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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 2: $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, 3: $ RWORK, INFO ) 4: * 5: * -- LAPACK driver 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 EQUED, FACT, UPLO 12: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 13: DOUBLE PRECISION RCOND 14: * .. 15: * .. Array Arguments .. 16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) 17: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 18: $ WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to 25: * compute the solution to a complex system of linear equations 26: * A * X = B, 27: * where A is an N-by-N Hermitian positive definite matrix and X and B 28: * are N-by-NRHS matrices. 29: * 30: * Error bounds on the solution and a condition estimate are also 31: * provided. 32: * 33: * Description 34: * =========== 35: * 36: * The following steps are performed: 37: * 38: * 1. If FACT = 'E', real scaling factors are computed to equilibrate 39: * the system: 40: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B 41: * Whether or not the system will be equilibrated depends on the 42: * scaling of the matrix A, but if equilibration is used, A is 43: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 44: * 45: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 46: * factor the matrix A (after equilibration if FACT = 'E') as 47: * A = U**H* U, if UPLO = 'U', or 48: * A = L * L**H, if UPLO = 'L', 49: * where U is an upper triangular matrix and L is a lower triangular 50: * matrix. 51: * 52: * 3. If the leading i-by-i principal minor is not positive definite, 53: * then the routine returns with INFO = i. Otherwise, the factored 54: * form of A is used to estimate the condition number of the matrix 55: * A. If the reciprocal of the condition number is less than machine 56: * precision, INFO = N+1 is returned as a warning, but the routine 57: * still goes on to solve for X and compute error bounds as 58: * described below. 59: * 60: * 4. The system of equations is solved for X using the factored form 61: * of A. 62: * 63: * 5. Iterative refinement is applied to improve the computed solution 64: * matrix and calculate error bounds and backward error estimates 65: * for it. 66: * 67: * 6. If equilibration was used, the matrix X is premultiplied by 68: * diag(S) so that it solves the original system before 69: * equilibration. 70: * 71: * Arguments 72: * ========= 73: * 74: * FACT (input) CHARACTER*1 75: * Specifies whether or not the factored form of the matrix A is 76: * supplied on entry, and if not, whether the matrix A should be 77: * equilibrated before it is factored. 78: * = 'F': On entry, AF contains the factored form of A. 79: * If EQUED = 'Y', the matrix A has been equilibrated 80: * with scaling factors given by S. A and AF will not 81: * be modified. 82: * = 'N': The matrix A will be copied to AF and factored. 83: * = 'E': The matrix A will be equilibrated if necessary, then 84: * copied to AF and factored. 85: * 86: * UPLO (input) CHARACTER*1 87: * = 'U': Upper triangle of A is stored; 88: * = 'L': Lower triangle of A is stored. 89: * 90: * N (input) INTEGER 91: * The number of linear equations, i.e., the order of the 92: * matrix A. N >= 0. 93: * 94: * NRHS (input) INTEGER 95: * The number of right hand sides, i.e., the number of columns 96: * of the matrices B and X. NRHS >= 0. 97: * 98: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 99: * On entry, the Hermitian matrix A, except if FACT = 'F' and 100: * EQUED = 'Y', then A must contain the equilibrated matrix 101: * diag(S)*A*diag(S). If UPLO = 'U', the leading 102: * N-by-N upper triangular part of A contains the upper 103: * triangular part of the matrix A, and the strictly lower 104: * triangular part of A is not referenced. If UPLO = 'L', the 105: * leading N-by-N lower triangular part of A contains the lower 106: * triangular part of the matrix A, and the strictly upper 107: * triangular part of A is not referenced. A is not modified if 108: * FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. 109: * 110: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 111: * diag(S)*A*diag(S). 112: * 113: * LDA (input) INTEGER 114: * The leading dimension of the array A. LDA >= max(1,N). 115: * 116: * AF (input or output) COMPLEX*16 array, dimension (LDAF,N) 117: * If FACT = 'F', then AF is an input argument and on entry 118: * contains the triangular factor U or L from the Cholesky 119: * factorization A = U**H*U or A = L*L**H, in the same storage 120: * format as A. If EQUED .ne. 'N', then AF is the factored form 121: * of the equilibrated matrix diag(S)*A*diag(S). 122: * 123: * If FACT = 'N', then AF is an output argument and on exit 124: * returns the triangular factor U or L from the Cholesky 125: * factorization A = U**H*U or A = L*L**H of the original 126: * matrix A. 127: * 128: * If FACT = 'E', then AF is an output argument and on exit 129: * returns the triangular factor U or L from the Cholesky 130: * factorization A = U**H*U or A = L*L**H of the equilibrated 131: * matrix A (see the description of A for the form of the 132: * equilibrated matrix). 133: * 134: * LDAF (input) INTEGER 135: * The leading dimension of the array AF. LDAF >= max(1,N). 136: * 137: * EQUED (input or output) CHARACTER*1 138: * Specifies the form of equilibration that was done. 139: * = 'N': No equilibration (always true if FACT = 'N'). 140: * = 'Y': Equilibration was done, i.e., A has been replaced by 141: * diag(S) * A * diag(S). 142: * EQUED is an input argument if FACT = 'F'; otherwise, it is an 143: * output argument. 144: * 145: * S (input or output) DOUBLE PRECISION array, dimension (N) 146: * The scale factors for A; not accessed if EQUED = 'N'. S is 147: * an input argument if FACT = 'F'; otherwise, S is an output 148: * argument. If FACT = 'F' and EQUED = 'Y', each element of S 149: * must be positive. 150: * 151: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) 152: * On entry, the N-by-NRHS righthand side matrix B. 153: * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', 154: * B is overwritten by diag(S) * B. 155: * 156: * LDB (input) INTEGER 157: * The leading dimension of the array B. LDB >= max(1,N). 158: * 159: * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 160: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to 161: * the original system of equations. Note that if EQUED = 'Y', 162: * A and B are modified on exit, and the solution to the 163: * equilibrated system is inv(diag(S))*X. 164: * 165: * LDX (input) INTEGER 166: * The leading dimension of the array X. LDX >= max(1,N). 167: * 168: * RCOND (output) DOUBLE PRECISION 169: * The estimate of the reciprocal condition number of the matrix 170: * A after equilibration (if done). If RCOND is less than the 171: * machine precision (in particular, if RCOND = 0), the matrix 172: * is singular to working precision. This condition is 173: * indicated by a return code of INFO > 0. 174: * 175: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 176: * The estimated forward error bound for each solution vector 177: * X(j) (the j-th column of the solution matrix X). 178: * If XTRUE is the true solution corresponding to X(j), FERR(j) 179: * is an estimated upper bound for the magnitude of the largest 180: * element in (X(j) - XTRUE) divided by the magnitude of the 181: * largest element in X(j). The estimate is as reliable as 182: * the estimate for RCOND, and is almost always a slight 183: * overestimate of the true error. 184: * 185: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 186: * The componentwise relative backward error of each solution 187: * vector X(j) (i.e., the smallest relative change in 188: * any element of A or B that makes X(j) an exact solution). 189: * 190: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 191: * 192: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 193: * 194: * INFO (output) INTEGER 195: * = 0: successful exit 196: * < 0: if INFO = -i, the i-th argument had an illegal value 197: * > 0: if INFO = i, and i is 198: * <= N: the leading minor of order i of A is 199: * not positive definite, so the factorization 200: * could not be completed, and the solution has not 201: * been computed. RCOND = 0 is returned. 202: * = N+1: U is nonsingular, but RCOND is less than machine 203: * precision, meaning that the matrix is singular 204: * to working precision. Nevertheless, the 205: * solution and error bounds are computed because 206: * there are a number of situations where the 207: * computed solution can be more accurate than the 208: * value of RCOND would suggest. 209: * 210: * ===================================================================== 211: * 212: * .. Parameters .. 213: DOUBLE PRECISION ZERO, ONE 214: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 215: * .. 216: * .. Local Scalars .. 217: LOGICAL EQUIL, NOFACT, RCEQU 218: INTEGER I, INFEQU, J 219: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM 220: * .. 221: * .. External Functions .. 222: LOGICAL LSAME 223: DOUBLE PRECISION DLAMCH, ZLANHE 224: EXTERNAL LSAME, DLAMCH, ZLANHE 225: * .. 226: * .. External Subroutines .. 227: EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS, 228: $ ZPOTRF, ZPOTRS 229: * .. 230: * .. Intrinsic Functions .. 231: INTRINSIC MAX, MIN 232: * .. 233: * .. Executable Statements .. 234: * 235: INFO = 0 236: NOFACT = LSAME( FACT, 'N' ) 237: EQUIL = LSAME( FACT, 'E' ) 238: IF( NOFACT .OR. EQUIL ) THEN 239: EQUED = 'N' 240: RCEQU = .FALSE. 241: ELSE 242: RCEQU = LSAME( EQUED, 'Y' ) 243: SMLNUM = DLAMCH( 'Safe minimum' ) 244: BIGNUM = ONE / SMLNUM 245: END IF 246: * 247: * Test the input parameters. 248: * 249: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) 250: $ THEN 251: INFO = -1 252: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 253: $ THEN 254: INFO = -2 255: ELSE IF( N.LT.0 ) THEN 256: INFO = -3 257: ELSE IF( NRHS.LT.0 ) THEN 258: INFO = -4 259: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 260: INFO = -6 261: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 262: INFO = -8 263: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 264: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 265: INFO = -9 266: ELSE 267: IF( RCEQU ) THEN 268: SMIN = BIGNUM 269: SMAX = ZERO 270: DO 10 J = 1, N 271: SMIN = MIN( SMIN, S( J ) ) 272: SMAX = MAX( SMAX, S( J ) ) 273: 10 CONTINUE 274: IF( SMIN.LE.ZERO ) THEN 275: INFO = -10 276: ELSE IF( N.GT.0 ) THEN 277: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 278: ELSE 279: SCOND = ONE 280: END IF 281: END IF 282: IF( INFO.EQ.0 ) THEN 283: IF( LDB.LT.MAX( 1, N ) ) THEN 284: INFO = -12 285: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 286: INFO = -14 287: END IF 288: END IF 289: END IF 290: * 291: IF( INFO.NE.0 ) THEN 292: CALL XERBLA( 'ZPOSVX', -INFO ) 293: RETURN 294: END IF 295: * 296: IF( EQUIL ) THEN 297: * 298: * Compute row and column scalings to equilibrate the matrix A. 299: * 300: CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU ) 301: IF( INFEQU.EQ.0 ) THEN 302: * 303: * Equilibrate the matrix. 304: * 305: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 306: RCEQU = LSAME( EQUED, 'Y' ) 307: END IF 308: END IF 309: * 310: * Scale the right hand side. 311: * 312: IF( RCEQU ) THEN 313: DO 30 J = 1, NRHS 314: DO 20 I = 1, N 315: B( I, J ) = S( I )*B( I, J ) 316: 20 CONTINUE 317: 30 CONTINUE 318: END IF 319: * 320: IF( NOFACT .OR. EQUIL ) THEN 321: * 322: * Compute the Cholesky factorization A = U'*U or A = L*L'. 323: * 324: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 325: CALL ZPOTRF( UPLO, N, AF, LDAF, INFO ) 326: * 327: * Return if INFO is non-zero. 328: * 329: IF( INFO.GT.0 )THEN 330: RCOND = ZERO 331: RETURN 332: END IF 333: END IF 334: * 335: * Compute the norm of the matrix A. 336: * 337: ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK ) 338: * 339: * Compute the reciprocal of the condition number of A. 340: * 341: CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO ) 342: * 343: * Compute the solution matrix X. 344: * 345: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 346: CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 347: * 348: * Use iterative refinement to improve the computed solution and 349: * compute error bounds and backward error estimates for it. 350: * 351: CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, 352: $ FERR, BERR, WORK, RWORK, INFO ) 353: * 354: * Transform the solution matrix X to a solution of the original 355: * system. 356: * 357: IF( RCEQU ) THEN 358: DO 50 J = 1, NRHS 359: DO 40 I = 1, N 360: X( I, J ) = S( I )*X( I, J ) 361: 40 CONTINUE 362: 50 CONTINUE 363: DO 60 J = 1, NRHS 364: FERR( J ) = FERR( J ) / SCOND 365: 60 CONTINUE 366: END IF 367: * 368: * Set INFO = N+1 if the matrix is singular to working precision. 369: * 370: IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 371: $ INFO = N + 1 372: * 373: RETURN 374: * 375: * End of ZPOSVX 376: * 377: END