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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, 2: $ X, LDX, FERR, BERR, WORK, RWORK, INFO ) 3: * 4: * -- LAPACK 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: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. 10: * 11: * .. Scalar Arguments .. 12: CHARACTER UPLO 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS 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: * ZSYRFS improves the computed solution to a system of linear 26: * equations when the coefficient matrix is symmetric indefinite, and 27: * provides error bounds and backward error estimates for the solution. 28: * 29: * Arguments 30: * ========= 31: * 32: * UPLO (input) CHARACTER*1 33: * = 'U': Upper triangle of A is stored; 34: * = 'L': Lower triangle of A is stored. 35: * 36: * N (input) INTEGER 37: * The order of the matrix A. N >= 0. 38: * 39: * NRHS (input) INTEGER 40: * The number of right hand sides, i.e., the number of columns 41: * of the matrices B and X. NRHS >= 0. 42: * 43: * A (input) COMPLEX*16 array, dimension (LDA,N) 44: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 45: * upper triangular part of A contains the upper triangular part 46: * of the matrix A, and the strictly lower triangular part of A 47: * is not referenced. If UPLO = 'L', the leading N-by-N lower 48: * triangular part of A contains the lower triangular part of 49: * the matrix A, and the strictly upper triangular part of A is 50: * not referenced. 51: * 52: * LDA (input) INTEGER 53: * The leading dimension of the array A. LDA >= max(1,N). 54: * 55: * AF (input) COMPLEX*16 array, dimension (LDAF,N) 56: * The factored form of the matrix A. AF contains the block 57: * diagonal matrix D and the multipliers used to obtain the 58: * factor U or L from the factorization A = U*D*U**T or 59: * A = L*D*L**T as computed by ZSYTRF. 60: * 61: * LDAF (input) INTEGER 62: * The leading dimension of the array AF. LDAF >= max(1,N). 63: * 64: * IPIV (input) INTEGER array, dimension (N) 65: * Details of the interchanges and the block structure of D 66: * as determined by ZSYTRF. 67: * 68: * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 69: * The right hand side matrix B. 70: * 71: * LDB (input) INTEGER 72: * The leading dimension of the array B. LDB >= max(1,N). 73: * 74: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 75: * On entry, the solution matrix X, as computed by ZSYTRS. 76: * On exit, the improved solution matrix X. 77: * 78: * LDX (input) INTEGER 79: * The leading dimension of the array X. LDX >= max(1,N). 80: * 81: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 82: * The estimated forward error bound for each solution vector 83: * X(j) (the j-th column of the solution matrix X). 84: * If XTRUE is the true solution corresponding to X(j), FERR(j) 85: * is an estimated upper bound for the magnitude of the largest 86: * element in (X(j) - XTRUE) divided by the magnitude of the 87: * largest element in X(j). The estimate is as reliable as 88: * the estimate for RCOND, and is almost always a slight 89: * overestimate of the true error. 90: * 91: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 92: * The componentwise relative backward error of each solution 93: * vector X(j) (i.e., the smallest relative change in 94: * any element of A or B that makes X(j) an exact solution). 95: * 96: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 97: * 98: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 99: * 100: * INFO (output) INTEGER 101: * = 0: successful exit 102: * < 0: if INFO = -i, the i-th argument had an illegal value 103: * 104: * Internal Parameters 105: * =================== 106: * 107: * ITMAX is the maximum number of steps of iterative refinement. 108: * 109: * ===================================================================== 110: * 111: * .. Parameters .. 112: INTEGER ITMAX 113: PARAMETER ( ITMAX = 5 ) 114: DOUBLE PRECISION ZERO 115: PARAMETER ( ZERO = 0.0D+0 ) 116: COMPLEX*16 ONE 117: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 118: DOUBLE PRECISION TWO 119: PARAMETER ( TWO = 2.0D+0 ) 120: DOUBLE PRECISION THREE 121: PARAMETER ( THREE = 3.0D+0 ) 122: * .. 123: * .. Local Scalars .. 124: LOGICAL UPPER 125: INTEGER COUNT, I, J, K, KASE, NZ 126: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 127: COMPLEX*16 ZDUM 128: * .. 129: * .. Local Arrays .. 130: INTEGER ISAVE( 3 ) 131: * .. 132: * .. External Subroutines .. 133: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSYMV, ZSYTRS 134: * .. 135: * .. Intrinsic Functions .. 136: INTRINSIC ABS, DBLE, DIMAG, MAX 137: * .. 138: * .. External Functions .. 139: LOGICAL LSAME 140: DOUBLE PRECISION DLAMCH 141: EXTERNAL LSAME, DLAMCH 142: * .. 143: * .. Statement Functions .. 144: DOUBLE PRECISION CABS1 145: * .. 146: * .. Statement Function definitions .. 147: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 148: * .. 149: * .. Executable Statements .. 150: * 151: * Test the input parameters. 152: * 153: INFO = 0 154: UPPER = LSAME( UPLO, 'U' ) 155: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 156: INFO = -1 157: ELSE IF( N.LT.0 ) THEN 158: INFO = -2 159: ELSE IF( NRHS.LT.0 ) THEN 160: INFO = -3 161: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 162: INFO = -5 163: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 164: INFO = -7 165: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 166: INFO = -10 167: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 168: INFO = -12 169: END IF 170: IF( INFO.NE.0 ) THEN 171: CALL XERBLA( 'ZSYRFS', -INFO ) 172: RETURN 173: END IF 174: * 175: * Quick return if possible 176: * 177: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 178: DO 10 J = 1, NRHS 179: FERR( J ) = ZERO 180: BERR( J ) = ZERO 181: 10 CONTINUE 182: RETURN 183: END IF 184: * 185: * NZ = maximum number of nonzero elements in each row of A, plus 1 186: * 187: NZ = N + 1 188: EPS = DLAMCH( 'Epsilon' ) 189: SAFMIN = DLAMCH( 'Safe minimum' ) 190: SAFE1 = NZ*SAFMIN 191: SAFE2 = SAFE1 / EPS 192: * 193: * Do for each right hand side 194: * 195: DO 140 J = 1, NRHS 196: * 197: COUNT = 1 198: LSTRES = THREE 199: 20 CONTINUE 200: * 201: * Loop until stopping criterion is satisfied. 202: * 203: * Compute residual R = B - A * X 204: * 205: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 206: CALL ZSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 ) 207: * 208: * Compute componentwise relative backward error from formula 209: * 210: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 211: * 212: * where abs(Z) is the componentwise absolute value of the matrix 213: * or vector Z. If the i-th component of the denominator is less 214: * than SAFE2, then SAFE1 is added to the i-th components of the 215: * numerator and denominator before dividing. 216: * 217: DO 30 I = 1, N 218: RWORK( I ) = CABS1( B( I, J ) ) 219: 30 CONTINUE 220: * 221: * Compute abs(A)*abs(X) + abs(B). 222: * 223: IF( UPPER ) THEN 224: DO 50 K = 1, N 225: S = ZERO 226: XK = CABS1( X( K, J ) ) 227: DO 40 I = 1, K - 1 228: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 229: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 230: 40 CONTINUE 231: RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK + S 232: 50 CONTINUE 233: ELSE 234: DO 70 K = 1, N 235: S = ZERO 236: XK = CABS1( X( K, J ) ) 237: RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK 238: DO 60 I = K + 1, N 239: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK 240: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) ) 241: 60 CONTINUE 242: RWORK( K ) = RWORK( K ) + S 243: 70 CONTINUE 244: END IF 245: S = ZERO 246: DO 80 I = 1, N 247: IF( RWORK( I ).GT.SAFE2 ) THEN 248: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 249: ELSE 250: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 251: $ ( RWORK( I )+SAFE1 ) ) 252: END IF 253: 80 CONTINUE 254: BERR( J ) = S 255: * 256: * Test stopping criterion. Continue iterating if 257: * 1) The residual BERR(J) is larger than machine epsilon, and 258: * 2) BERR(J) decreased by at least a factor of 2 during the 259: * last iteration, and 260: * 3) At most ITMAX iterations tried. 261: * 262: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 263: $ COUNT.LE.ITMAX ) THEN 264: * 265: * Update solution and try again. 266: * 267: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 268: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 269: LSTRES = BERR( J ) 270: COUNT = COUNT + 1 271: GO TO 20 272: END IF 273: * 274: * Bound error from formula 275: * 276: * norm(X - XTRUE) / norm(X) .le. FERR = 277: * norm( abs(inv(A))* 278: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 279: * 280: * where 281: * norm(Z) is the magnitude of the largest component of Z 282: * inv(A) is the inverse of A 283: * abs(Z) is the componentwise absolute value of the matrix or 284: * vector Z 285: * NZ is the maximum number of nonzeros in any row of A, plus 1 286: * EPS is machine epsilon 287: * 288: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 289: * is incremented by SAFE1 if the i-th component of 290: * abs(A)*abs(X) + abs(B) is less than SAFE2. 291: * 292: * Use ZLACN2 to estimate the infinity-norm of the matrix 293: * inv(A) * diag(W), 294: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 295: * 296: DO 90 I = 1, N 297: IF( RWORK( I ).GT.SAFE2 ) THEN 298: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 299: ELSE 300: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 301: $ SAFE1 302: END IF 303: 90 CONTINUE 304: * 305: KASE = 0 306: 100 CONTINUE 307: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 308: IF( KASE.NE.0 ) THEN 309: IF( KASE.EQ.1 ) THEN 310: * 311: * Multiply by diag(W)*inv(A'). 312: * 313: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 314: DO 110 I = 1, N 315: WORK( I ) = RWORK( I )*WORK( I ) 316: 110 CONTINUE 317: ELSE IF( KASE.EQ.2 ) THEN 318: * 319: * Multiply by inv(A)*diag(W). 320: * 321: DO 120 I = 1, N 322: WORK( I ) = RWORK( I )*WORK( I ) 323: 120 CONTINUE 324: CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO ) 325: END IF 326: GO TO 100 327: END IF 328: * 329: * Normalize error. 330: * 331: LSTRES = ZERO 332: DO 130 I = 1, N 333: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 334: 130 CONTINUE 335: IF( LSTRES.NE.ZERO ) 336: $ FERR( J ) = FERR( J ) / LSTRES 337: * 338: 140 CONTINUE 339: * 340: RETURN 341: * 342: * End of ZSYRFS 343: * 344: END