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authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
commitbaba851215b44ac3b60b9248eb02bcce7eb76247 (patch)
tree8c0f5c006875532a30d4409f5e94b0f310ff00a7 /SRC/zgebd2.f
Move LAPACK trunk into position.
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+ SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
+*
+* -- LAPACK routine (version 3.1) --
+* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+* November 2006
+*
+* .. Scalar Arguments ..
+ INTEGER INFO, LDA, M, N
+* ..
+* .. Array Arguments ..
+ DOUBLE PRECISION D( * ), E( * )
+ COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
+* ..
+*
+* Purpose
+* =======
+*
+* ZGEBD2 reduces a complex general m by n matrix A to upper or lower
+* real bidiagonal form B by a unitary transformation: Q' * A * P = B.
+*
+* If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
+*
+* Arguments
+* =========
+*
+* M (input) INTEGER
+* The number of rows in the matrix A. M >= 0.
+*
+* N (input) INTEGER
+* The number of columns in the matrix A. N >= 0.
+*
+* A (input/output) COMPLEX*16 array, dimension (LDA,N)
+* On entry, the m by n general matrix to be reduced.
+* On exit,
+* if m >= n, the diagonal and the first superdiagonal are
+* overwritten with the upper bidiagonal matrix B; the
+* elements below the diagonal, with the array TAUQ, represent
+* the unitary matrix Q as a product of elementary
+* reflectors, and the elements above the first superdiagonal,
+* with the array TAUP, represent the unitary matrix P as
+* a product of elementary reflectors;
+* if m < n, the diagonal and the first subdiagonal are
+* overwritten with the lower bidiagonal matrix B; the
+* elements below the first subdiagonal, with the array TAUQ,
+* represent the unitary matrix Q as a product of
+* elementary reflectors, and the elements above the diagonal,
+* with the array TAUP, represent the unitary matrix P as
+* a product of elementary reflectors.
+* See Further Details.
+*
+* LDA (input) INTEGER
+* The leading dimension of the array A. LDA >= max(1,M).
+*
+* D (output) DOUBLE PRECISION array, dimension (min(M,N))
+* The diagonal elements of the bidiagonal matrix B:
+* D(i) = A(i,i).
+*
+* E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
+* The off-diagonal elements of the bidiagonal matrix B:
+* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
+* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
+*
+* TAUQ (output) COMPLEX*16 array dimension (min(M,N))
+* The scalar factors of the elementary reflectors which
+* represent the unitary matrix Q. See Further Details.
+*
+* TAUP (output) COMPLEX*16 array, dimension (min(M,N))
+* The scalar factors of the elementary reflectors which
+* represent the unitary matrix P. See Further Details.
+*
+* WORK (workspace) COMPLEX*16 array, dimension (max(M,N))
+*
+* INFO (output) INTEGER
+* = 0: successful exit
+* < 0: if INFO = -i, the i-th argument had an illegal value.
+*
+* Further Details
+* ===============
+*
+* The matrices Q and P are represented as products of elementary
+* reflectors:
+*
+* If m >= n,
+*
+* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
+*
+* Each H(i) and G(i) has the form:
+*
+* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
+*
+* where tauq and taup are complex scalars, and v and u are complex
+* vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
+* A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
+* A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
+*
+* If m < n,
+*
+* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
+*
+* Each H(i) and G(i) has the form:
+*
+* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
+*
+* where tauq and taup are complex scalars, v and u are complex vectors;
+* v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
+* u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
+* tauq is stored in TAUQ(i) and taup in TAUP(i).
+*
+* The contents of A on exit are illustrated by the following examples:
+*
+* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
+*
+* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
+* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
+* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
+* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
+* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
+* ( v1 v2 v3 v4 v5 )
+*
+* where d and e denote diagonal and off-diagonal elements of B, vi
+* denotes an element of the vector defining H(i), and ui an element of
+* the vector defining G(i).
+*
+* =====================================================================
+*
+* .. Parameters ..
+ COMPLEX*16 ZERO, ONE
+ PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
+ $ ONE = ( 1.0D+0, 0.0D+0 ) )
+* ..
+* .. Local Scalars ..
+ INTEGER I
+ COMPLEX*16 ALPHA
+* ..
+* .. External Subroutines ..
+ EXTERNAL XERBLA, ZLACGV, ZLARF, ZLARFG
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC DCONJG, MAX, MIN
+* ..
+* .. Executable Statements ..
+*
+* Test the input parameters
+*
+ INFO = 0
+ IF( M.LT.0 ) THEN
+ INFO = -1
+ ELSE IF( N.LT.0 ) THEN
+ INFO = -2
+ ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
+ INFO = -4
+ END IF
+ IF( INFO.LT.0 ) THEN
+ CALL XERBLA( 'ZGEBD2', -INFO )
+ RETURN
+ END IF
+*
+ IF( M.GE.N ) THEN
+*
+* Reduce to upper bidiagonal form
+*
+ DO 10 I = 1, N
+*
+* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
+*
+ ALPHA = A( I, I )
+ CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
+ $ TAUQ( I ) )
+ D( I ) = ALPHA
+ A( I, I ) = ONE
+*
+* Apply H(i)' to A(i:m,i+1:n) from the left
+*
+ IF( I.LT.N )
+ $ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
+ $ DCONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK )
+ A( I, I ) = D( I )
+*
+ IF( I.LT.N ) THEN
+*
+* Generate elementary reflector G(i) to annihilate
+* A(i,i+2:n)
+*
+ CALL ZLACGV( N-I, A( I, I+1 ), LDA )
+ ALPHA = A( I, I+1 )
+ CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
+ $ TAUP( I ) )
+ E( I ) = ALPHA
+ A( I, I+1 ) = ONE
+*
+* Apply G(i) to A(i+1:m,i+1:n) from the right
+*
+ CALL ZLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
+ $ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
+ CALL ZLACGV( N-I, A( I, I+1 ), LDA )
+ A( I, I+1 ) = E( I )
+ ELSE
+ TAUP( I ) = ZERO
+ END IF
+ 10 CONTINUE
+ ELSE
+*
+* Reduce to lower bidiagonal form
+*
+ DO 20 I = 1, M
+*
+* Generate elementary reflector G(i) to annihilate A(i,i+1:n)
+*
+ CALL ZLACGV( N-I+1, A( I, I ), LDA )
+ ALPHA = A( I, I )
+ CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
+ $ TAUP( I ) )
+ D( I ) = ALPHA
+ A( I, I ) = ONE
+*
+* Apply G(i) to A(i+1:m,i:n) from the right
+*
+ IF( I.LT.M )
+ $ CALL ZLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
+ $ TAUP( I ), A( I+1, I ), LDA, WORK )
+ CALL ZLACGV( N-I+1, A( I, I ), LDA )
+ A( I, I ) = D( I )
+*
+ IF( I.LT.M ) THEN
+*
+* Generate elementary reflector H(i) to annihilate
+* A(i+2:m,i)
+*
+ ALPHA = A( I+1, I )
+ CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
+ $ TAUQ( I ) )
+ E( I ) = ALPHA
+ A( I+1, I ) = ONE
+*
+* Apply H(i)' to A(i+1:m,i+1:n) from the left
+*
+ CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1,
+ $ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,
+ $ WORK )
+ A( I+1, I ) = E( I )
+ ELSE
+ TAUQ( I ) = ZERO
+ END IF
+ 20 CONTINUE
+ END IF
+ RETURN
+*
+* End of ZGEBD2
+*
+ END