Numerical Methods Contents - SAM

Syntax of BLAS calls:
The functions have been implemented for different types, and are distinguished by the first letter of the
function name. E.g. sdot is the dot product implementation for single precision and ddot for double
precision.
BLAS LEVEL 1: vector operations, asymptotic complexity O(n), n ˆ= vector length
• dot product
ρ = x T y
xDOT(N,X,INCX,Y,INCY)
– x ∈ {S, D}, scalar type: S ˆ= type float, D ˆ= type double
– N ˆ= length of vector (modulo stride INCX)
– X ˆ= vector x: array of type x
– INCX ˆ= stride for traversing vector X
– Y ˆ= vector y: array of type x
– INCY ˆ= stride for traversing vector Y
• vector operations y = αx + y
xAXPY(N,ALPHA,X,INCX,Y,INCY)
Ôº ½º
– x ∈ {S, D, C, Z}, S ˆ= type float, D ˆ= type double, C ˆ= type complex
– N ˆ= length of vector (modulo stride INCX)
– ALPHA ˆ= scalar α
– X ˆ= vector x: array of type x
– INCX ˆ= stride for traversing vector X
– Y ˆ= vector y: array of type x
– INCY ˆ= stride for traversing vector Y
BLAS LEVEL 2: matrix-vector operations, asymptotic complexity O(mn), (m, n) ˆ= matrix size
• matrix×vector multiplication
y = αAx + βy
xGEMV(TRANS,M,N,ALPHA,A,LDA,X,
INCX,BETA,Y,INCY)
– x ∈ {S, D, C, Z}, scalar type: S ˆ= type float, D ˆ= type double, C ˆ= type complex
– M,N ˆ= size of matrix A
– ALPHA ˆ= scalar parameter α
– A ˆ= matrix A stored in linear array of length M · N (column major arrangement)
(A) i,j = A[N ∗ (j − 1) + i] .
– LDA ˆ= “leading dimension” of A ∈ K n,m , that is, the number n of rows.
– X ˆ= vector x: array of type x
– INCX ˆ= stride for traversing vector X
– BETA ˆ= scalar paramter β
– Y ˆ= vector y: array of type x
– INCY ˆ= stride for traversing vector Y
• BLAS LEVEL 3: matrix-matrix operations, asymptotic complexity O(mnk), (m, n,k) ˆ= matrix
sizes
– matrix×matrix multiplication C = αAB + βC
xGEMM(TRANSA,TRANSB,M,N,K,
ALPHA,A,LDA,X,B,LDB,
BETA,C,LDC)
(☞ meaning of arguments as above)
Example 1.4.1 (Gaining efficiency through use of BLAS).
Code 1.4.2: ColumnMajor Matrix Class in C++, Ex. 1.4.1
1 /∗ Author : Manfred Quack
2 ∗ ColumnMajorMatrix . h
3 ∗ This Class Implements a ColumnMajor M a trix S t r u c t u r e i n C++
4 ∗ − i t provides an access operator ( ) using ColumnMajor Access
5 ∗ − i t also provides 4 d i f f e r e n t implementations o f a Matrix−M a trix
M u l t i p l i c a t i o n
6 ∗/
7 #include
8 #include < c s t d l i b >
9 #include < s t d i o . h>
10 #include
11 # ifndef _USE_MKL
12 # i f d e f _MAC_OS
13 #include / / part of Accelerate Framework
14 # endif
15 # i f d e f _LINUX
16 extern "C" {
17 #include
18 }
19 # endif
20 #else
Ôº¼ ½º
Ôº¾ ½º
21 #include
Ôº½ ½º