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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE CGEEV( 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 REAL RWORK( * ) 00015 COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 00016 $ W( * ), WORK( * ) 00017 * .. 00018 * 00019 * Purpose 00020 * ======= 00021 * 00022 * CGEEV 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 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 array, dimension (N) 00057 * W contains the computed eigenvalues. 00058 * 00059 * VL (output) COMPLEX 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 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 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) REAL 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 REAL ZERO, ONE 00107 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 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 REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM 00115 COMPLEX TMP 00116 * .. 00117 * .. Local Arrays .. 00118 LOGICAL SELECT( 1 ) 00119 REAL DUM( 1 ) 00120 * .. 00121 * .. External Subroutines .. 00122 EXTERNAL CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL, 00123 $ CSCAL, CSSCAL, CTREVC, CUNGHR, SLABAD, XERBLA 00124 * .. 00125 * .. External Functions .. 00126 LOGICAL LSAME 00127 INTEGER ILAENV, ISAMAX 00128 REAL CLANGE, SCNRM2, SLAMCH 00129 EXTERNAL LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH 00130 * .. 00131 * .. Intrinsic Functions .. 00132 INTRINSIC AIMAG, CMPLX, CONJG, MAX, REAL, 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 * 00157 * Compute workspace 00158 * (Note: Comments in the code beginning "Workspace:" describe the 00159 * minimal amount of workspace needed at that point in the code, 00160 * as well as the preferred amount for good performance. 00161 * CWorkspace refers to complex workspace, and RWorkspace to real 00162 * workspace. NB refers to the optimal block size for the 00163 * immediately following subroutine, as returned by ILAENV. 00164 * HSWORK refers to the workspace preferred by CHSEQR, as 00165 * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, 00166 * the worst case.) 00167 * 00168 IF( INFO.EQ.0 ) THEN 00169 IF( N.EQ.0 ) THEN 00170 MINWRK = 1 00171 MAXWRK = 1 00172 ELSE 00173 MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 ) 00174 MINWRK = 2*N 00175 IF( WANTVL ) THEN 00176 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR', 00177 $ ' ', N, 1, N, -1 ) ) 00178 CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, 00179 $ WORK, -1, INFO ) 00180 ELSE IF( WANTVR ) THEN 00181 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR', 00182 $ ' ', N, 1, N, -1 ) ) 00183 CALL CHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, 00184 $ WORK, -1, INFO ) 00185 ELSE 00186 CALL CHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, 00187 $ WORK, -1, INFO ) 00188 END IF 00189 HSWORK = WORK( 1 ) 00190 MAXWRK = MAX( MAXWRK, HSWORK, MINWRK ) 00191 END IF 00192 WORK( 1 ) = MAXWRK 00193 * 00194 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 00195 INFO = -12 00196 END IF 00197 END IF 00198 * 00199 IF( INFO.NE.0 ) THEN 00200 CALL XERBLA( 'CGEEV ', -INFO ) 00201 RETURN 00202 ELSE IF( LQUERY ) THEN 00203 RETURN 00204 END IF 00205 * 00206 * Quick return if possible 00207 * 00208 IF( N.EQ.0 ) 00209 $ RETURN 00210 * 00211 * Get machine constants 00212 * 00213 EPS = SLAMCH( 'P' ) 00214 SMLNUM = SLAMCH( 'S' ) 00215 BIGNUM = ONE / SMLNUM 00216 CALL SLABAD( SMLNUM, BIGNUM ) 00217 SMLNUM = SQRT( SMLNUM ) / EPS 00218 BIGNUM = ONE / SMLNUM 00219 * 00220 * Scale A if max element outside range [SMLNUM,BIGNUM] 00221 * 00222 ANRM = CLANGE( 'M', N, N, A, LDA, DUM ) 00223 SCALEA = .FALSE. 00224 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00225 SCALEA = .TRUE. 00226 CSCALE = SMLNUM 00227 ELSE IF( ANRM.GT.BIGNUM ) THEN 00228 SCALEA = .TRUE. 00229 CSCALE = BIGNUM 00230 END IF 00231 IF( SCALEA ) 00232 $ CALL CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) 00233 * 00234 * Balance the matrix 00235 * (CWorkspace: none) 00236 * (RWorkspace: need N) 00237 * 00238 IBAL = 1 00239 CALL CGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR ) 00240 * 00241 * Reduce to upper Hessenberg form 00242 * (CWorkspace: need 2*N, prefer N+N*NB) 00243 * (RWorkspace: none) 00244 * 00245 ITAU = 1 00246 IWRK = ITAU + N 00247 CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), 00248 $ LWORK-IWRK+1, IERR ) 00249 * 00250 IF( WANTVL ) THEN 00251 * 00252 * Want left eigenvectors 00253 * Copy Householder vectors to VL 00254 * 00255 SIDE = 'L' 00256 CALL CLACPY( 'L', N, N, A, LDA, VL, LDVL ) 00257 * 00258 * Generate unitary matrix in VL 00259 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 00260 * (RWorkspace: none) 00261 * 00262 CALL CUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), 00263 $ LWORK-IWRK+1, IERR ) 00264 * 00265 * Perform QR iteration, accumulating Schur vectors in VL 00266 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00267 * (RWorkspace: none) 00268 * 00269 IWRK = ITAU 00270 CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL, 00271 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00272 * 00273 IF( WANTVR ) THEN 00274 * 00275 * Want left and right eigenvectors 00276 * Copy Schur vectors to VR 00277 * 00278 SIDE = 'B' 00279 CALL CLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) 00280 END IF 00281 * 00282 ELSE IF( WANTVR ) THEN 00283 * 00284 * Want right eigenvectors 00285 * Copy Householder vectors to VR 00286 * 00287 SIDE = 'R' 00288 CALL CLACPY( 'L', N, N, A, LDA, VR, LDVR ) 00289 * 00290 * Generate unitary matrix in VR 00291 * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 00292 * (RWorkspace: none) 00293 * 00294 CALL CUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), 00295 $ LWORK-IWRK+1, IERR ) 00296 * 00297 * Perform QR iteration, accumulating Schur vectors in VR 00298 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00299 * (RWorkspace: none) 00300 * 00301 IWRK = ITAU 00302 CALL CHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR, 00303 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00304 * 00305 ELSE 00306 * 00307 * Compute eigenvalues only 00308 * (CWorkspace: need 1, prefer HSWORK (see comments) ) 00309 * (RWorkspace: none) 00310 * 00311 IWRK = ITAU 00312 CALL CHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, 00313 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 00314 END IF 00315 * 00316 * If INFO > 0 from CHSEQR, then quit 00317 * 00318 IF( INFO.GT.0 ) 00319 $ GO TO 50 00320 * 00321 IF( WANTVL .OR. WANTVR ) THEN 00322 * 00323 * Compute left and/or right eigenvectors 00324 * (CWorkspace: need 2*N) 00325 * (RWorkspace: need 2*N) 00326 * 00327 IRWORK = IBAL + N 00328 CALL CTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 00329 $ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR ) 00330 END IF 00331 * 00332 IF( WANTVL ) THEN 00333 * 00334 * Undo balancing of left eigenvectors 00335 * (CWorkspace: none) 00336 * (RWorkspace: need N) 00337 * 00338 CALL CGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), N, VL, LDVL, 00339 $ IERR ) 00340 * 00341 * Normalize left eigenvectors and make largest component real 00342 * 00343 DO 20 I = 1, N 00344 SCL = ONE / SCNRM2( N, VL( 1, I ), 1 ) 00345 CALL CSSCAL( N, SCL, VL( 1, I ), 1 ) 00346 DO 10 K = 1, N 00347 RWORK( IRWORK+K-1 ) = REAL( VL( K, I ) )**2 + 00348 $ AIMAG( VL( K, I ) )**2 00349 10 CONTINUE 00350 K = ISAMAX( N, RWORK( IRWORK ), 1 ) 00351 TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 00352 CALL CSCAL( N, TMP, VL( 1, I ), 1 ) 00353 VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO ) 00354 20 CONTINUE 00355 END IF 00356 * 00357 IF( WANTVR ) THEN 00358 * 00359 * Undo balancing of right eigenvectors 00360 * (CWorkspace: none) 00361 * (RWorkspace: need N) 00362 * 00363 CALL CGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), N, VR, LDVR, 00364 $ IERR ) 00365 * 00366 * Normalize right eigenvectors and make largest component real 00367 * 00368 DO 40 I = 1, N 00369 SCL = ONE / SCNRM2( N, VR( 1, I ), 1 ) 00370 CALL CSSCAL( N, SCL, VR( 1, I ), 1 ) 00371 DO 30 K = 1, N 00372 RWORK( IRWORK+K-1 ) = REAL( VR( K, I ) )**2 + 00373 $ AIMAG( VR( K, I ) )**2 00374 30 CONTINUE 00375 K = ISAMAX( N, RWORK( IRWORK ), 1 ) 00376 TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 00377 CALL CSCAL( N, TMP, VR( 1, I ), 1 ) 00378 VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO ) 00379 40 CONTINUE 00380 END IF 00381 * 00382 * Undo scaling if necessary 00383 * 00384 50 CONTINUE 00385 IF( SCALEA ) THEN 00386 CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), 00387 $ MAX( N-INFO, 1 ), IERR ) 00388 IF( INFO.GT.0 ) THEN 00389 CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) 00390 END IF 00391 END IF 00392 * 00393 WORK( 1 ) = MAXWRK 00394 RETURN 00395 * 00396 * End of CGEEV 00397 * 00398 END
1.7.3 
