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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DGGHRD 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DGGHRD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgghrd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgghrd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgghrd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 00022 * LDQ, Z, LDZ, INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * CHARACTER COMPQ, COMPZ 00026 * INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N 00027 * .. 00028 * .. Array Arguments .. 00029 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00030 * $ Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> DGGHRD reduces a pair of real matrices (A,B) to generalized upper 00040 *> Hessenberg form using orthogonal transformations, where A is a 00041 *> general matrix and B is upper triangular. The form of the 00042 *> generalized eigenvalue problem is 00043 *> A*x = lambda*B*x, 00044 *> and B is typically made upper triangular by computing its QR 00045 *> factorization and moving the orthogonal matrix Q to the left side 00046 *> of the equation. 00047 *> 00048 *> This subroutine simultaneously reduces A to a Hessenberg matrix H: 00049 *> Q**T*A*Z = H 00050 *> and transforms B to another upper triangular matrix T: 00051 *> Q**T*B*Z = T 00052 *> in order to reduce the problem to its standard form 00053 *> H*y = lambda*T*y 00054 *> where y = Z**T*x. 00055 *> 00056 *> The orthogonal matrices Q and Z are determined as products of Givens 00057 *> rotations. They may either be formed explicitly, or they may be 00058 *> postmultiplied into input matrices Q1 and Z1, so that 00059 *> 00060 *> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T 00061 *> 00062 *> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T 00063 *> 00064 *> If Q1 is the orthogonal matrix from the QR factorization of B in the 00065 *> original equation A*x = lambda*B*x, then DGGHRD reduces the original 00066 *> problem to generalized Hessenberg form. 00067 *> \endverbatim 00068 * 00069 * Arguments: 00070 * ========== 00071 * 00072 *> \param[in] COMPQ 00073 *> \verbatim 00074 *> COMPQ is CHARACTER*1 00075 *> = 'N': do not compute Q; 00076 *> = 'I': Q is initialized to the unit matrix, and the 00077 *> orthogonal matrix Q is returned; 00078 *> = 'V': Q must contain an orthogonal matrix Q1 on entry, 00079 *> and the product Q1*Q is returned. 00080 *> \endverbatim 00081 *> 00082 *> \param[in] COMPZ 00083 *> \verbatim 00084 *> COMPZ is CHARACTER*1 00085 *> = 'N': do not compute Z; 00086 *> = 'I': Z is initialized to the unit matrix, and the 00087 *> orthogonal matrix Z is returned; 00088 *> = 'V': Z must contain an orthogonal matrix Z1 on entry, 00089 *> and the product Z1*Z is returned. 00090 *> \endverbatim 00091 *> 00092 *> \param[in] N 00093 *> \verbatim 00094 *> N is INTEGER 00095 *> The order of the matrices A and B. N >= 0. 00096 *> \endverbatim 00097 *> 00098 *> \param[in] ILO 00099 *> \verbatim 00100 *> ILO is INTEGER 00101 *> \endverbatim 00102 *> 00103 *> \param[in] IHI 00104 *> \verbatim 00105 *> IHI is INTEGER 00106 *> 00107 *> ILO and IHI mark the rows and columns of A which are to be 00108 *> reduced. It is assumed that A is already upper triangular 00109 *> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are 00110 *> normally set by a previous call to DGGBAL; otherwise they 00111 *> should be set to 1 and N respectively. 00112 *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 00113 *> \endverbatim 00114 *> 00115 *> \param[in,out] A 00116 *> \verbatim 00117 *> A is DOUBLE PRECISION array, dimension (LDA, N) 00118 *> On entry, the N-by-N general matrix to be reduced. 00119 *> On exit, the upper triangle and the first subdiagonal of A 00120 *> are overwritten with the upper Hessenberg matrix H, and the 00121 *> rest is set to zero. 00122 *> \endverbatim 00123 *> 00124 *> \param[in] LDA 00125 *> \verbatim 00126 *> LDA is INTEGER 00127 *> The leading dimension of the array A. LDA >= max(1,N). 00128 *> \endverbatim 00129 *> 00130 *> \param[in,out] B 00131 *> \verbatim 00132 *> B is DOUBLE PRECISION array, dimension (LDB, N) 00133 *> On entry, the N-by-N upper triangular matrix B. 00134 *> On exit, the upper triangular matrix T = Q**T B Z. The 00135 *> elements below the diagonal are set to zero. 00136 *> \endverbatim 00137 *> 00138 *> \param[in] LDB 00139 *> \verbatim 00140 *> LDB is INTEGER 00141 *> The leading dimension of the array B. LDB >= max(1,N). 00142 *> \endverbatim 00143 *> 00144 *> \param[in,out] Q 00145 *> \verbatim 00146 *> Q is DOUBLE PRECISION array, dimension (LDQ, N) 00147 *> On entry, if COMPQ = 'V', the orthogonal matrix Q1, 00148 *> typically from the QR factorization of B. 00149 *> On exit, if COMPQ='I', the orthogonal matrix Q, and if 00150 *> COMPQ = 'V', the product Q1*Q. 00151 *> Not referenced if COMPQ='N'. 00152 *> \endverbatim 00153 *> 00154 *> \param[in] LDQ 00155 *> \verbatim 00156 *> LDQ is INTEGER 00157 *> The leading dimension of the array Q. 00158 *> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. 00159 *> \endverbatim 00160 *> 00161 *> \param[in,out] Z 00162 *> \verbatim 00163 *> Z is DOUBLE PRECISION array, dimension (LDZ, N) 00164 *> On entry, if COMPZ = 'V', the orthogonal matrix Z1. 00165 *> On exit, if COMPZ='I', the orthogonal matrix Z, and if 00166 *> COMPZ = 'V', the product Z1*Z. 00167 *> Not referenced if COMPZ='N'. 00168 *> \endverbatim 00169 *> 00170 *> \param[in] LDZ 00171 *> \verbatim 00172 *> LDZ is INTEGER 00173 *> The leading dimension of the array Z. 00174 *> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. 00175 *> \endverbatim 00176 *> 00177 *> \param[out] INFO 00178 *> \verbatim 00179 *> INFO is INTEGER 00180 *> = 0: successful exit. 00181 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00182 *> \endverbatim 00183 * 00184 * Authors: 00185 * ======== 00186 * 00187 *> \author Univ. of Tennessee 00188 *> \author Univ. of California Berkeley 00189 *> \author Univ. of Colorado Denver 00190 *> \author NAG Ltd. 00191 * 00192 *> \date November 2011 00193 * 00194 *> \ingroup doubleOTHERcomputational 00195 * 00196 *> \par Further Details: 00197 * ===================== 00198 *> 00199 *> \verbatim 00200 *> 00201 *> This routine reduces A to Hessenberg and B to triangular form by 00202 *> an unblocked reduction, as described in _Matrix_Computations_, 00203 *> by Golub and Van Loan (Johns Hopkins Press.) 00204 *> \endverbatim 00205 *> 00206 * ===================================================================== 00207 SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 00208 $ LDQ, Z, LDZ, INFO ) 00209 * 00210 * -- LAPACK computational routine (version 3.4.0) -- 00211 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00212 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00213 * November 2011 00214 * 00215 * .. Scalar Arguments .. 00216 CHARACTER COMPQ, COMPZ 00217 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N 00218 * .. 00219 * .. Array Arguments .. 00220 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00221 $ Z( LDZ, * ) 00222 * .. 00223 * 00224 * ===================================================================== 00225 * 00226 * .. Parameters .. 00227 DOUBLE PRECISION ONE, ZERO 00228 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 00229 * .. 00230 * .. Local Scalars .. 00231 LOGICAL ILQ, ILZ 00232 INTEGER ICOMPQ, ICOMPZ, JCOL, JROW 00233 DOUBLE PRECISION C, S, TEMP 00234 * .. 00235 * .. External Functions .. 00236 LOGICAL LSAME 00237 EXTERNAL LSAME 00238 * .. 00239 * .. External Subroutines .. 00240 EXTERNAL DLARTG, DLASET, DROT, XERBLA 00241 * .. 00242 * .. Intrinsic Functions .. 00243 INTRINSIC MAX 00244 * .. 00245 * .. Executable Statements .. 00246 * 00247 * Decode COMPQ 00248 * 00249 IF( LSAME( COMPQ, 'N' ) ) THEN 00250 ILQ = .FALSE. 00251 ICOMPQ = 1 00252 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN 00253 ILQ = .TRUE. 00254 ICOMPQ = 2 00255 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN 00256 ILQ = .TRUE. 00257 ICOMPQ = 3 00258 ELSE 00259 ICOMPQ = 0 00260 END IF 00261 * 00262 * Decode COMPZ 00263 * 00264 IF( LSAME( COMPZ, 'N' ) ) THEN 00265 ILZ = .FALSE. 00266 ICOMPZ = 1 00267 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00268 ILZ = .TRUE. 00269 ICOMPZ = 2 00270 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00271 ILZ = .TRUE. 00272 ICOMPZ = 3 00273 ELSE 00274 ICOMPZ = 0 00275 END IF 00276 * 00277 * Test the input parameters. 00278 * 00279 INFO = 0 00280 IF( ICOMPQ.LE.0 ) THEN 00281 INFO = -1 00282 ELSE IF( ICOMPZ.LE.0 ) THEN 00283 INFO = -2 00284 ELSE IF( N.LT.0 ) THEN 00285 INFO = -3 00286 ELSE IF( ILO.LT.1 ) THEN 00287 INFO = -4 00288 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN 00289 INFO = -5 00290 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00291 INFO = -7 00292 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00293 INFO = -9 00294 ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN 00295 INFO = -11 00296 ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN 00297 INFO = -13 00298 END IF 00299 IF( INFO.NE.0 ) THEN 00300 CALL XERBLA( 'DGGHRD', -INFO ) 00301 RETURN 00302 END IF 00303 * 00304 * Initialize Q and Z if desired. 00305 * 00306 IF( ICOMPQ.EQ.3 ) 00307 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) 00308 IF( ICOMPZ.EQ.3 ) 00309 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) 00310 * 00311 * Quick return if possible 00312 * 00313 IF( N.LE.1 ) 00314 $ RETURN 00315 * 00316 * Zero out lower triangle of B 00317 * 00318 DO 20 JCOL = 1, N - 1 00319 DO 10 JROW = JCOL + 1, N 00320 B( JROW, JCOL ) = ZERO 00321 10 CONTINUE 00322 20 CONTINUE 00323 * 00324 * Reduce A and B 00325 * 00326 DO 40 JCOL = ILO, IHI - 2 00327 * 00328 DO 30 JROW = IHI, JCOL + 2, -1 00329 * 00330 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL) 00331 * 00332 TEMP = A( JROW-1, JCOL ) 00333 CALL DLARTG( TEMP, A( JROW, JCOL ), C, S, 00334 $ A( JROW-1, JCOL ) ) 00335 A( JROW, JCOL ) = ZERO 00336 CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA, 00337 $ A( JROW, JCOL+1 ), LDA, C, S ) 00338 CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB, 00339 $ B( JROW, JROW-1 ), LDB, C, S ) 00340 IF( ILQ ) 00341 $ CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, S ) 00342 * 00343 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1) 00344 * 00345 TEMP = B( JROW, JROW ) 00346 CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S, 00347 $ B( JROW, JROW ) ) 00348 B( JROW, JROW-1 ) = ZERO 00349 CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S ) 00350 CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C, 00351 $ S ) 00352 IF( ILZ ) 00353 $ CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S ) 00354 30 CONTINUE 00355 40 CONTINUE 00356 * 00357 RETURN 00358 * 00359 * End of DGGHRD 00360 * 00361 END