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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE ZGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, 00002 $ WORK, LWORK, RWORK, INFO ) 00003 * 00004 * -- LAPACK driver routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER JOBVL, JOBVR 00011 INTEGER INFO, LDA, LDVL, LDVR, LWORK, N 00012 * .. 00013 * .. Array Arguments .. 00014 DOUBLE PRECISION RWORK( * ) 00015 COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 00016 $ W( * ), WORK( * ) 00017 * .. 00018 * 00019 * Purpose 00020 * ======= 00021 * 00022 * ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the 00023 * eigenvalues and, optionally, the left and/or right eigenvectors. 00024 * 00025 * The right eigenvector v(j) of A satisfies 00026 * A * v(j) = lambda(j) * v(j) 00027 * where lambda(j) is its eigenvalue. 00028 * The left eigenvector u(j) of A satisfies 00029 * u(j)**H * A = lambda(j) * u(j)**H 00030 * where u(j)**H denotes the conjugate transpose of u(j). 00031 * 00032 * The computed eigenvectors are normalized to have Euclidean norm 00033 * equal to 1 and largest component real. 00034 * 00035 * Arguments 00036 * ========= 00037 * 00038 * JOBVL (input) CHARACTER*1 00039 * = 'N': left eigenvectors of A are not computed; 00040 * = 'V': left eigenvectors of are computed. 00041 * 00042 * JOBVR (input) CHARACTER*1 00043 * = 'N': right eigenvectors of A are not computed; 00044 * = 'V': right eigenvectors of A are computed. 00045 * 00046 * N (input) INTEGER 00047 * The order of the matrix A. N >= 0. 00048 * 00049 * A (input/output) COMPLEX*16 array, dimension (LDA,N) 00050 * On entry, the N-by-N matrix A. 00051 * On exit, A has been overwritten. 00052 * 00053 * LDA (input) INTEGER 00054 * The leading dimension of the array A. LDA >= max(1,N). 00055 * 00056 * W (output) COMPLEX*16 array, dimension (N) 00057 * W contains the computed eigenvalues. 00058 * 00059 * VL (output) COMPLEX*16 array, dimension (LDVL,N) 00060 * If JOBVL = 'V', the left eigenvectors u(j) are stored one 00061 * after another in the columns of VL, in the same order 00062 * as their eigenvalues. 00063 * If JOBVL = 'N', VL is not referenced. 00064 * u(j) = VL(:,j), the j-th column of VL. 00065 * 00066 * LDVL (input) INTEGER 00067 * The leading dimension of the array VL. LDVL >= 1; if 00068 * JOBVL = 'V', LDVL >= N. 00069 * 00070 * VR (output) COMPLEX*16 array, dimension (LDVR,N) 00071 * If JOBVR = 'V', the right eigenvectors v(j) are stored one 00072 * after another in the columns of VR, in the same order 00073 * as their eigenvalues. 00074 * If JOBVR = 'N', VR is not referenced. 00075 * v(j) = VR(:,j), the j-th column of VR. 00076 * 00077 * LDVR (input) INTEGER 00078 * The leading dimension of the array VR. LDVR >= 1; if 00079 * JOBVR = 'V', LDVR >= N. 00080 * 00081 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) 00082 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00083 * 00084 * LWORK (input) INTEGER 00085 * The dimension of the array WORK. LWORK >= max(1,2*N). 00086 * For good performance, LWORK must generally be larger. 00087 * 00088 * If LWORK = -1, then a workspace query is assumed; the routine 00089 * only calculates the optimal size of the WORK array, returns 00090 * this value as the first entry of the WORK array, and no error 00091 * message related to LWORK is issued by XERBLA. 00092 * 00093 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) 00094 * 00095 * INFO (output) INTEGER 00096 * = 0: successful exit 00097 * < 0: if INFO = -i, the i-th argument had an illegal value. 00098 * > 0: if INFO = i, the QR algorithm failed to compute all the 00099 * eigenvalues, and no eigenvectors have been computed; 00100 * elements and i+1:N of W contain eigenvalues which have 00101 * converged. 00102 * 00103 * ===================================================================== 00104 * 00105 * .. Parameters .. 00106 DOUBLE PRECISION ZERO, ONE 00107 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00108 * .. 00109 * .. Local Scalars .. 00110 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR 00111 CHARACTER SIDE 00112 INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU, 00113 $ IWRK, K, MAXWRK, MINWRK, NOUT 00114 DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM 00115 COMPLEX*16 TMP 00116 * .. 00117 * .. Local Arrays .. 00118 LOGICAL SELECT( 1 ) 00119 DOUBLE PRECISION DUM( 1 ) 00120 * .. 00121 * .. External Subroutines .. 00122 EXTERNAL DLABAD, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, ZGEHRD, 00123 $ ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC, ZUNGHR 00124 * .. 00125 * .. External Functions .. 00126 LOGICAL LSAME 00127 INTEGER IDAMAX, ILAENV 00128 DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE 00129 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE 00130 * .. 00131 * .. Intrinsic Functions .. 00132 INTRINSIC DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT 00133 * .. 00134 * .. Executable Statements .. 00135 * 00136 * Test the input arguments 00137 * 00138 INFO = 0 00139 LQUERY = ( LWORK.EQ.-1 ) 00140 WANTVL = LSAME( JOBVL, 'V' ) 00141 WANTVR = LSAME( JOBVR, 'V' ) 00142 IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN 00143 INFO = -1 00144 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN 00145 INFO = -2 00146 ELSE IF( N.LT.0 ) THEN 00147 INFO = -3 00148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00149 INFO = -5 00150 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN 00151 INFO = -8 00152 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN 00153 INFO = -10 00154 END IF 00155 * 00156 * Compute workspace 00157 * (Note: Comments in the code beginning "Workspace:" describe the 00158 * minimal amount of workspace needed at that point in the code, 00159 * as well as the preferred amount for good performance. 00160 * CWorkspace refers to complex workspace, and RWorkspace to real 00161 * workspace. NB refers to the optimal block size for the 00162 * immediately following subroutine, as returned by ILAENV. 00163 * HSWORK refers to the workspace preferred by ZHSEQR, as 00164 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, 00165 * the worst case.) 00166 * 00167 IF( INFO.EQ.0 ) THEN 00168 IF( N.EQ.0 ) THEN 00169 MINWRK = 1 00170 MAXWRK = 1 00171 ELSE 00172 MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 ) 00173 MINWRK = 2*N 00174 IF( WANTVL ) THEN 00175 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', 00176 $ ' ', N, 1, N, -1 ) ) 00177 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, 00178 $ WORK, -1, INFO ) 00179 ELSE IF( WANTVR ) THEN 00180 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', 00181 $ ' ', N, 1, N, -1 ) ) 00182 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, 00183 $ WORK, -1, INFO ) 00184 ELSE 00185 CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, 00186 $ WORK, -1, INFO ) 00187 END IF 00188 HSWORK = WORK( 1 ) 00189 MAXWRK = MAX( MAXWRK, HSWORK, MINWRK ) 00190 END IF 00191 WORK( 1 ) = MAXWRK 00192 * 00193 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 00194 INFO = -12 00195 END IF 00196 END IF 00197 * 00198 IF( INFO.NE.0 ) THEN 00199 CALL XERBLA( 'ZGEEV ', -INFO ) 00200 RETURN 00201 ELSE IF( LQUERY ) THEN 00202 RETURN 00203 END IF 00204 * 00205 * Quick return if possible 00206 * 00207 IF( N.EQ.0 ) 00208 $ RETURN 00209 * 00210 * Get machine constants 00211 * 00212 EPS = DLAMCH( 'P' ) 00213 SMLNUM = DLAMCH( 'S' ) 00214 BIGNUM = ONE / SMLNUM 00215 CALL DLABAD( SMLNUM, BIGNUM ) 00216 SMLNUM = SQRT( SMLNUM ) / EPS 00217 BIGNUM = ONE / SMLNUM 00218 * 00219 * Scale A if max element outside range [SMLNUM,BIGNUM] 00220 * 00221 ANRM = ZLANGE( 'M', N, N, A, LDA, DUM ) 00222 SCALEA = .FALSE. 00223 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00224 SCALEA = .TRUE. 00225 CSCALE = SMLNUM 00226 ELSE IF( ANRM.GT.BIGNUM ) THEN 00227 SCALEA = .TRUE. 00228 CSCALE = BIGNUM 00229 END IF 00230 IF( SCALEA ) 00231 $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) 00232 * 00233 * Balance the matrix 00234 * (CWorkspace: none) 00235 * (RWorkspace: need N) 00236 * 00237 IBAL = 1 00238 CALL ZGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR ) 00239 * 00240 * Reduce to upper Hessenberg form 00241 * (CWorkspace: need 2*N, prefer N+N*NB) 00242 * (RWorkspace: none) 00243 * 00244 ITAU = 1 00245 IWRK = ITAU + N 00246 CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), 00247 $ LWORK-IWRK+1, IERR ) 00248 * 00249 IF( WANTVL ) THEN 00250 * 00251 * Want left eigenvectors 00252 * Copy Householder vectors to VL 00253 * 00254 SIDE = 'L' 00255 CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL ) 00256 * 00257 * Generate unitary matrix in VL 00258 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 00259 * (RWorkspace: none) 00260 * 00261 CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), 00262 $ LWORK-IWRK+1, IERR ) 00263 * 00264 * Perform QR iteration, accumulating Schur vectors in VL 00265 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00266 * (RWorkspace: none) 00267 * 00268 IWRK = ITAU 00269 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL, 00270 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00271 * 00272 IF( WANTVR ) THEN 00273 * 00274 * Want left and right eigenvectors 00275 * Copy Schur vectors to VR 00276 * 00277 SIDE = 'B' 00278 CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) 00279 END IF 00280 * 00281 ELSE IF( WANTVR ) THEN 00282 * 00283 * Want right eigenvectors 00284 * Copy Householder vectors to VR 00285 * 00286 SIDE = 'R' 00287 CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR ) 00288 * 00289 * Generate unitary matrix in VR 00290 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 00291 * (RWorkspace: none) 00292 * 00293 CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), 00294 $ LWORK-IWRK+1, IERR ) 00295 * 00296 * Perform QR iteration, accumulating Schur vectors in VR 00297 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00298 * (RWorkspace: none) 00299 * 00300 IWRK = ITAU 00301 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR, 00302 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00303 * 00304 ELSE 00305 * 00306 * Compute eigenvalues only 00307 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00308 * (RWorkspace: none) 00309 * 00310 IWRK = ITAU 00311 CALL ZHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, 00312 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00313 END IF 00314 * 00315 * If INFO > 0 from ZHSEQR, then quit 00316 * 00317 IF( INFO.GT.0 ) 00318 $ GO TO 50 00319 * 00320 IF( WANTVL .OR. WANTVR ) THEN 00321 * 00322 * Compute left and/or right eigenvectors 00323 * (CWorkspace: need 2*N) 00324 * (RWorkspace: need 2*N) 00325 * 00326 IRWORK = IBAL + N 00327 CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 00328 $ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR ) 00329 END IF 00330 * 00331 IF( WANTVL ) THEN 00332 * 00333 * Undo balancing of left eigenvectors 00334 * (CWorkspace: none) 00335 * (RWorkspace: need N) 00336 * 00337 CALL ZGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL, 00338 $ IERR ) 00339 * 00340 * Normalize left eigenvectors and make largest component real 00341 * 00342 DO 20 I = 1, N 00343 SCL = ONE / DZNRM2( N, VL( 1, I ), 1 ) 00344 CALL ZDSCAL( N, SCL, VL( 1, I ), 1 ) 00345 DO 10 K = 1, N 00346 RWORK( IRWORK+K-1 ) = DBLE( VL( K, I ) )**2 + 00347 $ DIMAG( VL( K, I ) )**2 00348 10 CONTINUE 00349 K = IDAMAX( N, RWORK( IRWORK ), 1 ) 00350 TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 00351 CALL ZSCAL( N, TMP, VL( 1, I ), 1 ) 00352 VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO ) 00353 20 CONTINUE 00354 END IF 00355 * 00356 IF( WANTVR ) THEN 00357 * 00358 * Undo balancing of right eigenvectors 00359 * (CWorkspace: none) 00360 * (RWorkspace: need N) 00361 * 00362 CALL ZGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR, 00363 $ IERR ) 00364 * 00365 * Normalize right eigenvectors and make largest component real 00366 * 00367 DO 40 I = 1, N 00368 SCL = ONE / DZNRM2( N, VR( 1, I ), 1 ) 00369 CALL ZDSCAL( N, SCL, VR( 1, I ), 1 ) 00370 DO 30 K = 1, N 00371 RWORK( IRWORK+K-1 ) = DBLE( VR( K, I ) )**2 + 00372 $ DIMAG( VR( K, I ) )**2 00373 30 CONTINUE 00374 K = IDAMAX( N, RWORK( IRWORK ), 1 ) 00375 TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 00376 CALL ZSCAL( N, TMP, VR( 1, I ), 1 ) 00377 VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO ) 00378 40 CONTINUE 00379 END IF 00380 * 00381 * Undo scaling if necessary 00382 * 00383 50 CONTINUE 00384 IF( SCALEA ) THEN 00385 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), 00386 $ MAX( N-INFO, 1 ), IERR ) 00387 IF( INFO.GT.0 ) THEN 00388 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) 00389 END IF 00390 END IF 00391 * 00392 WORK( 1 ) = MAXWRK 00393 RETURN 00394 * 00395 * End of ZGEEV 00396 * 00397 END
1.7.3 
