Overview of the BLAS

The BLAS give us a standardized set of basic codes for performing operations on vectors and matrices. BLAS take advantage of the Fortran storage structure and the structure of the mathematical system wherever possible. Additionally, many computers have the BLAS library optimized to their system. Here we use five routines:

1. SCOPY: copies a vector onto another vector
2. SAXPY: adds vector (multiplied by a scalar) to vector
3. SGEMV: general matrix vector product
4. STRMV: matrix vector product when the matrix is triangular
5. STRSV: solves for triangular matrix

The prefix ``S'' denotes single precision. This prefix may be changed to ``D'', ``C'', or ``Z'', giving the routine double, complex, or double complex precision. (Of course, the declarations would also have to be changed.) It is important to note that putting double precision into single variables works, but single into double will cause errors.

If we define a(i,j) and = x(i), we can see what the code is doing:

• ALPHA = SDOT( N, X, 1, Y, 1 ) computes the inner product of two vectors and , putting the result in scalar .

  The corresponding Fortran segment is

  ALPHA = 0.0
  DO I = 1, N
    ALPHA = ALPHA  + X(I)*Y(I)
  ENDDO

• CALL SAXPY( N, ALPHA, X, 1, Y ) multiplies a vector of length by the scalar , then adds the result to the vector , putting the result in .

  The corresponding Fortran segment is

  DO I = 1, N
    Y(I) = ALPHA*X(I) + Y(I)
  ENDDO

• CALL SGEMV( 'N', M, N, ONE, A, LDA, X, 1, ONE, B, 1 ) computes the matrix-vector product plus vector , putting the resulting vector in .

  The corresponding Fortran segment:

  DO J = 1, N
     DO I = 1, M
        B(I) = A(I,J)*X(J) + B(I)
     ENDDO
  ENDDO

  This illustrates a feature of the BLAS that often requires close attention. For example, we will use this routine to compute the residual vector , where is our current approximation to the solution (merely change the fourth argument to -1.0E0). Vector will be overwritten with the residual vector; thus, if we need it later, we will first copy it to temporary storage.

• CALL STRMV( 'U', 'N', 'N', N, A, LDA, X, 1 ) computes the matrix-vector product , putting the resulting vector in , for upper triangular matrix .

  The corresponding Fortran segment is

  DO J = 1, N
     TEMP = X(J)
     DO I = 1, J
        X(I) = X(I) + TEMP*A(I,J)
     ENDDO
  ENDDO

Note that the parameters in single quotes are for descriptions such as 'U' for `UPPER TRIANGULAR', 'N' for `No Transpose'. This feature will be used extensively, resulting in storage savings (among other advantages).

The variable LDA is critical for addressing the array correctly. LDA is the leading dimension of the two-dimensional array A, that is, LDA is the declared (or allocated) number of rows of the two-dimensional array .