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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZHESVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, 2: $ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, 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 FACT, UPLO 12: INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS 13: DOUBLE PRECISION RCOND 14: * .. 15: * .. Array Arguments .. 16: INTEGER IPIV( * ) 17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 18: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 19: $ WORK( * ), X( LDX, * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * ZHESVX uses the diagonal pivoting factorization to compute the 26: * solution to a complex system of linear equations A * X = B, 27: * where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS 28: * 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 = 'N', the diagonal pivoting method is used to factor A. 39: * The form of the factorization is 40: * A = U * D * U**H, if UPLO = 'U', or 41: * A = L * D * L**H, if UPLO = 'L', 42: * where U (or L) is a product of permutation and unit upper (lower) 43: * triangular matrices, and D is Hermitian and block diagonal with 44: * 1-by-1 and 2-by-2 diagonal blocks. 45: * 46: * 2. If some D(i,i)=0, so that D is exactly singular, then the routine 47: * returns with INFO = i. Otherwise, the factored form of A is used 48: * to estimate the condition number of the matrix A. If the 49: * reciprocal of the condition number is less than machine precision, 50: * INFO = N+1 is returned as a warning, but the routine still goes on 51: * to solve for X and compute error bounds as described below. 52: * 53: * 3. The system of equations is solved for X using the factored form 54: * of A. 55: * 56: * 4. Iterative refinement is applied to improve the computed solution 57: * matrix and calculate error bounds and backward error estimates 58: * for it. 59: * 60: * Arguments 61: * ========= 62: * 63: * FACT (input) CHARACTER*1 64: * Specifies whether or not the factored form of A has been 65: * supplied on entry. 66: * = 'F': On entry, AF and IPIV contain the factored form 67: * of A. A, AF and IPIV will not be modified. 68: * = 'N': The matrix A will be copied to AF and factored. 69: * 70: * UPLO (input) CHARACTER*1 71: * = 'U': Upper triangle of A is stored; 72: * = 'L': Lower triangle of A is stored. 73: * 74: * N (input) INTEGER 75: * The number of linear equations, i.e., the order of the 76: * matrix A. N >= 0. 77: * 78: * NRHS (input) INTEGER 79: * The number of right hand sides, i.e., the number of columns 80: * of the matrices B and X. NRHS >= 0. 81: * 82: * A (input) COMPLEX*16 array, dimension (LDA,N) 83: * The Hermitian matrix A. If UPLO = 'U', the leading N-by-N 84: * upper triangular part of A contains the upper triangular part 85: * of the matrix A, and the strictly lower triangular part of A 86: * is not referenced. If UPLO = 'L', the leading N-by-N lower 87: * triangular part of A contains the lower triangular part of 88: * the matrix A, and the strictly upper triangular part of A is 89: * not referenced. 90: * 91: * LDA (input) INTEGER 92: * The leading dimension of the array A. LDA >= max(1,N). 93: * 94: * AF (input or output) COMPLEX*16 array, dimension (LDAF,N) 95: * If FACT = 'F', then AF is an input argument and on entry 96: * contains the block diagonal matrix D and the multipliers used 97: * to obtain the factor U or L from the factorization 98: * A = U*D*U**H or A = L*D*L**H as computed by ZHETRF. 99: * 100: * If FACT = 'N', then AF is an output argument and on exit 101: * returns the block diagonal matrix D and the multipliers used 102: * to obtain the factor U or L from the factorization 103: * A = U*D*U**H or A = L*D*L**H. 104: * 105: * LDAF (input) INTEGER 106: * The leading dimension of the array AF. LDAF >= max(1,N). 107: * 108: * IPIV (input or output) INTEGER array, dimension (N) 109: * If FACT = 'F', then IPIV is an input argument and on entry 110: * contains details of the interchanges and the block structure 111: * of D, as determined by ZHETRF. 112: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were 113: * interchanged and D(k,k) is a 1-by-1 diagonal block. 114: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 115: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 116: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 117: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 118: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 119: * 120: * If FACT = 'N', then IPIV is an output argument and on exit 121: * contains details of the interchanges and the block structure 122: * of D, as determined by ZHETRF. 123: * 124: * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 125: * The N-by-NRHS right hand side matrix B. 126: * 127: * LDB (input) INTEGER 128: * The leading dimension of the array B. LDB >= max(1,N). 129: * 130: * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 131: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 132: * 133: * LDX (input) INTEGER 134: * The leading dimension of the array X. LDX >= max(1,N). 135: * 136: * RCOND (output) DOUBLE PRECISION 137: * The estimate of the reciprocal condition number of the matrix 138: * A. If RCOND is less than the machine precision (in 139: * particular, if RCOND = 0), the matrix is singular to working 140: * precision. This condition is indicated by a return code of 141: * INFO > 0. 142: * 143: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 144: * The estimated forward error bound for each solution vector 145: * X(j) (the j-th column of the solution matrix X). 146: * If XTRUE is the true solution corresponding to X(j), FERR(j) 147: * is an estimated upper bound for the magnitude of the largest 148: * element in (X(j) - XTRUE) divided by the magnitude of the 149: * largest element in X(j). The estimate is as reliable as 150: * the estimate for RCOND, and is almost always a slight 151: * overestimate of the true error. 152: * 153: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 154: * The componentwise relative backward error of each solution 155: * vector X(j) (i.e., the smallest relative change in 156: * any element of A or B that makes X(j) an exact solution). 157: * 158: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) 159: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 160: * 161: * LWORK (input) INTEGER 162: * The length of WORK. LWORK >= max(1,2*N), and for best 163: * performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where 164: * NB is the optimal blocksize for ZHETRF. 165: * 166: * If LWORK = -1, then a workspace query is assumed; the routine 167: * only calculates the optimal size of the WORK array, returns 168: * this value as the first entry of the WORK array, and no error 169: * message related to LWORK is issued by XERBLA. 170: * 171: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 172: * 173: * INFO (output) INTEGER 174: * = 0: successful exit 175: * < 0: if INFO = -i, the i-th argument had an illegal value 176: * > 0: if INFO = i, and i is 177: * <= N: D(i,i) is exactly zero. The factorization 178: * has been completed but the factor D is exactly 179: * singular, so the solution and error bounds could 180: * not be computed. RCOND = 0 is returned. 181: * = N+1: D is nonsingular, but RCOND is less than machine 182: * precision, meaning that the matrix is singular 183: * to working precision. Nevertheless, the 184: * solution and error bounds are computed because 185: * there are a number of situations where the 186: * computed solution can be more accurate than the 187: * value of RCOND would suggest. 188: * 189: * ===================================================================== 190: * 191: * .. Parameters .. 192: DOUBLE PRECISION ZERO 193: PARAMETER ( ZERO = 0.0D+0 ) 194: * .. 195: * .. Local Scalars .. 196: LOGICAL LQUERY, NOFACT 197: INTEGER LWKOPT, NB 198: DOUBLE PRECISION ANORM 199: * .. 200: * .. External Functions .. 201: LOGICAL LSAME 202: INTEGER ILAENV 203: DOUBLE PRECISION DLAMCH, ZLANHE 204: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE 205: * .. 206: * .. External Subroutines .. 207: EXTERNAL XERBLA, ZHECON, ZHERFS, ZHETRF, ZHETRS, ZLACPY 208: * .. 209: * .. Intrinsic Functions .. 210: INTRINSIC MAX 211: * .. 212: * .. Executable Statements .. 213: * 214: * Test the input parameters. 215: * 216: INFO = 0 217: NOFACT = LSAME( FACT, 'N' ) 218: LQUERY = ( LWORK.EQ.-1 ) 219: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN 220: INFO = -1 221: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) 222: $ THEN 223: INFO = -2 224: ELSE IF( N.LT.0 ) THEN 225: INFO = -3 226: ELSE IF( NRHS.LT.0 ) THEN 227: INFO = -4 228: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 229: INFO = -6 230: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 231: INFO = -8 232: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 233: INFO = -11 234: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 235: INFO = -13 236: ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN 237: INFO = -18 238: END IF 239: * 240: IF( INFO.EQ.0 ) THEN 241: LWKOPT = MAX( 1, 2*N ) 242: IF( NOFACT ) THEN 243: NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 ) 244: LWKOPT = MAX( LWKOPT, N*NB ) 245: END IF 246: WORK( 1 ) = LWKOPT 247: END IF 248: * 249: IF( INFO.NE.0 ) THEN 250: CALL XERBLA( 'ZHESVX', -INFO ) 251: RETURN 252: ELSE IF( LQUERY ) THEN 253: RETURN 254: END IF 255: * 256: IF( NOFACT ) THEN 257: * 258: * Compute the factorization A = U*D*U' or A = L*D*L'. 259: * 260: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 261: CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO ) 262: * 263: * Return if INFO is non-zero. 264: * 265: IF( INFO.GT.0 )THEN 266: RCOND = ZERO 267: RETURN 268: END IF 269: END IF 270: * 271: * Compute the norm of the matrix A. 272: * 273: ANORM = ZLANHE( 'I', UPLO, N, A, LDA, RWORK ) 274: * 275: * Compute the reciprocal of the condition number of A. 276: * 277: CALL ZHECON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, INFO ) 278: * 279: * Compute the solution vectors X. 280: * 281: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 282: CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 283: * 284: * Use iterative refinement to improve the computed solutions and 285: * compute error bounds and backward error estimates for them. 286: * 287: CALL ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, 288: $ LDX, FERR, BERR, WORK, RWORK, INFO ) 289: * 290: * Set INFO = N+1 if the matrix is singular to working precision. 291: * 292: IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) 293: $ INFO = N + 1 294: * 295: WORK( 1 ) = LWKOPT 296: * 297: RETURN 298: * 299: * End of ZHESVX 300: * 301: END