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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DGEBD2 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DGEBD2 + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebd2.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebd2.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebd2.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 00022 * 00023 * .. Scalar Arguments .. 00024 * INTEGER INFO, LDA, M, N 00025 * .. 00026 * .. Array Arguments .. 00027 * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), 00028 * $ TAUQ( * ), WORK( * ) 00029 * .. 00030 * 00031 * 00032 *> \par Purpose: 00033 * ============= 00034 *> 00035 *> \verbatim 00036 *> 00037 *> DGEBD2 reduces a real general m by n matrix A to upper or lower 00038 *> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. 00039 *> 00040 *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 00041 *> \endverbatim 00042 * 00043 * Arguments: 00044 * ========== 00045 * 00046 *> \param[in] M 00047 *> \verbatim 00048 *> M is INTEGER 00049 *> The number of rows in the matrix A. M >= 0. 00050 *> \endverbatim 00051 *> 00052 *> \param[in] N 00053 *> \verbatim 00054 *> N is INTEGER 00055 *> The number of columns in the matrix A. N >= 0. 00056 *> \endverbatim 00057 *> 00058 *> \param[in,out] A 00059 *> \verbatim 00060 *> A is DOUBLE PRECISION array, dimension (LDA,N) 00061 *> On entry, the m by n general matrix to be reduced. 00062 *> On exit, 00063 *> if m >= n, the diagonal and the first superdiagonal are 00064 *> overwritten with the upper bidiagonal matrix B; the 00065 *> elements below the diagonal, with the array TAUQ, represent 00066 *> the orthogonal matrix Q as a product of elementary 00067 *> reflectors, and the elements above the first superdiagonal, 00068 *> with the array TAUP, represent the orthogonal matrix P as 00069 *> a product of elementary reflectors; 00070 *> if m < n, the diagonal and the first subdiagonal are 00071 *> overwritten with the lower bidiagonal matrix B; the 00072 *> elements below the first subdiagonal, with the array TAUQ, 00073 *> represent the orthogonal matrix Q as a product of 00074 *> elementary reflectors, and the elements above the diagonal, 00075 *> with the array TAUP, represent the orthogonal matrix P as 00076 *> a product of elementary reflectors. 00077 *> See Further Details. 00078 *> \endverbatim 00079 *> 00080 *> \param[in] LDA 00081 *> \verbatim 00082 *> LDA is INTEGER 00083 *> The leading dimension of the array A. LDA >= max(1,M). 00084 *> \endverbatim 00085 *> 00086 *> \param[out] D 00087 *> \verbatim 00088 *> D is DOUBLE PRECISION array, dimension (min(M,N)) 00089 *> The diagonal elements of the bidiagonal matrix B: 00090 *> D(i) = A(i,i). 00091 *> \endverbatim 00092 *> 00093 *> \param[out] E 00094 *> \verbatim 00095 *> E is DOUBLE PRECISION array, dimension (min(M,N)-1) 00096 *> The off-diagonal elements of the bidiagonal matrix B: 00097 *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 00098 *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 00099 *> \endverbatim 00100 *> 00101 *> \param[out] TAUQ 00102 *> \verbatim 00103 *> TAUQ is DOUBLE PRECISION array dimension (min(M,N)) 00104 *> The scalar factors of the elementary reflectors which 00105 *> represent the orthogonal matrix Q. See Further Details. 00106 *> \endverbatim 00107 *> 00108 *> \param[out] TAUP 00109 *> \verbatim 00110 *> TAUP is DOUBLE PRECISION array, dimension (min(M,N)) 00111 *> The scalar factors of the elementary reflectors which 00112 *> represent the orthogonal matrix P. See Further Details. 00113 *> \endverbatim 00114 *> 00115 *> \param[out] WORK 00116 *> \verbatim 00117 *> WORK is DOUBLE PRECISION array, dimension (max(M,N)) 00118 *> \endverbatim 00119 *> 00120 *> \param[out] INFO 00121 *> \verbatim 00122 *> INFO is INTEGER 00123 *> = 0: successful exit. 00124 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00125 *> \endverbatim 00126 * 00127 * Authors: 00128 * ======== 00129 * 00130 *> \author Univ. of Tennessee 00131 *> \author Univ. of California Berkeley 00132 *> \author Univ. of Colorado Denver 00133 *> \author NAG Ltd. 00134 * 00135 *> \date November 2011 00136 * 00137 *> \ingroup doubleGEcomputational 00138 * 00139 *> \par Further Details: 00140 * ===================== 00141 *> 00142 *> \verbatim 00143 *> 00144 *> The matrices Q and P are represented as products of elementary 00145 *> reflectors: 00146 *> 00147 *> If m >= n, 00148 *> 00149 *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 00150 *> 00151 *> Each H(i) and G(i) has the form: 00152 *> 00153 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T 00154 *> 00155 *> where tauq and taup are real scalars, and v and u are real vectors; 00156 *> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); 00157 *> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); 00158 *> tauq is stored in TAUQ(i) and taup in TAUP(i). 00159 *> 00160 *> If m < n, 00161 *> 00162 *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 00163 *> 00164 *> Each H(i) and G(i) has the form: 00165 *> 00166 *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T 00167 *> 00168 *> where tauq and taup are real scalars, and v and u are real vectors; 00169 *> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 00170 *> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 00171 *> tauq is stored in TAUQ(i) and taup in TAUP(i). 00172 *> 00173 *> The contents of A on exit are illustrated by the following examples: 00174 *> 00175 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00176 *> 00177 *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 00178 *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 00179 *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 00180 *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 00181 *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 00182 *> ( v1 v2 v3 v4 v5 ) 00183 *> 00184 *> where d and e denote diagonal and off-diagonal elements of B, vi 00185 *> denotes an element of the vector defining H(i), and ui an element of 00186 *> the vector defining G(i). 00187 *> \endverbatim 00188 *> 00189 * ===================================================================== 00190 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 00191 * 00192 * -- LAPACK computational routine (version 3.4.0) -- 00193 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00195 * November 2011 00196 * 00197 * .. Scalar Arguments .. 00198 INTEGER INFO, LDA, M, N 00199 * .. 00200 * .. Array Arguments .. 00201 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), 00202 $ TAUQ( * ), WORK( * ) 00203 * .. 00204 * 00205 * ===================================================================== 00206 * 00207 * .. Parameters .. 00208 DOUBLE PRECISION ZERO, ONE 00209 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00210 * .. 00211 * .. Local Scalars .. 00212 INTEGER I 00213 * .. 00214 * .. External Subroutines .. 00215 EXTERNAL DLARF, DLARFG, XERBLA 00216 * .. 00217 * .. Intrinsic Functions .. 00218 INTRINSIC MAX, MIN 00219 * .. 00220 * .. Executable Statements .. 00221 * 00222 * Test the input parameters 00223 * 00224 INFO = 0 00225 IF( M.LT.0 ) THEN 00226 INFO = -1 00227 ELSE IF( N.LT.0 ) THEN 00228 INFO = -2 00229 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00230 INFO = -4 00231 END IF 00232 IF( INFO.LT.0 ) THEN 00233 CALL XERBLA( 'DGEBD2', -INFO ) 00234 RETURN 00235 END IF 00236 * 00237 IF( M.GE.N ) THEN 00238 * 00239 * Reduce to upper bidiagonal form 00240 * 00241 DO 10 I = 1, N 00242 * 00243 * Generate elementary reflector H(i) to annihilate A(i+1:m,i) 00244 * 00245 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, 00246 $ TAUQ( I ) ) 00247 D( I ) = A( I, I ) 00248 A( I, I ) = ONE 00249 * 00250 * Apply H(i) to A(i:m,i+1:n) from the left 00251 * 00252 IF( I.LT.N ) 00253 $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), 00254 $ A( I, I+1 ), LDA, WORK ) 00255 A( I, I ) = D( I ) 00256 * 00257 IF( I.LT.N ) THEN 00258 * 00259 * Generate elementary reflector G(i) to annihilate 00260 * A(i,i+2:n) 00261 * 00262 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), 00263 $ LDA, TAUP( I ) ) 00264 E( I ) = A( I, I+1 ) 00265 A( I, I+1 ) = ONE 00266 * 00267 * Apply G(i) to A(i+1:m,i+1:n) from the right 00268 * 00269 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 00270 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 00271 A( I, I+1 ) = E( I ) 00272 ELSE 00273 TAUP( I ) = ZERO 00274 END IF 00275 10 CONTINUE 00276 ELSE 00277 * 00278 * Reduce to lower bidiagonal form 00279 * 00280 DO 20 I = 1, M 00281 * 00282 * Generate elementary reflector G(i) to annihilate A(i,i+1:n) 00283 * 00284 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, 00285 $ TAUP( I ) ) 00286 D( I ) = A( I, I ) 00287 A( I, I ) = ONE 00288 * 00289 * Apply G(i) to A(i+1:m,i:n) from the right 00290 * 00291 IF( I.LT.M ) 00292 $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 00293 $ TAUP( I ), A( I+1, I ), LDA, WORK ) 00294 A( I, I ) = D( I ) 00295 * 00296 IF( I.LT.M ) THEN 00297 * 00298 * Generate elementary reflector H(i) to annihilate 00299 * A(i+2:m,i) 00300 * 00301 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, 00302 $ TAUQ( I ) ) 00303 E( I ) = A( I+1, I ) 00304 A( I+1, I ) = ONE 00305 * 00306 * Apply H(i) to A(i+1:m,i+1:n) from the left 00307 * 00308 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), 00309 $ A( I+1, I+1 ), LDA, WORK ) 00310 A( I+1, I ) = E( I ) 00311 ELSE 00312 TAUQ( I ) = ZERO 00313 END IF 00314 20 CONTINUE 00315 END IF 00316 RETURN 00317 * 00318 * End of DGEBD2 00319 * 00320 END