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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b ZGGHRD 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGGHRD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgghrd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgghrd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgghrd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGGHRD( 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 * COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00030 * $ Z( LDZ, * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper 00040 *> Hessenberg form using unitary 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 unitary matrix Q to the left side 00046 *> of the equation. 00047 *> 00048 *> This subroutine simultaneously reduces A to a Hessenberg matrix H: 00049 *> Q**H*A*Z = H 00050 *> and transforms B to another upper triangular matrix T: 00051 *> Q**H*B*Z = T 00052 *> in order to reduce the problem to its standard form 00053 *> H*y = lambda*T*y 00054 *> where y = Z**H*x. 00055 *> 00056 *> The unitary 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 *> Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H 00060 *> Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H 00061 *> If Q1 is the unitary matrix from the QR factorization of B in the 00062 *> original equation A*x = lambda*B*x, then ZGGHRD reduces the original 00063 *> problem to generalized Hessenberg form. 00064 *> \endverbatim 00065 * 00066 * Arguments: 00067 * ========== 00068 * 00069 *> \param[in] COMPQ 00070 *> \verbatim 00071 *> COMPQ is CHARACTER*1 00072 *> = 'N': do not compute Q; 00073 *> = 'I': Q is initialized to the unit matrix, and the 00074 *> unitary matrix Q is returned; 00075 *> = 'V': Q must contain a unitary matrix Q1 on entry, 00076 *> and the product Q1*Q is returned. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] COMPZ 00080 *> \verbatim 00081 *> COMPZ is CHARACTER*1 00082 *> = 'N': do not compute Q; 00083 *> = 'I': Q is initialized to the unit matrix, and the 00084 *> unitary matrix Q is returned; 00085 *> = 'V': Q must contain a unitary matrix Q1 on entry, 00086 *> and the product Q1*Q is returned. 00087 *> \endverbatim 00088 *> 00089 *> \param[in] N 00090 *> \verbatim 00091 *> N is INTEGER 00092 *> The order of the matrices A and B. N >= 0. 00093 *> \endverbatim 00094 *> 00095 *> \param[in] ILO 00096 *> \verbatim 00097 *> ILO is INTEGER 00098 *> \endverbatim 00099 *> 00100 *> \param[in] IHI 00101 *> \verbatim 00102 *> IHI is INTEGER 00103 *> 00104 *> ILO and IHI mark the rows and columns of A which are to be 00105 *> reduced. It is assumed that A is already upper triangular 00106 *> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are 00107 *> normally set by a previous call to ZGGBAL; otherwise they 00108 *> should be set to 1 and N respectively. 00109 *> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. 00110 *> \endverbatim 00111 *> 00112 *> \param[in,out] A 00113 *> \verbatim 00114 *> A is COMPLEX*16 array, dimension (LDA, N) 00115 *> On entry, the N-by-N general matrix to be reduced. 00116 *> On exit, the upper triangle and the first subdiagonal of A 00117 *> are overwritten with the upper Hessenberg matrix H, and the 00118 *> rest is set to zero. 00119 *> \endverbatim 00120 *> 00121 *> \param[in] LDA 00122 *> \verbatim 00123 *> LDA is INTEGER 00124 *> The leading dimension of the array A. LDA >= max(1,N). 00125 *> \endverbatim 00126 *> 00127 *> \param[in,out] B 00128 *> \verbatim 00129 *> B is COMPLEX*16 array, dimension (LDB, N) 00130 *> On entry, the N-by-N upper triangular matrix B. 00131 *> On exit, the upper triangular matrix T = Q**H B Z. The 00132 *> elements below the diagonal are set to zero. 00133 *> \endverbatim 00134 *> 00135 *> \param[in] LDB 00136 *> \verbatim 00137 *> LDB is INTEGER 00138 *> The leading dimension of the array B. LDB >= max(1,N). 00139 *> \endverbatim 00140 *> 00141 *> \param[in,out] Q 00142 *> \verbatim 00143 *> Q is COMPLEX*16 array, dimension (LDQ, N) 00144 *> On entry, if COMPQ = 'V', the unitary matrix Q1, typically 00145 *> from the QR factorization of B. 00146 *> On exit, if COMPQ='I', the unitary matrix Q, and if 00147 *> COMPQ = 'V', the product Q1*Q. 00148 *> Not referenced if COMPQ='N'. 00149 *> \endverbatim 00150 *> 00151 *> \param[in] LDQ 00152 *> \verbatim 00153 *> LDQ is INTEGER 00154 *> The leading dimension of the array Q. 00155 *> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. 00156 *> \endverbatim 00157 *> 00158 *> \param[in,out] Z 00159 *> \verbatim 00160 *> Z is COMPLEX*16 array, dimension (LDZ, N) 00161 *> On entry, if COMPZ = 'V', the unitary matrix Z1. 00162 *> On exit, if COMPZ='I', the unitary matrix Z, and if 00163 *> COMPZ = 'V', the product Z1*Z. 00164 *> Not referenced if COMPZ='N'. 00165 *> \endverbatim 00166 *> 00167 *> \param[in] LDZ 00168 *> \verbatim 00169 *> LDZ is INTEGER 00170 *> The leading dimension of the array Z. 00171 *> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. 00172 *> \endverbatim 00173 *> 00174 *> \param[out] INFO 00175 *> \verbatim 00176 *> INFO is INTEGER 00177 *> = 0: successful exit. 00178 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00179 *> \endverbatim 00180 * 00181 * Authors: 00182 * ======== 00183 * 00184 *> \author Univ. of Tennessee 00185 *> \author Univ. of California Berkeley 00186 *> \author Univ. of Colorado Denver 00187 *> \author NAG Ltd. 00188 * 00189 *> \date November 2011 00190 * 00191 *> \ingroup complex16OTHERcomputational 00192 * 00193 *> \par Further Details: 00194 * ===================== 00195 *> 00196 *> \verbatim 00197 *> 00198 *> This routine reduces A to Hessenberg and B to triangular form by 00199 *> an unblocked reduction, as described in _Matrix_Computations_, 00200 *> by Golub and van Loan (Johns Hopkins Press). 00201 *> \endverbatim 00202 *> 00203 * ===================================================================== 00204 SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 00205 $ LDQ, Z, LDZ, INFO ) 00206 * 00207 * -- LAPACK computational routine (version 3.4.0) -- 00208 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00209 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00210 * November 2011 00211 * 00212 * .. Scalar Arguments .. 00213 CHARACTER COMPQ, COMPZ 00214 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N 00215 * .. 00216 * .. Array Arguments .. 00217 COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), 00218 $ Z( LDZ, * ) 00219 * .. 00220 * 00221 * ===================================================================== 00222 * 00223 * .. Parameters .. 00224 COMPLEX*16 CONE, CZERO 00225 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ), 00226 $ CZERO = ( 0.0D+0, 0.0D+0 ) ) 00227 * .. 00228 * .. Local Scalars .. 00229 LOGICAL ILQ, ILZ 00230 INTEGER ICOMPQ, ICOMPZ, JCOL, JROW 00231 DOUBLE PRECISION C 00232 COMPLEX*16 CTEMP, S 00233 * .. 00234 * .. External Functions .. 00235 LOGICAL LSAME 00236 EXTERNAL LSAME 00237 * .. 00238 * .. External Subroutines .. 00239 EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT 00240 * .. 00241 * .. Intrinsic Functions .. 00242 INTRINSIC DCONJG, MAX 00243 * .. 00244 * .. Executable Statements .. 00245 * 00246 * Decode COMPQ 00247 * 00248 IF( LSAME( COMPQ, 'N' ) ) THEN 00249 ILQ = .FALSE. 00250 ICOMPQ = 1 00251 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN 00252 ILQ = .TRUE. 00253 ICOMPQ = 2 00254 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN 00255 ILQ = .TRUE. 00256 ICOMPQ = 3 00257 ELSE 00258 ICOMPQ = 0 00259 END IF 00260 * 00261 * Decode COMPZ 00262 * 00263 IF( LSAME( COMPZ, 'N' ) ) THEN 00264 ILZ = .FALSE. 00265 ICOMPZ = 1 00266 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00267 ILZ = .TRUE. 00268 ICOMPZ = 2 00269 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00270 ILZ = .TRUE. 00271 ICOMPZ = 3 00272 ELSE 00273 ICOMPZ = 0 00274 END IF 00275 * 00276 * Test the input parameters. 00277 * 00278 INFO = 0 00279 IF( ICOMPQ.LE.0 ) THEN 00280 INFO = -1 00281 ELSE IF( ICOMPZ.LE.0 ) THEN 00282 INFO = -2 00283 ELSE IF( N.LT.0 ) THEN 00284 INFO = -3 00285 ELSE IF( ILO.LT.1 ) THEN 00286 INFO = -4 00287 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN 00288 INFO = -5 00289 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00290 INFO = -7 00291 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00292 INFO = -9 00293 ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN 00294 INFO = -11 00295 ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN 00296 INFO = -13 00297 END IF 00298 IF( INFO.NE.0 ) THEN 00299 CALL XERBLA( 'ZGGHRD', -INFO ) 00300 RETURN 00301 END IF 00302 * 00303 * Initialize Q and Z if desired. 00304 * 00305 IF( ICOMPQ.EQ.3 ) 00306 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) 00307 IF( ICOMPZ.EQ.3 ) 00308 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ ) 00309 * 00310 * Quick return if possible 00311 * 00312 IF( N.LE.1 ) 00313 $ RETURN 00314 * 00315 * Zero out lower triangle of B 00316 * 00317 DO 20 JCOL = 1, N - 1 00318 DO 10 JROW = JCOL + 1, N 00319 B( JROW, JCOL ) = CZERO 00320 10 CONTINUE 00321 20 CONTINUE 00322 * 00323 * Reduce A and B 00324 * 00325 DO 40 JCOL = ILO, IHI - 2 00326 * 00327 DO 30 JROW = IHI, JCOL + 2, -1 00328 * 00329 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL) 00330 * 00331 CTEMP = A( JROW-1, JCOL ) 00332 CALL ZLARTG( CTEMP, A( JROW, JCOL ), C, S, 00333 $ A( JROW-1, JCOL ) ) 00334 A( JROW, JCOL ) = CZERO 00335 CALL ZROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA, 00336 $ A( JROW, JCOL+1 ), LDA, C, S ) 00337 CALL ZROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB, 00338 $ B( JROW, JROW-1 ), LDB, C, S ) 00339 IF( ILQ ) 00340 $ CALL ZROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, 00341 $ DCONJG( S ) ) 00342 * 00343 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1) 00344 * 00345 CTEMP = B( JROW, JROW ) 00346 CALL ZLARTG( CTEMP, B( JROW, JROW-1 ), C, S, 00347 $ B( JROW, JROW ) ) 00348 B( JROW, JROW-1 ) = CZERO 00349 CALL ZROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S ) 00350 CALL ZROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C, 00351 $ S ) 00352 IF( ILZ ) 00353 $ CALL ZROT( 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 ZGGHRD 00360 * 00361 END