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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, 2: $ LDB, X, LDX, FERR, BERR, WORK, IWORK, 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 DLACN2 in place of DLACON, 5 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: INTEGER IWORK( * ) 17: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * DPBRFS improves the computed solution to a system of linear 25: * equations when the coefficient matrix is symmetric 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 symmetric 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) DOUBLE PRECISION array, dimension (LDAFB,N) 58: * The triangular factor U or L from the Cholesky factorization 59: * A = U**T*U or A = L*L**T of the band matrix A as computed by 60: * DPBTRF, 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) 72: * On entry, the solution matrix X, as computed by DPBTRS. 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) DOUBLE PRECISION array, dimension (3*N) 94: * 95: * IWORK (workspace) INTEGER 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: DOUBLE PRECISION ONE 114: PARAMETER ( ONE = 1.0D+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: * .. 125: * .. Local Arrays .. 126: INTEGER ISAVE( 3 ) 127: * .. 128: * .. External Subroutines .. 129: EXTERNAL DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV, XERBLA 130: * .. 131: * .. Intrinsic Functions .. 132: INTRINSIC ABS, MAX, MIN 133: * .. 134: * .. External Functions .. 135: LOGICAL LSAME 136: DOUBLE PRECISION DLAMCH 137: EXTERNAL LSAME, DLAMCH 138: * .. 139: * .. Executable Statements .. 140: * 141: * Test the input parameters. 142: * 143: INFO = 0 144: UPPER = LSAME( UPLO, 'U' ) 145: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 146: INFO = -1 147: ELSE IF( N.LT.0 ) THEN 148: INFO = -2 149: ELSE IF( KD.LT.0 ) THEN 150: INFO = -3 151: ELSE IF( NRHS.LT.0 ) THEN 152: INFO = -4 153: ELSE IF( LDAB.LT.KD+1 ) THEN 154: INFO = -6 155: ELSE IF( LDAFB.LT.KD+1 ) THEN 156: INFO = -8 157: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 158: INFO = -10 159: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 160: INFO = -12 161: END IF 162: IF( INFO.NE.0 ) THEN 163: CALL XERBLA( 'DPBRFS', -INFO ) 164: RETURN 165: END IF 166: * 167: * Quick return if possible 168: * 169: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 170: DO 10 J = 1, NRHS 171: FERR( J ) = ZERO 172: BERR( J ) = ZERO 173: 10 CONTINUE 174: RETURN 175: END IF 176: * 177: * NZ = maximum number of nonzero elements in each row of A, plus 1 178: * 179: NZ = MIN( N+1, 2*KD+2 ) 180: EPS = DLAMCH( 'Epsilon' ) 181: SAFMIN = DLAMCH( 'Safe minimum' ) 182: SAFE1 = NZ*SAFMIN 183: SAFE2 = SAFE1 / EPS 184: * 185: * Do for each right hand side 186: * 187: DO 140 J = 1, NRHS 188: * 189: COUNT = 1 190: LSTRES = THREE 191: 20 CONTINUE 192: * 193: * Loop until stopping criterion is satisfied. 194: * 195: * Compute residual R = B - A * X 196: * 197: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 198: CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE, 199: $ WORK( N+1 ), 1 ) 200: * 201: * Compute componentwise relative backward error from formula 202: * 203: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 204: * 205: * where abs(Z) is the componentwise absolute value of the matrix 206: * or vector Z. If the i-th component of the denominator is less 207: * than SAFE2, then SAFE1 is added to the i-th components of the 208: * numerator and denominator before dividing. 209: * 210: DO 30 I = 1, N 211: WORK( I ) = ABS( B( I, J ) ) 212: 30 CONTINUE 213: * 214: * Compute abs(A)*abs(X) + abs(B). 215: * 216: IF( UPPER ) THEN 217: DO 50 K = 1, N 218: S = ZERO 219: XK = ABS( X( K, J ) ) 220: L = KD + 1 - K 221: DO 40 I = MAX( 1, K-KD ), K - 1 222: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK 223: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 224: 40 CONTINUE 225: WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S 226: 50 CONTINUE 227: ELSE 228: DO 70 K = 1, N 229: S = ZERO 230: XK = ABS( X( K, J ) ) 231: WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK 232: L = 1 - K 233: DO 60 I = K + 1, MIN( N, K+KD ) 234: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK 235: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) ) 236: 60 CONTINUE 237: WORK( K ) = WORK( K ) + S 238: 70 CONTINUE 239: END IF 240: S = ZERO 241: DO 80 I = 1, N 242: IF( WORK( I ).GT.SAFE2 ) THEN 243: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 244: ELSE 245: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 246: $ ( WORK( I )+SAFE1 ) ) 247: END IF 248: 80 CONTINUE 249: BERR( J ) = S 250: * 251: * Test stopping criterion. Continue iterating if 252: * 1) The residual BERR(J) is larger than machine epsilon, and 253: * 2) BERR(J) decreased by at least a factor of 2 during the 254: * last iteration, and 255: * 3) At most ITMAX iterations tried. 256: * 257: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 258: $ COUNT.LE.ITMAX ) THEN 259: * 260: * Update solution and try again. 261: * 262: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, 263: $ INFO ) 264: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 265: LSTRES = BERR( J ) 266: COUNT = COUNT + 1 267: GO TO 20 268: END IF 269: * 270: * Bound error from formula 271: * 272: * norm(X - XTRUE) / norm(X) .le. FERR = 273: * norm( abs(inv(A))* 274: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 275: * 276: * where 277: * norm(Z) is the magnitude of the largest component of Z 278: * inv(A) is the inverse of A 279: * abs(Z) is the componentwise absolute value of the matrix or 280: * vector Z 281: * NZ is the maximum number of nonzeros in any row of A, plus 1 282: * EPS is machine epsilon 283: * 284: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 285: * is incremented by SAFE1 if the i-th component of 286: * abs(A)*abs(X) + abs(B) is less than SAFE2. 287: * 288: * Use DLACN2 to estimate the infinity-norm of the matrix 289: * inv(A) * diag(W), 290: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 291: * 292: DO 90 I = 1, N 293: IF( WORK( I ).GT.SAFE2 ) THEN 294: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 295: ELSE 296: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 297: END IF 298: 90 CONTINUE 299: * 300: KASE = 0 301: 100 CONTINUE 302: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 303: $ KASE, ISAVE ) 304: IF( KASE.NE.0 ) THEN 305: IF( KASE.EQ.1 ) THEN 306: * 307: * Multiply by diag(W)*inv(A'). 308: * 309: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, 310: $ INFO ) 311: DO 110 I = 1, N 312: WORK( N+I ) = WORK( N+I )*WORK( I ) 313: 110 CONTINUE 314: ELSE IF( KASE.EQ.2 ) THEN 315: * 316: * Multiply by inv(A)*diag(W). 317: * 318: DO 120 I = 1, N 319: WORK( N+I ) = WORK( N+I )*WORK( I ) 320: 120 CONTINUE 321: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N, 322: $ INFO ) 323: END IF 324: GO TO 100 325: END IF 326: * 327: * Normalize error. 328: * 329: LSTRES = ZERO 330: DO 130 I = 1, N 331: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 332: 130 CONTINUE 333: IF( LSTRES.NE.ZERO ) 334: $ FERR( J ) = FERR( J ) / LSTRES 335: * 336: 140 CONTINUE 337: * 338: RETURN 339: * 340: * End of DPBRFS 341: * 342: END