLEX (cgbmv )DOUBLE COMPLEX (zgbmv )

DIMENSION trans = 'N' 'n'(1 + (n - 1)*abs(incx))(1 + (m - 1)*abs(incx)) x x

incx INTEGERxincx

beta REAL (sgbmv )DOUBLE PRECISION (dgbmv )COMPLEX (cgbmv )DOUBLE COMPLEX (zgbmv )

betabetay

2-26

BLAS BLAS 2y REAL (sgbmv )

DOUBLE PRECISION (dgbmv )COMPLEX (cgbmv )OUBLE COMPLEX (zgbmv )

DIMENSIONtrans ='N' 'n'(1 + (m - 1)*abs(incy))(1 + (n - 1)*abs(incy)) y y

incy INTEGERyincy

y y

?gemv

call sgemv ( trans, m, n, alpha, a, lda, x, incx,beta, y, incy )

call dgemv ( trans, m, n, alpha, a, lda, x, incx,beta, y, incy )

call cgemv ( trans, m, n, alpha, a, lda, x, incx,beta, y, incy )

call zgemv ( trans, m, n, alpha, a, lda, x, incx,beta, y, incy )

?gemvy := alpha*a*x + beta*y

2-27

y := alpha*a'*x + beta*y

BLAS BLAS 2y := alpha*conjg(a')*x + beta*y

alpha beta

x y

am n

trans CHARACTER*1

m INTEGER am

n INTEGER an

alpha REAL (sgemv )DOUBLE PRECISION (dgemv )COMPLEX (cgemv )DOUBLE COMPLEX (zgemv )

alpha

a REAL (sgemv )DOUBLE PRECISION (dgemv )COMPLEX (cgemv )DOUBLE COMPLEX (zgemv )

DIMENSION (lda, n) a m n

trans

N n y:= alpha*a*x + beta*y

T t y:= alpha*a'*x + beta*y

C c y:= alpha*conjg(a')*x +beta*y

2-28

BLAS BLAS 2lda INTEGER ( )

a 1lda max(1, m)

x REAL (sgemv )DOUBLE PRECISION (dgemv )OMPLEX (cgemv )DOUBLE COMPLEX (zgemv )

DIMENSIONtrans = 'N' 'n'(1+(n-1)*abs(incx))(1+(m - 1)*abs(incx)) x x

incx INTEGERxincx

beta REAL (sgemv )DOUBLE PRECISION (dgemv )COMPLEX (cgemv )DOUBLE COMPLEX (zgemv )

betabetay

y REAL (sgemv )DOUBLE PRECISION (dgemv )COMPLEX (cgemv )DOUBLE COMPLEX (zgemv )

DIMENSIONtrans = 'N' 'n'(1+(m-1)*abs(incy))(1+(n-1)*abs(incy))beta y y

incy INTEGERyincy

y y

2-29

BLAS BLAS 2

?ger 1

call sger ( m, n, alpha, x, incx, y, incy, a, lda )

call dger ( m, n, alpha, x, incx, y, incy, a, lda )

?gera := alpha*x*y' + a

alpha

xm

yn

am n

m INTEGER am

n INTEGER an

alpha REAL (sger )DOUBLE PRECISION (dger )

alpha

x REAL (sger )DOUBLE PRECISION (dger )

DIMENSION (1 + (m - 1)*abs(incx)) x m x

2-30

BLAS BLAS 2incx INTEGERxincx

y REAL (sger )DOUBLE PRECISION (dger )

DIMENSION (1 + (n - 1)*abs(incy)) y n y

incy INTEGERyincy

a REAL (sger )DOUBLE PRECISION (dger )


lda INTEGER ( )a 1lda max(1, m)

a

?gerc 1 ( )

call cgerc ( m, n, alpha, x, incx, y, incy, a, lda )

call zgerc ( m, n, alpha, x, incx, y, incy, a, lda )

?gerca := alpha*x*conjg(y') + a

2-31

BLAS BLAS 2alpha

xm

yn

am n

m INTEGER am

n INTEGER an

alpha SINGLE PRECISION COMPLEX (cgerc )DOUBLE PRECISION COMPLEX (zgerc )

alpha

x SINGLE PRECISION COMPLEX (cgerc )DOUBLE PRECISION COMPLEX (zgerc )

DIMENSION (1 + (m - 1)*abs(incx)) x m x

incx INTEGERxincx

y COMPLEX (cgerc )DOUBLE COMPLEX (zgerc )


incy INTEGERyincy

a COMPLEX (cgerc )DOUBLE COMPLEX (zgerc )

2-32

BLAS BLAS 2DIMENSION (lda, n) a m n


a

?geru 1 ( )

call cgeru ( m, n, alpha, x, incx, y, incy, a, lda )

call zgeru ( m, n, alpha, x, incx, y, incy, a, lda )

?gerua:= alpha*x*y' + a

alpha

xm

yn

am n

2-33

BLAS BLAS 2m INTEGER am

n INTEGER an

alpha COMPLEX (cgeru )DOUBLE COMPLEX (zgeru )

alpha

x COMPLEX (cgeru )DOUBLE COMPLEX (zgeru )

DIMENSION(1 + (m - 1)*abs(incx))xmx

incx INTEGERxincx

y COMPLEX (cgeru )DOUBLE COMPLEX (zgeru )

DIMENSION(1 + (n - 1)*abs(incy))yny

incy INTEGERyincy

a COMPLEX (cgeru )DOUBLE COMPLEX (zgeru )



a

2-34

BLAS BLAS 2

?hbmv

call chbmv ( uplo, n, k, alpha, a, lda, x, incx, beta, y, incy )

call zhbmv ( uplo, n, k, alpha, a, lda, x, incx, beta, y, incy )

?hbmvy := alpha*a*x + beta*y

alpha beta

x yn

ak n n

uplo character*1 a

n INTEGER an

k INTEGER ak0 k

uplo a

U u a

L l a

2-35

BLAS BLAS 2alpha COMPLEX (chbmv )

DOUBLE COMPLEX (zhbmv )

alpha

a COMPLEX (chbmv )DOUBLE COMPLEX (zhbmv )

DIMENSION (lda, n)uplo = 'U' 'u' a (k + 1) n (k + 1) k 2 a k k

do 20, j = 1, n m = k + 1 - j do 10, i = max(1, j - k), j a(m + i, j) = matrix(i, j) 10 continue20 continue

uplo = 'L' 'l' a (k + 1) n 1 2 1 a k k

do 20, j = 1, n m = 1 - j do 10, i = j, min( n, j + k )

2-36

BLAS BLAS 2 a( m + i, j ) = matrix( i, j ) 10 continue20 continue

lda INTEGER ( )a 1lda (k + 1)

x COMPLEX (chbmv )DOUBLE COMPLEX (zhbmv )

DIMENSION (1 + (n - 1)*abs(incx)) x x

incx INTEGERxincx

beta COMPLEX (chbmv )DOUBLE COMPLEX (zhbmv )

beta

y COMPLEX (chbmv )DOUBLE COMPLEX (zhbmv )

DIMENSION(1 + (n - 1)*abs(incy))yy

incy INTEGERyincy

y y

2-37

BLAS BLAS 2

?hemv

call chemv ( uplo, n, alpha, a, lda, x, incx, beta, y, incy )

call zhemv ( uplo, n, alpha, a, lda, x, incx, beta, y, incy )

?hemvy := alpha*a*x + beta*y

alpha beta

x yn

an n

uplo CHARACTER*1 a

n INTEGER an

uplo aU u a

L l a

2-38

BLAS BLAS 2alpha COMPLEX (chemv )

DOUBLE COMPLEX (zhemv )

alpha

a COMPLEX (chemv )DOUBLE COMPLEX (zhemv )

DIMENSION (lda, n)uplo = 'U' 'u' a n n auplo = 'L' 'l' a n n a

lda INTEGER ( )a 1lda max(1, n)

x COMPLEX (chemv )DOUBLE COMPLEX (zhemv )

DIMENSION(1 + (n - 1)*abs(incx))xnx

incx INTEGERxincx

beta COMPLEX (chemv )DOUBLE COMPLEX (zhemv )

betabetay

2-39

BLAS BLAS 2y COMPLEX (chemv )

DOUBLE COMPLEX (zhemv )


incy INTEGERyincy

y y

?her 1

call cher ( uplo, n, alpha, x, incx, a, lda )

call zher ( uplo, n, alpha, x, incx, a, lda )

?hera := alpha*x*conjg(x') + a

alpha

xn

an n

2-40

BLAS BLAS 2uplo CHARACTER*1 a

n INTEGER an

alpha REAL (cher )DOUBLE PRECISION (zher )

alpha

x COMPLEX (cher )DOUBLE COMPLEX (zher )

(1 + (n - 1)*abs(incx)) x n x

incx INTEGERxincx

a COMPLEX (cher )DOUBLE COMPLEX (zher )

DIMENSION (lda, n)uplo = 'U' 'u' a n n a

uplo = 'L' 'l' a n n a

uplo a

U u a

L l a

2-41

BLAS BLAS 2


a uplo = 'U' 'u' a

uplo = 'L' 'l' a

?her2 2

call cher2 ( uplo, n, alpha, x, incx, y, incy, a, lda )

call zher2 ( uplo, n, alpha, x, incx, y, incy, a, lda )

?her2a := alpha*x*conjg(y') + conjg(alpha)*y*conjg(x') + a

alpha

x yn

an n

2-42


n INTEGER an

alpha COMPLEX (cher2 )DOUBLE COMPLEX (zher2 )

alpha

x COMPLEX (cher2 )DOUBLE COMPLEX (zher2 )


incx INTEGERxincx

y COMPLEX (cher2 )DOUBLE COMPLEX (zher2 )


incy INTEGERyincy

a COMPLEX (cher2 )DOUBLE COMPLEX (zher2 )

uplo a

U u aL l a

2-43

BLAS BLAS 2DIMENSION (lda, n)uplo = 'U' 'u' a n n a



a uplo = 'U' 'u' a

uplo = 'L' 'l' a

?hpmv

call chpmv ( uplo, n, alpha, ap, x, incx, beta, y, incy )

call zhpmv ( uplo, n, alpha, ap, x, incx, beta, y, incy )

2-44

BLAS BLAS 2

?hpmvy := alpha*a*x + beta*y

alpha beta

x yn

a n n

uplo CHARACTER*1 a

ap

n INTEGER an

alpha COMPLEX (chpmv )DOUBLE COMPLEX (zhpmv )

alpha

ap COMPLEX (chpmv )DOUBLE COMPLEX (zhpmv )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1, 1)ap(2) a(1, 2)ap(3) a(2, 2) apuplo = 'L' 'l'ap(1)a(1, 1)ap(2) a(2, 1)ap(3) a(3, 1) ap

uplo aU u a ap

L l a ap

2-45

BLAS BLAS 2

x COMPLEX (chpmv )DOUBLE PRECISION COMPLEX (zhpmv )


incx INTEGERxincx

beta COMPLEX (chpmv )DOUBLE COMPLEX (zhpmv )

betabetay

y COMPLEX (chpmv )DOUBLE COMPLEX (zhpmv )


incy INTEGERyincy

y y

2-46

BLAS BLAS 2

?hpr 1

call chpr ( uplo, n, alpha, x, incx, ap )

call zhpr ( uplo, n, alpha, x, incx, ap )

?hpra := alpha*x*conjg(x') + a

alpha

xn

a n n

uplo CHARACTER*1 a

ap

n INTEGER a n

alpha REAL (chpr )DOUBLE PRECISION (zhpr )

alpha

uplo a

U u a ap

L l a ap

2-47

BLAS BLAS 2x COMPLEX (chpr )

DOUBLE COMPLEX (zhpr )


incx INTEGERxincx

ap COMPLEX (chpr )DOUBLE COMPLEX (zhpr )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'AP(1) a(1, 1)ap(2) a(1, 2)ap(3) a(2, 2) ap

uplo = 'L' 'l'ap(1) a(1, 1)ap(2) a(2, 1)ap(3) a(3, 1) ap

ap uplo = 'U' 'u'

uplo = 'L' 'l'

2-48

BLAS BLAS 2

?hpr2 2

call chpr2 ( uplo, n, alpha, x, incx, y, incy, ap )

call zhpr2 ( uplo, n, alpha, x, incx, y, incy, ap )

?hpr2a := alpha*x*conjg(y') + conjg(alpha)*y*conjg(x') + a

alpha

x yn

a n n

uplo CHARACTER*1 a

ap

n INTEGER an

alpha COMPLEX (chpr2 )DOUBLE COMPLEX (zhpr2 )

alpha

uplo a

U u a ap

L l a ap

2-49

BLAS BLAS 2x COMPLEX (chpr2 )

DOUBLE COMPLEX (zhpr2 )

(1 + (n - 1)*abs(incx)) x n x

incx INTEGER xincx

y COMPLEX (chpr2 )DOUBLE COMPLEX (zhpr2 )


incy INTEGER yincy

ap COMPLEX (chpr2 )DOUBLE COMPLEX (zhpr2 )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1, 1)ap(2) a(1, 2)ap(3) a(2, 2) ap

uplo = 'L' 'l'ap(1) a(1, 1)ap(2) A(2, 1)ap(3) A(3, 1) ap

2-50

BLAS BLAS 2ap uplo = 'U' 'u'

uplo = 'L' 'l'

?sbmv

call ssbmv ( uplo, n, k, alpha, a, lda, x, incx, beta, y, incy )

call dsbmv ( uplo, n, k, alpha, a, lda, x, incx, beta, y, incy )

?sbmvy := alpha*a*x + beta*y

alpha beta

x yn

ak n n

2-51


n INTEGER an

k INTEGER ak0 k

alpha REAL (ssbmv )DOUBLE PRECISION (dsbmv )

alpha

a REAL (ssbmv )DOUBLE PRECISION (dsbmv )


do 20, j = 1, n m = k + 1 - j do 10, i = max( 1, j - k ), j a( m + i, j ) = matrix( i, j ) 10 continue20 continue

uplo a

U u a

L l a

2-52

BLAS BLAS 2uplo = 'L' 'l' a (k + 1) n 1 2 1 a k k

do 20, j = 1, n m = 1 - j do 10, i = j, min( n, j + k ) a( m + i, j ) = matrix( i, j ) 10 continue20 continue


x REAL (ssbmv )DOUBLE PRECISION (dsbmv )

DIMENSION(1 + (n - 1)*abs(incx))xx

incx INTEGERx incx

beta REAL (ssbmv )DOUBLE PRECISION (dsbmv )

beta

y REAL (ssbmv )DOUBLE PRECISION (dsbmv )

DIMENSION(1 + (n - 1)*abs(incy))yy

2-53

BLAS BLAS 2incy INTEGERyincy

y y

?spmv

call sspmv ( uplo, n, alpha, ap, x, incx, beta, y, incy )

call dspmv ( uplo, n, alpha, ap, x, incx, beta, y, incy )

?spmvy := alpha*a*x + beta*y

alpha beta

x yn

a n n

uplo CHARACTER*1 a

ap

uplo a

U u a ap

L l a ap

2-54

BLAS BLAS 2n INTEGER an

alpha REAL (sspmv )DOUBLE PRECISION (dspmv )

alpha

ap REAL (sspmv )DOUBLE PRECISION (dspmv )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1, 1)ap(2) a(1, 2)ap(3) a(2, 2) apuplo = 'L' 'l'ap(1) a(1, 1)ap(2) a(2, 1)ap(3) a(3, 1) ap

x REAL (sspmv )DOUBLE PRECISION (dspmv )

DIMENSION (1 + (n - 1)*abs(incx)) x n x

incx INTEGERxincx

beta REAL (sspmv )DOUBLE PRECISION (dspmv )

betabetay

y REAL (sspmv )DOUBLE PRECISION (dspmv )


2-55

BLAS BLAS 2incy INTEGER yincy

y y

?spr 1

call sspr( uplo, n, alpha, x, incx, ap )

call dspr( uplo, n, alpha, x, incx, ap )

?spra:= alpha*x*x' + a

alpha

x n

a n n

uplo CHARACTER*1 a

ap

uplo a

U u a ap

L l a ap

2-56


alpha REAL (sspr )DOUBLE PRECISION (dspr )

alpha

x REAL (sspr )DOUBLE PRECISION (dspr )


incx INTEGER xincx

ap REAL (sspr )DOUBLE PRECISION (dspr )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1,1)ap(2) a(1,2)ap(3) a(2,2) ap

uplo = 'L' 'l' ap(1) a(1,1)ap(2) a(2,1)ap(3) a(3,1) ap

ap uplo = 'U' 'u'

uplo = 'L' 'l'

2-57

BLAS BLAS 2

?spr2 2

call sspr2( uplo, n, alpha, x, incx, y, incy, ap )

call dspr2( uplo, n, alpha, x, incx, y, incy, ap )

?spr2 a:= alpha*x*y' + alpha*y*x' + a

alpha

x yn

a n n

uplo CHARACTER*1 a

ap

n INTEGER a n

alpha REAL (sspr2 )DOUBLE PRECISION (dspr2 )

alpha

uplo a

U u a ap

L l a ap

2-58

BLAS BLAS 2x REAL (sspr2 )

DOUBLE PRECISION (dspr2 )

DIMENSION(1 + (n - 1)*abs(incx))xn x

incx INTEGERxincx

y REAL (sspr2 )DOUBLE PRECISION (dspr2 )


incy INTEGER yincy

ap REAL (sspr2 )DOUBLE PRECISION (dspr2 )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1,1)ap(2) a(1,2)ap(3) a(2,2) ap

uplo = 'L' 'l'ap(1) a(1,1)ap(2) a(2,1)ap(3) a(3,1) ap

ap uplo = 'U' 'u'

uplo = 'L' 'l'

2-59

BLAS BLAS 2

?symv

call ssymv ( uplo, n, alpha, a, lda, x, incx, beta, y,

incy )

call dsymv ( uplo, n, alpha, a, lda, x, incx, beta, y,

incy )

?symv y := alpha*a*x + beta*y

alpha beta

x yn

an n

uplo CHARACTER*1 a

n INTEGER a n

uplo a

U u a

L l a

2-60

BLAS BLAS 2alpha REAL (ssymv )

DOUBLE PRECISION (dsymv )

alpha

a REAL (ssymv )DOUBLE PRECISION (dsymv )

DIMENSION (lda, n)uplo = 'U' 'u' a n nauplo = 'L''l' an na

lda INTEGER ( )a 1lda max(1,n)

x REAL (ssymv )DOUBLE PRECISION (dsymv )


incx INTEGERxincx

beta REAL (ssymv )DOUBLE PRECISION (dsymv )

betabetay

y REAL (ssymv )DOUBLE PRECISION (dsymv )


2-61

BLAS BLAS 2incy INTEGERyincy

y y

?syr 1

call ssyr( uplo, n, alpha, x, incx, a, lda )

call dsyr( uplo, n, alpha, x, incx, a, lda )

?syr

a := alpha*x*x' + a

alpha

xn

an n

uplo CHARACTER*1 a

uplo a

U u a

L l a

2-62


alpha REAL (ssyr )DOUBLE PRECISION (dsyr )

alpha

x REAL (ssyr )DOUBLE PRECISION (dsyr )

DIMENSION(1 + (n - 1)*abs(incx)) xnx

incx INTEGERxincx

a REAL (ssyr )DOUBLE PRECISION (dsyr )

DIMENSION (lda, n)uplo = 'U' 'u' a n na

uplo = 'L' 'l' a n na


a uplo = 'U' 'u' a

uplo = 'L' 'l' a

2-63

BLAS BLAS 2

?syr2 2

call ssyr2( uplo, n, alpha, x, incx, y, incy, a, lda )

call dsyr2( uplo, n, alpha, x, incx, y, incy, a, lda )

?syr2 a := alpha*x*y' + alpha*y*x' + a

alpha

x yn

an n

uplo CHARACTER*1 a

n INTEGER an

alpha REAL (ssyr2 )DOUBLE PRECISION (dsyr2 )

alpha

uplo a

U u a

L l a

2-64

BLAS BLAS 2x REAL (ssyr2 )

DOUBLE PRECISION (dsyr2 )


incx INTEGERxincx

y REAL (ssyr2 )DOUBLE PRECISION (dsyr2 )


incy INTEGER yincy

a REAL (ssyr2 )DOUBLE PRECISION (dsyr2 )

DIMENSION (lda, n)uplo = 'U' 'u' a n na



2-65

BLAS BLAS 2a uplo = 'U' 'u' a

uplo = 'L' 'l' a

?tbmv

call stbmv ( uplo, trans, diag, n, k, a, lda, x, incx )

call dtbmv ( uplo, trans, diag, n, k, a, lda, x, incx )

call ctbmv ( uplo, trans, diag, n, k, a, lda, x, incx )

call ztbmv ( uplo, trans, diag, n, k, a, lda, x, incx )

?tbmv

x := a*x x := a'*x x := conjg(a')*x

xn

an n(k + 1)

uplo CHARACTER*1

2-66

BLAS BLAS 2

trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

k INTEGER uplo = 'U' 'u'k auplo = 'L' 'l'k ak 0 k

a REAL (stbmv )DOUBLE PRECISION (dtbmv )COMPLEX (ctbmv )DOUBLE COMPLEX (ztbmv )

DIMENSION (lda, n)uplo = 'U' 'u' a (k + 1) n (k + 1) k 2

uplo a

U u

L l

trans

N n x := a*x

T t x := a'*x

C c x := conjg(a')*x

diag a

U u a

N n a

2-67

a k k

BLAS BLAS 2 do 20, j = 1, n m = k + 1 - j

do 10, i = max(1, j - k), j a(m + i, j) = matrix(i, j) 10 continue20 continue

uplo = 'L' 'l' a (k + 1) n 1 2 1 a k k

do 20, j = 1, n m = 1 - j do 10, i = j, min(n, j + k) a(m + i, j) = matrix (i, j) 10 continue20 continue

diag = 'U' 'u' a1


2-68

BLAS BLAS 2x REAL (stbmv )

DOUBLE PRECISION (dtbmv )COMPLEX (ctbmv )DOUBLE COMPLEX (ztbmv )

DIMENSION (1 + (n - 1)*abs(incx))xnx

incx INTEGER xincx

x x

?tbsv 1

call stbsv ( uplo, trans, diag, n, k, a, lda, x, incx )

call dtbsv ( uplo, trans, diag, n, k, a, lda, x, incx )

call ctbsv ( uplo, trans, diag, n, k, a, lda, x, incx )

call ztbsv ( uplo, trans, diag, n, k, a, lda, x, incx )

?tbsv

a*x = b, a'*x = b, conjg(a')*x = b

b xn

an n(k + 1)

2-69

BLAS BLAS 2

uplo CHARACTER*1

trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

k INTEGERuplo = 'U' 'u'k auplo = 'L''l'k ak0 k

uplo a

U u

L l

trans

N n a*x = b

T t a'*x = b

C c conjg(a')*x = b

diag a

U u a

N n a

2-70

BLAS BLAS 2a REAL (stbsv )

DOUBLE PRECISION (dtbsv )COMPLEX (ctbsv )DOUBLE COMPLEX (ztbsv )


do 20, j = 1, n m = k + 1 - j do 10, i = max(1, j - k), j a(m + i, j) = matrix (i, j) 10 continue20 continue

uplo = 'L' 'l' a (k + 1) n 1 2 1 a k k

do 20, j = 1, n m = 1 - j do 10, i = j, min(n, j + k) a(m + i, j) = matrix (i, j) 10 continue20 continue

2-71

BLAS BLAS 2diag = 'U' 'u' a1


x REAL (stbsv )DOUBLE PRECISION (dtbsv )COMPLEX (ctbsv )DOUBLE COMPLEX (ztbsv )

DIMENSION (1 + (n - 1)*abs(incx)) x n b

incx INTEGER xincx

x x

?tpmv

call stpmv ( uplo, trans, diag, n, ap, x, incx )

call dtpmv ( uplo, trans, diag, n, ap, x, incx )

call ctpmv ( uplo, trans, diag, n, ap, x, incx )

call ztpmv ( uplo, trans, diag, n, ap, x, incx )

?tpmv


2-72

BLAS BLAS 2xn

an n

uplo CHARACTER*1 a

trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

ap REAL (stpmv )DOUBLE PRECISION (dtpmv )COMPLEX (ctpmv )DOUBLE COMPLEX (ztpmv )

uplo a

U u

L l

trans

N n x := a*x

T t x := a'*x


diag a

U u a

N n a

2-73

BLAS BLAS 2DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1,1)ap(2) a(1,2)ap(3) a(2,2) apuplo = 'L' 'l'ap(1) a(1,1)ap(2)a(2,1)ap(3) a(3,1) apdiag = 'U' 'u'a1

x REAL (stpmv )DOUBLE PRECISION (dtpmv )COMPLEX (ctpmv )DOUBLE COMPLEX (ztpmv )

DIMENSION (1 + (n - 1)*abs(incx)) x n x

incx INTEGER xincx

x x

2-74

BLAS BLAS 2

?tpsv 1

call stpsv ( uplo, trans, diag, n, ap, x, incx )

call dtpsv ( uplo, trans, diag, n, ap, x, incx )

call ctpsv ( uplo, trans, diag, n, ap, x, incx )

call ztpsv ( uplo, trans, diag, n, ap, x, incx )

?tpsv

a*x = b a'*x = b conjg(a')*x = b

b xn

an n

uplo CHARACTER*1 a

uplo a

U u

L l

2-75

BLAS BLAS 2trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

ap REAL (stpsv )DOUBLE PRECISION (dtpsv )COMPLEX (ctpsv )DOUBLE COMPLEX (ztpsv )

DIMENSION ((n*(n + 1))/2)uplo = 'U''u'ap(1) a(1, 1)ap(2) a(1, 2)ap(3) a(2, 2) apuplo = 'L' 'l'ap(1) a(1, 1)ap(2)a(2, 1)ap(3) a(3, 1) apdiag = 'U' 'u'a1

trans

N n a*x = b

T t a'*x = b

C c conjg(a')*x = b

diag a

U u a

N n a

2-76

BLAS BLAS 2x REAL (stpsv )

DOUBLE PRECISION (dtpsv )COMPLEX (ctpsv )DOUBLE COMPLEX (ztpsv )

DIMENSION(1 + (n - 1)*abs(incx))xnb

incx INTEGER xincx

x x

?trmv

call strmv ( uplo, trans, diag, n, a, lda, x, incx )

call dtrmv ( uplo, trans, diag, n, a, lda, x, incx )

call ctrmv ( uplo, trans, diag, n, a, lda, x, incx )

call ztrmv ( uplo, trans, diag, n, a, lda, x, incx )

?trmv


xn

an n

2-77


trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

a REAL (strmv )DOUBLE PRECISION (dtrmv )COMPLEX (ctrmv )DOUBLE COMPLEX (ztrmv )

DIMENSION (lda,n)uplo = 'U' 'u' a n nauplo = 'L' 'l'

uplo a

U u

L l

trans

N n x := a*x

T t x := a'*x


diag a

U u a

N n a

2-78

a n nadiag = 'U' 'u'a1

BLAS BLAS 2lda INTEGER ( )

a 1lda max(1, n)

x REAL (strmv )DOUBLE PRECISION (dtrmv )COMPLEX (ctrmv )OUBLE COMPLEX (ztrmv )


incx INTEGERxincx

x x

?trsv 1

call strsv ( uplo, trans, diag, n, a, lda, x, incx )

call dtrsv ( uplo, trans, diag, n, a, lda, x, incx )

call ctrsv ( uplo, trans, diag, n, a, lda, x, incx )

call ztrsv ( uplo, trans, diag, n, a, lda, x, incx )

2-79

BLAS BLAS 2

?trsv

a*x = b a'*x = b conjg(a')*x = b

b xn

an n

uplo CHARACTER*1

trans CHARACTER*1

diag CHARACTER*1a

n INTEGER an

uplo a

U u

L l

trans

N n a*x = b

T t a'*x = b

C c conjg(a')*x = b

diag a

U u a

N n a

2-80

BLAS BLAS 2a REAL (strsv )

DOUBLE PRECISION (dtrsv )COMPLEX (ctrsv )OUBLE COMPLEX (ztrsv )

DIMENSION (lda,n)UPLO = 'U' 'u' a n nauplo= 'L' 'l' a n nadiag = 'U' 'u'a1


x REAL (strsv )DOUBLE PRECISION (dtrsv )COMPLEX (ctrsv )DOUBLE COMPLEX (ztrsv )

DIMENSION(1 + (n - 1)*abs(incx))xnb

incx INTEGER xincx

x x

2-81

BLAS BLAS 2BLAS 3

BLAS 3 2-3BLAS 3

MKL

BLAS 3MKL (SMP)

MKL

BLAS 3

2-3 BLAS 3

?gemm s, d, c, z

?hemm c, z

?herk c, z k

?her2k c, z 2k

?symm s, d, c, z

?syrk s, d, c, z k

?syr2k s, d, c, z 2k

?trmm s, d, c, z

?trsm s, d, c, z 1

2-82

BLAS BLAS 2 12

?gemm

call sgemm ( transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc )

call dgemm ( transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc )

call cgemm ( transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc )

call zgemm ( transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc )

?gemmc := alpha*op(a)*op(b) + beta*c

op(x)op(x) = x op(x) = x' op(x) = conjg(x')

alpha beta

ab c

op(a)m k

op(b)k n

2-83

cm n

BLAS BLAS 2transa CHARACTER*1

op(a)

transb CHARACTER*1op(b)

m INTEGER op(a) cm

n INTEGER op(b) cn

k INTEGER op(a) op(b)k

alpha REAL(sgemm )DOUBLE PRECISION (dgemm )COMPLEX (cgemm )DOUBLE COMPLEX (zgemm )

alpha

transa op(a)

N n op(a) = a

T t op(a) = a'

C c op(a) = conjg(A')

transb op(b)

N n op(b) = b

T t op(b) = b'

C c op(b) = conjg(b')

2-84

BLAS BLAS 2a REAL (sgemm )

DOUBLE PRECISION (dgemm )COMPLEX (cgemm )DOUBLE COMPLEX (zgemm )

DIMENSION (lda, ka)ka transa = 'N' 'n' k mtransa = 'N' 'n' a m k a a k m a

lda INTEGER ( )a 1transa = 'N' 'n'lda max(1, m)lda max(1, k)

b REAL (sgemm )DOUBLE PRECISION (dgemm )COMPLEX (cgemm )DOUBLE COMPLEX (zgemm )

DIMENSION (ldb, kb)kb transb = 'N' 'n' n ktransb = 'N' 'n' b k n b b n k b

ldb INTEGER ( )b 1transb = 'N' 'n'ldb max

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インテル(R) 数値演算ライブラリ(MKL) リファレンス・マニュアル · インテル® 数値演算ライブラリ－リファレンス・マニュアル－目次