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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, 2: $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, 3: $ IWORK, INFO ) 4: * 5: * -- LAPACK routine (version 3.2) -- 6: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 8: * November 2006 9: * 10: * .. Scalar Arguments .. 11: CHARACTER HOWMNY, JOB 12: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N 13: * .. 14: * .. Array Arguments .. 15: LOGICAL SELECT( * ) 16: INTEGER IWORK( * ) 17: DOUBLE PRECISION DIF( * ), S( * ) 18: COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ), 19: $ VR( LDVR, * ), WORK( * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * ZTGSNA estimates reciprocal condition numbers for specified 26: * eigenvalues and/or eigenvectors of a matrix pair (A, B). 27: * 28: * (A, B) must be in generalized Schur canonical form, that is, A and 29: * B are both upper triangular. 30: * 31: * Arguments 32: * ========= 33: * 34: * JOB (input) CHARACTER*1 35: * Specifies whether condition numbers are required for 36: * eigenvalues (S) or eigenvectors (DIF): 37: * = 'E': for eigenvalues only (S); 38: * = 'V': for eigenvectors only (DIF); 39: * = 'B': for both eigenvalues and eigenvectors (S and DIF). 40: * 41: * HOWMNY (input) CHARACTER*1 42: * = 'A': compute condition numbers for all eigenpairs; 43: * = 'S': compute condition numbers for selected eigenpairs 44: * specified by the array SELECT. 45: * 46: * SELECT (input) LOGICAL array, dimension (N) 47: * If HOWMNY = 'S', SELECT specifies the eigenpairs for which 48: * condition numbers are required. To select condition numbers 49: * for the corresponding j-th eigenvalue and/or eigenvector, 50: * SELECT(j) must be set to .TRUE.. 51: * If HOWMNY = 'A', SELECT is not referenced. 52: * 53: * N (input) INTEGER 54: * The order of the square matrix pair (A, B). N >= 0. 55: * 56: * A (input) COMPLEX*16 array, dimension (LDA,N) 57: * The upper triangular matrix A in the pair (A,B). 58: * 59: * LDA (input) INTEGER 60: * The leading dimension of the array A. LDA >= max(1,N). 61: * 62: * B (input) COMPLEX*16 array, dimension (LDB,N) 63: * The upper triangular matrix B in the pair (A, B). 64: * 65: * LDB (input) INTEGER 66: * The leading dimension of the array B. LDB >= max(1,N). 67: * 68: * VL (input) COMPLEX*16 array, dimension (LDVL,M) 69: * IF JOB = 'E' or 'B', VL must contain left eigenvectors of 70: * (A, B), corresponding to the eigenpairs specified by HOWMNY 71: * and SELECT. The eigenvectors must be stored in consecutive 72: * columns of VL, as returned by ZTGEVC. 73: * If JOB = 'V', VL is not referenced. 74: * 75: * LDVL (input) INTEGER 76: * The leading dimension of the array VL. LDVL >= 1; and 77: * If JOB = 'E' or 'B', LDVL >= N. 78: * 79: * VR (input) COMPLEX*16 array, dimension (LDVR,M) 80: * IF JOB = 'E' or 'B', VR must contain right eigenvectors of 81: * (A, B), corresponding to the eigenpairs specified by HOWMNY 82: * and SELECT. The eigenvectors must be stored in consecutive 83: * columns of VR, as returned by ZTGEVC. 84: * If JOB = 'V', VR is not referenced. 85: * 86: * LDVR (input) INTEGER 87: * The leading dimension of the array VR. LDVR >= 1; 88: * If JOB = 'E' or 'B', LDVR >= N. 89: * 90: * S (output) DOUBLE PRECISION array, dimension (MM) 91: * If JOB = 'E' or 'B', the reciprocal condition numbers of the 92: * selected eigenvalues, stored in consecutive elements of the 93: * array. 94: * If JOB = 'V', S is not referenced. 95: * 96: * DIF (output) DOUBLE PRECISION array, dimension (MM) 97: * If JOB = 'V' or 'B', the estimated reciprocal condition 98: * numbers of the selected eigenvectors, stored in consecutive 99: * elements of the array. 100: * If the eigenvalues cannot be reordered to compute DIF(j), 101: * DIF(j) is set to 0; this can only occur when the true value 102: * would be very small anyway. 103: * For each eigenvalue/vector specified by SELECT, DIF stores 104: * a Frobenius norm-based estimate of Difl. 105: * If JOB = 'E', DIF is not referenced. 106: * 107: * MM (input) INTEGER 108: * The number of elements in the arrays S and DIF. MM >= M. 109: * 110: * M (output) INTEGER 111: * The number of elements of the arrays S and DIF used to store 112: * the specified condition numbers; for each selected eigenvalue 113: * one element is used. If HOWMNY = 'A', M is set to N. 114: * 115: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) 116: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 117: * 118: * LWORK (input) INTEGER 119: * The dimension of the array WORK. LWORK >= max(1,N). 120: * If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). 121: * 122: * IWORK (workspace) INTEGER array, dimension (N+2) 123: * If JOB = 'E', IWORK is not referenced. 124: * 125: * INFO (output) INTEGER 126: * = 0: Successful exit 127: * < 0: If INFO = -i, the i-th argument had an illegal value 128: * 129: * Further Details 130: * =============== 131: * 132: * The reciprocal of the condition number of the i-th generalized 133: * eigenvalue w = (a, b) is defined as 134: * 135: * S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v)) 136: * 137: * where u and v are the right and left eigenvectors of (A, B) 138: * corresponding to w; |z| denotes the absolute value of the complex 139: * number, and norm(u) denotes the 2-norm of the vector u. The pair 140: * (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the 141: * matrix pair (A, B). If both a and b equal zero, then (A,B) is 142: * singular and S(I) = -1 is returned. 143: * 144: * An approximate error bound on the chordal distance between the i-th 145: * computed generalized eigenvalue w and the corresponding exact 146: * eigenvalue lambda is 147: * 148: * chord(w, lambda) <= EPS * norm(A, B) / S(I), 149: * 150: * where EPS is the machine precision. 151: * 152: * The reciprocal of the condition number of the right eigenvector u 153: * and left eigenvector v corresponding to the generalized eigenvalue w 154: * is defined as follows. Suppose 155: * 156: * (A, B) = ( a * ) ( b * ) 1 157: * ( 0 A22 ),( 0 B22 ) n-1 158: * 1 n-1 1 n-1 159: * 160: * Then the reciprocal condition number DIF(I) is 161: * 162: * Difl[(a, b), (A22, B22)] = sigma-min( Zl ) 163: * 164: * where sigma-min(Zl) denotes the smallest singular value of 165: * 166: * Zl = [ kron(a, In-1) -kron(1, A22) ] 167: * [ kron(b, In-1) -kron(1, B22) ]. 168: * 169: * Here In-1 is the identity matrix of size n-1 and X' is the conjugate 170: * transpose of X. kron(X, Y) is the Kronecker product between the 171: * matrices X and Y. 172: * 173: * We approximate the smallest singular value of Zl with an upper 174: * bound. This is done by ZLATDF. 175: * 176: * An approximate error bound for a computed eigenvector VL(i) or 177: * VR(i) is given by 178: * 179: * EPS * norm(A, B) / DIF(i). 180: * 181: * See ref. [2-3] for more details and further references. 182: * 183: * Based on contributions by 184: * Bo Kagstrom and Peter Poromaa, Department of Computing Science, 185: * Umea University, S-901 87 Umea, Sweden. 186: * 187: * References 188: * ========== 189: * 190: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the 191: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in 192: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and 193: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. 194: * 195: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified 196: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition 197: * Estimation: Theory, Algorithms and Software, Report 198: * UMINF - 94.04, Department of Computing Science, Umea University, 199: * S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. 200: * To appear in Numerical Algorithms, 1996. 201: * 202: * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software 203: * for Solving the Generalized Sylvester Equation and Estimating the 204: * Separation between Regular Matrix Pairs, Report UMINF - 93.23, 205: * Department of Computing Science, Umea University, S-901 87 Umea, 206: * Sweden, December 1993, Revised April 1994, Also as LAPACK Working 207: * Note 75. 208: * To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. 209: * 210: * ===================================================================== 211: * 212: * .. Parameters .. 213: DOUBLE PRECISION ZERO, ONE 214: INTEGER IDIFJB 215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 ) 216: * .. 217: * .. Local Scalars .. 218: LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS 219: INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2 220: DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM 221: COMPLEX*16 YHAX, YHBX 222: * .. 223: * .. Local Arrays .. 224: COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 ) 225: * .. 226: * .. External Functions .. 227: LOGICAL LSAME 228: DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2 229: COMPLEX*16 ZDOTC 230: EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC 231: * .. 232: * .. External Subroutines .. 233: EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL 234: * .. 235: * .. Intrinsic Functions .. 236: INTRINSIC ABS, DCMPLX, MAX 237: * .. 238: * .. Executable Statements .. 239: * 240: * Decode and test the input parameters 241: * 242: WANTBH = LSAME( JOB, 'B' ) 243: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH 244: WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH 245: * 246: SOMCON = LSAME( HOWMNY, 'S' ) 247: * 248: INFO = 0 249: LQUERY = ( LWORK.EQ.-1 ) 250: * 251: IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN 252: INFO = -1 253: ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN 254: INFO = -2 255: ELSE IF( N.LT.0 ) THEN 256: INFO = -4 257: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 258: INFO = -6 259: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 260: INFO = -8 261: ELSE IF( WANTS .AND. LDVL.LT.N ) THEN 262: INFO = -10 263: ELSE IF( WANTS .AND. LDVR.LT.N ) THEN 264: INFO = -12 265: ELSE 266: * 267: * Set M to the number of eigenpairs for which condition numbers 268: * are required, and test MM. 269: * 270: IF( SOMCON ) THEN 271: M = 0 272: DO 10 K = 1, N 273: IF( SELECT( K ) ) 274: $ M = M + 1 275: 10 CONTINUE 276: ELSE 277: M = N 278: END IF 279: * 280: IF( N.EQ.0 ) THEN 281: LWMIN = 1 282: ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN 283: LWMIN = 2*N*N 284: ELSE 285: LWMIN = N 286: END IF 287: WORK( 1 ) = LWMIN 288: * 289: IF( MM.LT.M ) THEN 290: INFO = -15 291: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 292: INFO = -18 293: END IF 294: END IF 295: * 296: IF( INFO.NE.0 ) THEN 297: CALL XERBLA( 'ZTGSNA', -INFO ) 298: RETURN 299: ELSE IF( LQUERY ) THEN 300: RETURN 301: END IF 302: * 303: * Quick return if possible 304: * 305: IF( N.EQ.0 ) 306: $ RETURN 307: * 308: * Get machine constants 309: * 310: EPS = DLAMCH( 'P' ) 311: SMLNUM = DLAMCH( 'S' ) / EPS 312: BIGNUM = ONE / SMLNUM 313: CALL DLABAD( SMLNUM, BIGNUM ) 314: KS = 0 315: DO 20 K = 1, N 316: * 317: * Determine whether condition numbers are required for the k-th 318: * eigenpair. 319: * 320: IF( SOMCON ) THEN 321: IF( .NOT.SELECT( K ) ) 322: $ GO TO 20 323: END IF 324: * 325: KS = KS + 1 326: * 327: IF( WANTS ) THEN 328: * 329: * Compute the reciprocal condition number of the k-th 330: * eigenvalue. 331: * 332: RNRM = DZNRM2( N, VR( 1, KS ), 1 ) 333: LNRM = DZNRM2( N, VL( 1, KS ), 1 ) 334: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA, 335: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 ) 336: YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 ) 337: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB, 338: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 ) 339: YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 ) 340: COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) ) 341: IF( COND.EQ.ZERO ) THEN 342: S( KS ) = -ONE 343: ELSE 344: S( KS ) = COND / ( RNRM*LNRM ) 345: END IF 346: END IF 347: * 348: IF( WANTDF ) THEN 349: IF( N.EQ.1 ) THEN 350: DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) ) 351: ELSE 352: * 353: * Estimate the reciprocal condition number of the k-th 354: * eigenvectors. 355: * 356: * Copy the matrix (A, B) to the array WORK and move the 357: * (k,k)th pair to the (1,1) position. 358: * 359: CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N ) 360: CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N ) 361: IFST = K 362: ILST = 1 363: * 364: CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), 365: $ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR ) 366: * 367: IF( IERR.GT.0 ) THEN 368: * 369: * Ill-conditioned problem - swap rejected. 370: * 371: DIF( KS ) = ZERO 372: ELSE 373: * 374: * Reordering successful, solve generalized Sylvester 375: * equation for R and L, 376: * A22 * R - L * A11 = A12 377: * B22 * R - L * B11 = B12, 378: * and compute estimate of Difl[(A11,B11), (A22, B22)]. 379: * 380: N1 = 1 381: N2 = N - N1 382: I = N*N + 1 383: CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ), 384: $ N, WORK, N, WORK( N1+1 ), N, 385: $ WORK( N*N1+N1+I ), N, WORK( I ), N, 386: $ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY, 387: $ 1, IWORK, IERR ) 388: END IF 389: END IF 390: END IF 391: * 392: 20 CONTINUE 393: WORK( 1 ) = LWMIN 394: RETURN 395: * 396: * End of ZTGSNA 397: * 398: END