1 *> \brief \b DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
3 * =========== DOCUMENTATION ===========
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21 * SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, M, N
26 * .. Array Arguments ..
27 * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
28 * $ TAUQ( * ), WORK( * )
37 *> DGEBD2 reduces a real general m by n matrix A to upper or lower
38 *> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
40 *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
49 *> The number of rows in the matrix A. M >= 0.
55 *> The number of columns in the matrix A. N >= 0.
60 *> A is DOUBLE PRECISION array, dimension (LDA,N)
61 *> On entry, the m by n general matrix to be reduced.
63 *> if m >= n, the diagonal and the first superdiagonal are
64 *> overwritten with the upper bidiagonal matrix B; the
65 *> elements below the diagonal, with the array TAUQ, represent
66 *> the orthogonal matrix Q as a product of elementary
67 *> reflectors, and the elements above the first superdiagonal,
68 *> with the array TAUP, represent the orthogonal matrix P as
69 *> a product of elementary reflectors;
70 *> if m < n, the diagonal and the first subdiagonal are
71 *> overwritten with the lower bidiagonal matrix B; the
72 *> elements below the first subdiagonal, with the array TAUQ,
73 *> represent the orthogonal matrix Q as a product of
74 *> elementary reflectors, and the elements above the diagonal,
75 *> with the array TAUP, represent the orthogonal matrix P as
76 *> a product of elementary reflectors.
77 *> See Further Details.
83 *> The leading dimension of the array A. LDA >= max(1,M).
88 *> D is DOUBLE PRECISION array, dimension (min(M,N))
89 *> The diagonal elements of the bidiagonal matrix B:
95 *> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
96 *> The off-diagonal elements of the bidiagonal matrix B:
97 *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
98 *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
103 *> TAUQ is DOUBLE PRECISION array dimension (min(M,N))
104 *> The scalar factors of the elementary reflectors which
105 *> represent the orthogonal matrix Q. See Further Details.
110 *> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
111 *> The scalar factors of the elementary reflectors which
112 *> represent the orthogonal matrix P. See Further Details.
117 *> WORK is DOUBLE PRECISION array, dimension (max(M,N))
123 *> = 0: successful exit.
124 *> < 0: if INFO = -i, the i-th argument had an illegal value.
130 *> \author Univ. of Tennessee
131 *> \author Univ. of California Berkeley
132 *> \author Univ. of Colorado Denver
135 *> \date September 2012
137 *> \ingroup doubleGEcomputational
139 *> \par Further Details:
140 * =====================
144 *> The matrices Q and P are represented as products of elementary
149 *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
151 *> Each H(i) and G(i) has the form:
153 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
155 *> where tauq and taup are real scalars, and v and u are real vectors;
156 *> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
157 *> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
158 *> tauq is stored in TAUQ(i) and taup in TAUP(i).
162 *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
164 *> Each H(i) and G(i) has the form:
166 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
168 *> where tauq and taup are real scalars, and v and u are real vectors;
169 *> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
170 *> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
171 *> tauq is stored in TAUQ(i) and taup in TAUP(i).
173 *> The contents of A on exit are illustrated by the following examples:
175 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
177 *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
178 *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
179 *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
180 *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
181 *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
182 *> ( v1 v2 v3 v4 v5 )
184 *> where d and e denote diagonal and off-diagonal elements of B, vi
185 *> denotes an element of the vector defining H(i), and ui an element of
186 *> the vector defining G(i).
189 * =====================================================================
190 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
192 * -- LAPACK computational routine (version 3.4.2) --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197 * .. Scalar Arguments ..
198 INTEGER INFO, LDA, M, N
200 * .. Array Arguments ..
201 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
202 $ TAUQ( * ), WORK( * )
205 * =====================================================================
208 DOUBLE PRECISION ZERO, ONE
209 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
211 * .. Local Scalars ..
214 * .. External Subroutines ..
215 EXTERNAL DLARF, DLARFG, XERBLA
217 * .. Intrinsic Functions ..
220 * .. Executable Statements ..
222 * Test the input parameters
227 ELSE IF( N.LT.0 ) THEN
229 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
233 CALL XERBLA( 'DGEBD2', -INFO )
239 * Reduce to upper bidiagonal form
243 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
245 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
250 * Apply H(i) to A(i:m,i+1:n) from the left
253 $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ),
254 $ A( I, I+1 ), LDA, WORK )
259 * Generate elementary reflector G(i) to annihilate
262 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
267 * Apply G(i) to A(i+1:m,i+1:n) from the right
269 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
270 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
278 * Reduce to lower bidiagonal form
282 * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
284 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
289 * Apply G(i) to A(i+1:m,i:n) from the right
292 $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
293 $ TAUP( I ), A( I+1, I ), LDA, WORK )
298 * Generate elementary reflector H(i) to annihilate
301 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
306 * Apply H(i) to A(i+1:m,i+1:n) from the left
308 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ),
309 $ A( I+1, I+1 ), LDA, WORK )