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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, 2: $ LDVR, WORK, LWORK, INFO ) 3: * 4: * -- LAPACK driver routine (version 3.2) -- 5: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 7: * November 2006 8: * 9: * .. Scalar Arguments .. 10: CHARACTER JOBVL, JOBVR 11: INTEGER INFO, LDA, LDVL, LDVR, LWORK, N 12: * .. 13: * .. Array Arguments .. 14: DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 15: $ WI( * ), WORK( * ), WR( * ) 16: * .. 17: * 18: * Purpose 19: * ======= 20: * 21: * DGEEV computes for an N-by-N real nonsymmetric matrix A, the 22: * eigenvalues and, optionally, the left and/or right eigenvectors. 23: * 24: * The right eigenvector v(j) of A satisfies 25: * A * v(j) = lambda(j) * v(j) 26: * where lambda(j) is its eigenvalue. 27: * The left eigenvector u(j) of A satisfies 28: * u(j)**H * A = lambda(j) * u(j)**H 29: * where u(j)**H denotes the conjugate transpose of u(j). 30: * 31: * The computed eigenvectors are normalized to have Euclidean norm 32: * equal to 1 and largest component real. 33: * 34: * Arguments 35: * ========= 36: * 37: * JOBVL (input) CHARACTER*1 38: * = 'N': left eigenvectors of A are not computed; 39: * = 'V': left eigenvectors of A are computed. 40: * 41: * JOBVR (input) CHARACTER*1 42: * = 'N': right eigenvectors of A are not computed; 43: * = 'V': right eigenvectors of A are computed. 44: * 45: * N (input) INTEGER 46: * The order of the matrix A. N >= 0. 47: * 48: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 49: * On entry, the N-by-N matrix A. 50: * On exit, A has been overwritten. 51: * 52: * LDA (input) INTEGER 53: * The leading dimension of the array A. LDA >= max(1,N). 54: * 55: * WR (output) DOUBLE PRECISION array, dimension (N) 56: * WI (output) DOUBLE PRECISION array, dimension (N) 57: * WR and WI contain the real and imaginary parts, 58: * respectively, of the computed eigenvalues. Complex 59: * conjugate pairs of eigenvalues appear consecutively 60: * with the eigenvalue having the positive imaginary part 61: * first. 62: * 63: * VL (output) DOUBLE PRECISION array, dimension (LDVL,N) 64: * If JOBVL = 'V', the left eigenvectors u(j) are stored one 65: * after another in the columns of VL, in the same order 66: * as their eigenvalues. 67: * If JOBVL = 'N', VL is not referenced. 68: * If the j-th eigenvalue is real, then u(j) = VL(:,j), 69: * the j-th column of VL. 70: * If the j-th and (j+1)-st eigenvalues form a complex 71: * conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and 72: * u(j+1) = VL(:,j) - i*VL(:,j+1). 73: * 74: * LDVL (input) INTEGER 75: * The leading dimension of the array VL. LDVL >= 1; if 76: * JOBVL = 'V', LDVL >= N. 77: * 78: * VR (output) DOUBLE PRECISION array, dimension (LDVR,N) 79: * If JOBVR = 'V', the right eigenvectors v(j) are stored one 80: * after another in the columns of VR, in the same order 81: * as their eigenvalues. 82: * If JOBVR = 'N', VR is not referenced. 83: * If the j-th eigenvalue is real, then v(j) = VR(:,j), 84: * the j-th column of VR. 85: * If the j-th and (j+1)-st eigenvalues form a complex 86: * conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and 87: * v(j+1) = VR(:,j) - i*VR(:,j+1). 88: * 89: * LDVR (input) INTEGER 90: * The leading dimension of the array VR. LDVR >= 1; if 91: * JOBVR = 'V', LDVR >= N. 92: * 93: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 94: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 95: * 96: * LWORK (input) INTEGER 97: * The dimension of the array WORK. LWORK >= max(1,3*N), and 98: * if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good 99: * performance, LWORK must generally be larger. 100: * 101: * If LWORK = -1, then a workspace query is assumed; the routine 102: * only calculates the optimal size of the WORK array, returns 103: * this value as the first entry of the WORK array, and no error 104: * message related to LWORK is issued by XERBLA. 105: * 106: * INFO (output) INTEGER 107: * = 0: successful exit 108: * < 0: if INFO = -i, the i-th argument had an illegal value. 109: * > 0: if INFO = i, the QR algorithm failed to compute all the 110: * eigenvalues, and no eigenvectors have been computed; 111: * elements i+1:N of WR and WI contain eigenvalues which 112: * have converged. 113: * 114: * ===================================================================== 115: * 116: * .. Parameters .. 117: DOUBLE PRECISION ZERO, ONE 118: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 119: * .. 120: * .. Local Scalars .. 121: LOGICAL LQUERY, SCALEA, WANTVL, WANTVR 122: CHARACTER SIDE 123: INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K, 124: $ MAXWRK, MINWRK, NOUT 125: DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM, 126: $ SN 127: * .. 128: * .. Local Arrays .. 129: LOGICAL SELECT( 1 ) 130: DOUBLE PRECISION DUM( 1 ) 131: * .. 132: * .. External Subroutines .. 133: EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY, 134: $ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC, 135: $ XERBLA 136: * .. 137: * .. External Functions .. 138: LOGICAL LSAME 139: INTEGER IDAMAX, ILAENV 140: DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2 141: EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2, 142: $ DNRM2 143: * .. 144: * .. Intrinsic Functions .. 145: INTRINSIC MAX, SQRT 146: * .. 147: * .. Executable Statements .. 148: * 149: * Test the input arguments 150: * 151: INFO = 0 152: LQUERY = ( LWORK.EQ.-1 ) 153: WANTVL = LSAME( JOBVL, 'V' ) 154: WANTVR = LSAME( JOBVR, 'V' ) 155: IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN 156: INFO = -1 157: ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN 158: INFO = -2 159: ELSE IF( N.LT.0 ) THEN 160: INFO = -3 161: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 162: INFO = -5 163: ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN 164: INFO = -9 165: ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN 166: INFO = -11 167: END IF 168: * 169: * Compute workspace 170: * (Note: Comments in the code beginning "Workspace:" describe the 171: * minimal amount of workspace needed at that point in the code, 172: * as well as the preferred amount for good performance. 173: * NB refers to the optimal block size for the immediately 174: * following subroutine, as returned by ILAENV. 175: * HSWORK refers to the workspace preferred by DHSEQR, as 176: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, 177: * the worst case.) 178: * 179: IF( INFO.EQ.0 ) THEN 180: IF( N.EQ.0 ) THEN 181: MINWRK = 1 182: MAXWRK = 1 183: ELSE 184: MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) 185: IF( WANTVL ) THEN 186: MINWRK = 4*N 187: MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, 188: $ 'DORGHR', ' ', N, 1, N, -1 ) ) 189: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL, 190: $ WORK, -1, INFO ) 191: HSWORK = WORK( 1 ) 192: MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) 193: MAXWRK = MAX( MAXWRK, 4*N ) 194: ELSE IF( WANTVR ) THEN 195: MINWRK = 4*N 196: MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, 197: $ 'DORGHR', ' ', N, 1, N, -1 ) ) 198: CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR, 199: $ WORK, -1, INFO ) 200: HSWORK = WORK( 1 ) 201: MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) 202: MAXWRK = MAX( MAXWRK, 4*N ) 203: ELSE 204: MINWRK = 3*N 205: CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR, 206: $ WORK, -1, INFO ) 207: HSWORK = WORK( 1 ) 208: MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK ) 209: END IF 210: MAXWRK = MAX( MAXWRK, MINWRK ) 211: END IF 212: WORK( 1 ) = MAXWRK 213: * 214: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 215: INFO = -13 216: END IF 217: END IF 218: * 219: IF( INFO.NE.0 ) THEN 220: CALL XERBLA( 'DGEEV ', -INFO ) 221: RETURN 222: ELSE IF( LQUERY ) THEN 223: RETURN 224: END IF 225: * 226: * Quick return if possible 227: * 228: IF( N.EQ.0 ) 229: $ RETURN 230: * 231: * Get machine constants 232: * 233: EPS = DLAMCH( 'P' ) 234: SMLNUM = DLAMCH( 'S' ) 235: BIGNUM = ONE / SMLNUM 236: CALL DLABAD( SMLNUM, BIGNUM ) 237: SMLNUM = SQRT( SMLNUM ) / EPS 238: BIGNUM = ONE / SMLNUM 239: * 240: * Scale A if max element outside range [SMLNUM,BIGNUM] 241: * 242: ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) 243: SCALEA = .FALSE. 244: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 245: SCALEA = .TRUE. 246: CSCALE = SMLNUM 247: ELSE IF( ANRM.GT.BIGNUM ) THEN 248: SCALEA = .TRUE. 249: CSCALE = BIGNUM 250: END IF 251: IF( SCALEA ) 252: $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) 253: * 254: * Balance the matrix 255: * (Workspace: need N) 256: * 257: IBAL = 1 258: CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) 259: * 260: * Reduce to upper Hessenberg form 261: * (Workspace: need 3*N, prefer 2*N+N*NB) 262: * 263: ITAU = IBAL + N 264: IWRK = ITAU + N 265: CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), 266: $ LWORK-IWRK+1, IERR ) 267: * 268: IF( WANTVL ) THEN 269: * 270: * Want left eigenvectors 271: * Copy Householder vectors to VL 272: * 273: SIDE = 'L' 274: CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL ) 275: * 276: * Generate orthogonal matrix in VL 277: * (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) 278: * 279: CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), 280: $ LWORK-IWRK+1, IERR ) 281: * 282: * Perform QR iteration, accumulating Schur vectors in VL 283: * (Workspace: need N+1, prefer N+HSWORK (see comments) ) 284: * 285: IWRK = ITAU 286: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL, 287: $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 288: * 289: IF( WANTVR ) THEN 290: * 291: * Want left and right eigenvectors 292: * Copy Schur vectors to VR 293: * 294: SIDE = 'B' 295: CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) 296: END IF 297: * 298: ELSE IF( WANTVR ) THEN 299: * 300: * Want right eigenvectors 301: * Copy Householder vectors to VR 302: * 303: SIDE = 'R' 304: CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR ) 305: * 306: * Generate orthogonal matrix in VR 307: * (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) 308: * 309: CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), 310: $ LWORK-IWRK+1, IERR ) 311: * 312: * Perform QR iteration, accumulating Schur vectors in VR 313: * (Workspace: need N+1, prefer N+HSWORK (see comments) ) 314: * 315: IWRK = ITAU 316: CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, 317: $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 318: * 319: ELSE 320: * 321: * Compute eigenvalues only 322: * (Workspace: need N+1, prefer N+HSWORK (see comments) ) 323: * 324: IWRK = ITAU 325: CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR, 326: $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 327: END IF 328: * 329: * If INFO > 0 from DHSEQR, then quit 330: * 331: IF( INFO.GT.0 ) 332: $ GO TO 50 333: * 334: IF( WANTVL .OR. WANTVR ) THEN 335: * 336: * Compute left and/or right eigenvectors 337: * (Workspace: need 4*N) 338: * 339: CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 340: $ N, NOUT, WORK( IWRK ), IERR ) 341: END IF 342: * 343: IF( WANTVL ) THEN 344: * 345: * Undo balancing of left eigenvectors 346: * (Workspace: need N) 347: * 348: CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL, 349: $ IERR ) 350: * 351: * Normalize left eigenvectors and make largest component real 352: * 353: DO 20 I = 1, N 354: IF( WI( I ).EQ.ZERO ) THEN 355: SCL = ONE / DNRM2( N, VL( 1, I ), 1 ) 356: CALL DSCAL( N, SCL, VL( 1, I ), 1 ) 357: ELSE IF( WI( I ).GT.ZERO ) THEN 358: SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ), 359: $ DNRM2( N, VL( 1, I+1 ), 1 ) ) 360: CALL DSCAL( N, SCL, VL( 1, I ), 1 ) 361: CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 ) 362: DO 10 K = 1, N 363: WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2 364: 10 CONTINUE 365: K = IDAMAX( N, WORK( IWRK ), 1 ) 366: CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R ) 367: CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN ) 368: VL( K, I+1 ) = ZERO 369: END IF 370: 20 CONTINUE 371: END IF 372: * 373: IF( WANTVR ) THEN 374: * 375: * Undo balancing of right eigenvectors 376: * (Workspace: need N) 377: * 378: CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR, 379: $ IERR ) 380: * 381: * Normalize right eigenvectors and make largest component real 382: * 383: DO 40 I = 1, N 384: IF( WI( I ).EQ.ZERO ) THEN 385: SCL = ONE / DNRM2( N, VR( 1, I ), 1 ) 386: CALL DSCAL( N, SCL, VR( 1, I ), 1 ) 387: ELSE IF( WI( I ).GT.ZERO ) THEN 388: SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ), 389: $ DNRM2( N, VR( 1, I+1 ), 1 ) ) 390: CALL DSCAL( N, SCL, VR( 1, I ), 1 ) 391: CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 ) 392: DO 30 K = 1, N 393: WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2 394: 30 CONTINUE 395: K = IDAMAX( N, WORK( IWRK ), 1 ) 396: CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R ) 397: CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN ) 398: VR( K, I+1 ) = ZERO 399: END IF 400: 40 CONTINUE 401: END IF 402: * 403: * Undo scaling if necessary 404: * 405: 50 CONTINUE 406: IF( SCALEA ) THEN 407: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ), 408: $ MAX( N-INFO, 1 ), IERR ) 409: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ), 410: $ MAX( N-INFO, 1 ), IERR ) 411: IF( INFO.GT.0 ) THEN 412: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N, 413: $ IERR ) 414: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, 415: $ IERR ) 416: END IF 417: END IF 418: * 419: WORK( 1 ) = MAXWRK 420: RETURN 421: * 422: * End of DGEEV 423: * 424: END