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CHAPTER 5. LAPACK AND THE BLAS |
dsyr2k.
Directly after the loop there is a call to the ‘unblocked dsytrd’ named dsytd2 to deal with the first/last few (<NB) rows/columns of the matrix. This excerpt concerns the situation when the upper triangle of the matrix A is stored. In that routine the mentioned loop looks very much the way we derived the formulae.
ELSE
*
*Reduce the lower triangle of A
DO 20 I = 1, N - 1
*Generate elementary reflector H(i) = I - tau * v * v’
*to annihilate A(i+2:n,i)
* |
|
|
|
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, |
|
$ |
|
TAUI ) |
|
E( I ) = A( I+1, I ) |
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* |
|
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IF( TAUI.NE.ZERO ) THEN |
|
* |
|
|
* |
Apply H(i) from both sides to A(i+1:n,i+1:n) |
|
* |
|
|
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A( I+1, I ) = ONE |
|
* |
|
|
* |
Compute |
x := tau * A * v storing y in TAU(i:n-1) |
* |
|
|
|
CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, |
|
$ |
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A( I+1, I ), 1, ZERO, TAU( I ), 1 ) |
* |
|
|
* |
Compute |
w := x - 1/2 * tau * (x’*v) * v |
* |
|
|
|
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), |
|
$ |
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1 ) |
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CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) |
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* |
|
|
* |
Apply the transformation as a rank-2 update: |
|
* |
A := A - v * w’ - w * v’ |
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* |
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|
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CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, |
|
$ |
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A( I+1, I+1 ), LDA ) |
* |
|
|
A( I+1, I ) = E( I )
END IF
D( I ) = A( I, I )
TAU( I ) = TAUI
20CONTINUE
D( N ) = A( N, N )
END IF
Bibliography
[1]E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. D. CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV,
AND D. SORENSEN, LAPACK Users’ Guide - Release 2.0, SIAM, Philadelphia, PA, 1994. (Software and guide are available from Netlib at URL http://www.netlib.org/lapack/).
BIBLIOGRAPHY |
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S. BLACKFORD, J. CHOI, |
A. CLEARY, E. D’AZEVEDO, J. |
DEMMEL, |
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DHILLON, J. DONGARRA, |
S. HAMMARLING, G. HENRY, A. |
PETITET, |
K. STANLEY, D. WALKER, AND R. C. WHALEY, ScaLAPACK Users’ Guide, SIAM, Philadelphia, PA, 1997. (Software and guide are available at URL http://www.netlib.org/scalapack/).
[3]J. J. DONGARRA, J. R. BUNCH, C. B. MOLER, AND G. W. STEWART, LINPACK
Users’ Guide, SIAM, Philadelphia, PA, 1979.
[4]J. J. DONGARRA, J. D. CROZ, I. DUFF, AND S. HAMMARLING, A proposal for a
set of level 3 basic linear algebra subprograms, ACM SIGNUM Newsletter, 22 (1987).
[5], A set of level 3 basic linear algebra subprograms, ACM Trans. Math. Softw., 16 (1990), pp. 1–17.
[6]J. J. DONGARRA, J. DU CROZ, S. HAMMARLING, AND R. J. HANSON, An extended
set of fortran basic linear algebra subprograms, ACM Trans. Math. Softw., 14 (1988),
pp.1–17.
[7], An extended set of fortran basic linear algebra subprograms: Model implementation and test programs, ACM Transactions on Mathematical Software, 14 (1988),
pp.18–32.
[8]B. S. GARBOW, J. M. BOYLE, J. J. DONGARRA, AND C. B. MOLER, Matrix Eigen-
system Routines – EISPACK Guide Extension, Lecture Notes in Computer Science 51, Springer-Verlag, Berlin, 1977.
[9]C. LAWSON, R. HANSON, D. KINCAID, AND F. KROGH, Basic linear algebra sub-
programs for Fortran usage, ACM Trans. Math. Softw., 5 (1979), pp. 308–325.
[10]B. T. SMITH, J. M. BOYLE, J. J. DONGARRA, B. S. GARBOW, Y. IKEBE, V. C. KLEMA, AND C. B. MOLER, Matrix Eigensystem Routines – EISPACK Guide, Lecture Notes in Computer Science 6, Springer-Verlag, Berlin, 2nd ed., 1976.
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CHAPTER 5. LAPACK AND THE BLAS |