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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, 2: $ LDB, 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, KD, LDAB, LDAFB, LDB, LDX, N, NRHS 14: * .. 15: * .. Array Arguments .. 16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 17: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 18: $ WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * ZPBRFS improves the computed solution to a system of linear 25: * equations when the coefficient matrix is Hermitian positive definite 26: * and banded, and provides error bounds and backward error estimates 27: * 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: * KD (input) INTEGER 40: * The number of superdiagonals of the matrix A if UPLO = 'U', 41: * or the number of subdiagonals if UPLO = 'L'. KD >= 0. 42: * 43: * NRHS (input) INTEGER 44: * The number of right hand sides, i.e., the number of columns 45: * of the matrices B and X. NRHS >= 0. 46: * 47: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N) 48: * The upper or lower triangle of the Hermitian band matrix A, 49: * stored in the first KD+1 rows of the array. The j-th column 50: * of A is stored in the j-th column of the array AB as follows: 51: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 52: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 53: * 54: * LDAB (input) INTEGER 55: * The leading dimension of the array AB. LDAB >= KD+1. 56: * 57: * AFB (input) COMPLEX*16 array, dimension (LDAFB,N) 58: * The triangular factor U or L from the Cholesky factorization 59: * A = U**H*U or A = L*L**H of the band matrix A as computed by 60: * ZPBTRF, in the same storage format as A (see AB). 61: * 62: * LDAFB (input) INTEGER 63: * The leading dimension of the array AFB. LDAFB >= KD+1. 64: * 65: * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 66: * The right hand side matrix B. 67: * 68: * LDB (input) INTEGER 69: * The leading dimension of the array B. LDB >= max(1,N). 70: * 71: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 72: * On entry, the solution matrix X, as computed by ZPBTRS. 73: * On exit, the improved solution matrix X. 74: * 75: * LDX (input) INTEGER 76: * The leading dimension of the array X. LDX >= max(1,N). 77: * 78: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 79: * The estimated forward error bound for each solution vector 80: * X(j) (the j-th column of the solution matrix X). 81: * If XTRUE is the true solution corresponding to X(j), FERR(j) 82: * is an estimated upper bound for the magnitude of the largest 83: * element in (X(j) - XTRUE) divided by the magnitude of the 84: * largest element in X(j). The estimate is as reliable as 85: * the estimate for RCOND, and is almost always a slight 86: * overestimate of the true error. 87: * 88: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 89: * The componentwise relative backward error of each solution 90: * vector X(j) (i.e., the smallest relative change in 91: * any element of A or B that makes X(j) an exact solution). 92: * 93: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 94: * 95: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 96: * 97: * INFO (output) INTEGER 98: * = 0: successful exit 99: * < 0: if INFO = -i, the i-th argument had an illegal value 100: * 101: * Internal Parameters 102: * =================== 103: * 104: * ITMAX is the maximum number of steps of iterative refinement. 105: * 106: * ===================================================================== 107: * 108: * .. Parameters .. 109: INTEGER ITMAX 110: PARAMETER ( ITMAX = 5 ) 111: DOUBLE PRECISION ZERO 112: PARAMETER ( ZERO = 0.0D+0 ) 113: COMPLEX*16 ONE 114: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 115: DOUBLE PRECISION TWO 116: PARAMETER ( TWO = 2.0D+0 ) 117: DOUBLE PRECISION THREE 118: PARAMETER ( THREE = 3.0D+0 ) 119: * .. 120: * .. Local Scalars .. 121: LOGICAL UPPER 122: INTEGER COUNT, I, J, K, KASE, L, NZ 123: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 124: COMPLEX*16 ZDUM 125: * .. 126: * .. Local Arrays .. 127: INTEGER ISAVE( 3 ) 128: * .. 129: * .. External Subroutines .. 130: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHBMV, ZLACN2, ZPBTRS 131: * .. 132: * .. Intrinsic Functions .. 133: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN 134: * .. 135: * .. External Functions .. 136: LOGICAL LSAME 137: DOUBLE PRECISION DLAMCH 138: EXTERNAL LSAME, DLAMCH 139: * .. 140: * .. Statement Functions .. 141: DOUBLE PRECISION CABS1 142: * .. 143: * .. Statement Function definitions .. 144: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 145: * .. 146: * .. Executable Statements .. 147: * 148: * Test the input parameters. 149: * 150: INFO = 0 151: UPPER = LSAME( UPLO, 'U' ) 152: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 153: INFO = -1 154: ELSE IF( N.LT.0 ) THEN 155: INFO = -2 156: ELSE IF( KD.LT.0 ) THEN 157: INFO = -3 158: ELSE IF( NRHS.LT.0 ) THEN 159: INFO = -4 160: ELSE IF( LDAB.LT.KD+1 ) THEN 161: INFO = -6 162: ELSE IF( LDAFB.LT.KD+1 ) THEN 163: INFO = -8 164: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 165: INFO = -10 166: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 167: INFO = -12 168: END IF 169: IF( INFO.NE.0 ) THEN 170: CALL XERBLA( 'ZPBRFS', -INFO ) 171: RETURN 172: END IF 173: * 174: * Quick return if possible 175: * 176: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 177: DO 10 J = 1, NRHS 178: FERR( J ) = ZERO 179: BERR( J ) = ZERO 180: 10 CONTINUE 181: RETURN 182: END IF 183: * 184: * NZ = maximum number of nonzero elements in each row of A, plus 1 185: * 186: NZ = MIN( N+1, 2*KD+2 ) 187: EPS = DLAMCH( 'Epsilon' ) 188: SAFMIN = DLAMCH( 'Safe minimum' ) 189: SAFE1 = NZ*SAFMIN 190: SAFE2 = SAFE1 / EPS 191: * 192: * Do for each right hand side 193: * 194: DO 140 J = 1, NRHS 195: * 196: COUNT = 1 197: LSTRES = THREE 198: 20 CONTINUE 199: * 200: * Loop until stopping criterion is satisfied. 201: * 202: * Compute residual R = B - A * X 203: * 204: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 205: CALL ZHBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE, 206: $ 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: L = KD + 1 - K 228: DO 40 I = MAX( 1, K-KD ), K - 1 229: RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK 230: S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) ) 231: 40 CONTINUE 232: RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( KD+1, K ) ) )* 233: $ XK + S 234: 50 CONTINUE 235: ELSE 236: DO 70 K = 1, N 237: S = ZERO 238: XK = CABS1( X( K, J ) ) 239: RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( 1, K ) ) )*XK 240: L = 1 - K 241: DO 60 I = K + 1, MIN( N, K+KD ) 242: RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK 243: S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) ) 244: 60 CONTINUE 245: RWORK( K ) = RWORK( K ) + S 246: 70 CONTINUE 247: END IF 248: S = ZERO 249: DO 80 I = 1, N 250: IF( RWORK( I ).GT.SAFE2 ) THEN 251: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 252: ELSE 253: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 254: $ ( RWORK( I )+SAFE1 ) ) 255: END IF 256: 80 CONTINUE 257: BERR( J ) = S 258: * 259: * Test stopping criterion. Continue iterating if 260: * 1) The residual BERR(J) is larger than machine epsilon, and 261: * 2) BERR(J) decreased by at least a factor of 2 during the 262: * last iteration, and 263: * 3) At most ITMAX iterations tried. 264: * 265: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 266: $ COUNT.LE.ITMAX ) THEN 267: * 268: * Update solution and try again. 269: * 270: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO ) 271: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 272: LSTRES = BERR( J ) 273: COUNT = COUNT + 1 274: GO TO 20 275: END IF 276: * 277: * Bound error from formula 278: * 279: * norm(X - XTRUE) / norm(X) .le. FERR = 280: * norm( abs(inv(A))* 281: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 282: * 283: * where 284: * norm(Z) is the magnitude of the largest component of Z 285: * inv(A) is the inverse of A 286: * abs(Z) is the componentwise absolute value of the matrix or 287: * vector Z 288: * NZ is the maximum number of nonzeros in any row of A, plus 1 289: * EPS is machine epsilon 290: * 291: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 292: * is incremented by SAFE1 if the i-th component of 293: * abs(A)*abs(X) + abs(B) is less than SAFE2. 294: * 295: * Use ZLACN2 to estimate the infinity-norm of the matrix 296: * inv(A) * diag(W), 297: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 298: * 299: DO 90 I = 1, N 300: IF( RWORK( I ).GT.SAFE2 ) THEN 301: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 302: ELSE 303: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 304: $ SAFE1 305: END IF 306: 90 CONTINUE 307: * 308: KASE = 0 309: 100 CONTINUE 310: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 311: IF( KASE.NE.0 ) THEN 312: IF( KASE.EQ.1 ) THEN 313: * 314: * Multiply by diag(W)*inv(A'). 315: * 316: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO ) 317: DO 110 I = 1, N 318: WORK( I ) = RWORK( I )*WORK( I ) 319: 110 CONTINUE 320: ELSE IF( KASE.EQ.2 ) THEN 321: * 322: * Multiply by inv(A)*diag(W). 323: * 324: DO 120 I = 1, N 325: WORK( I ) = RWORK( I )*WORK( I ) 326: 120 CONTINUE 327: CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO ) 328: END IF 329: GO TO 100 330: END IF 331: * 332: * Normalize error. 333: * 334: LSTRES = ZERO 335: DO 130 I = 1, N 336: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 337: 130 CONTINUE 338: IF( LSTRES.NE.ZERO ) 339: $ FERR( J ) = FERR( J ) / LSTRES 340: * 341: 140 CONTINUE 342: * 343: RETURN 344: * 345: * End of ZPBRFS 346: * 347: END