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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, 2: $ BERR, WORK, 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: * .. Scalar Arguments .. 10: INTEGER INFO, LDB, LDX, N, NRHS 11: * .. 12: * .. Array Arguments .. 13: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), 14: $ E( * ), EF( * ), FERR( * ), WORK( * ), 15: $ X( LDX, * ) 16: * .. 17: * 18: * Purpose 19: * ======= 20: * 21: * DPTRFS improves the computed solution to a system of linear 22: * equations when the coefficient matrix is symmetric positive definite 23: * and tridiagonal, and provides error bounds and backward error 24: * estimates for the solution. 25: * 26: * Arguments 27: * ========= 28: * 29: * N (input) INTEGER 30: * The order of the matrix A. N >= 0. 31: * 32: * NRHS (input) INTEGER 33: * The number of right hand sides, i.e., the number of columns 34: * of the matrix B. NRHS >= 0. 35: * 36: * D (input) DOUBLE PRECISION array, dimension (N) 37: * The n diagonal elements of the tridiagonal matrix A. 38: * 39: * E (input) DOUBLE PRECISION array, dimension (N-1) 40: * The (n-1) subdiagonal elements of the tridiagonal matrix A. 41: * 42: * DF (input) DOUBLE PRECISION array, dimension (N) 43: * The n diagonal elements of the diagonal matrix D from the 44: * factorization computed by DPTTRF. 45: * 46: * EF (input) DOUBLE PRECISION array, dimension (N-1) 47: * The (n-1) subdiagonal elements of the unit bidiagonal factor 48: * L from the factorization computed by DPTTRF. 49: * 50: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 51: * The right hand side matrix B. 52: * 53: * LDB (input) INTEGER 54: * The leading dimension of the array B. LDB >= max(1,N). 55: * 56: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) 57: * On entry, the solution matrix X, as computed by DPTTRS. 58: * On exit, the improved solution matrix X. 59: * 60: * LDX (input) INTEGER 61: * The leading dimension of the array X. LDX >= max(1,N). 62: * 63: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 64: * The forward error bound for each solution vector 65: * X(j) (the j-th column of the solution matrix X). 66: * If XTRUE is the true solution corresponding to X(j), FERR(j) 67: * is an estimated upper bound for the magnitude of the largest 68: * element in (X(j) - XTRUE) divided by the magnitude of the 69: * largest element in X(j). 70: * 71: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 72: * The componentwise relative backward error of each solution 73: * vector X(j) (i.e., the smallest relative change in 74: * any element of A or B that makes X(j) an exact solution). 75: * 76: * WORK (workspace) DOUBLE PRECISION array, dimension (2*N) 77: * 78: * INFO (output) INTEGER 79: * = 0: successful exit 80: * < 0: if INFO = -i, the i-th argument had an illegal value 81: * 82: * Internal Parameters 83: * =================== 84: * 85: * ITMAX is the maximum number of steps of iterative refinement. 86: * 87: * ===================================================================== 88: * 89: * .. Parameters .. 90: INTEGER ITMAX 91: PARAMETER ( ITMAX = 5 ) 92: DOUBLE PRECISION ZERO 93: PARAMETER ( ZERO = 0.0D+0 ) 94: DOUBLE PRECISION ONE 95: PARAMETER ( ONE = 1.0D+0 ) 96: DOUBLE PRECISION TWO 97: PARAMETER ( TWO = 2.0D+0 ) 98: DOUBLE PRECISION THREE 99: PARAMETER ( THREE = 3.0D+0 ) 100: * .. 101: * .. Local Scalars .. 102: INTEGER COUNT, I, IX, J, NZ 103: DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2, 104: $ SAFMIN 105: * .. 106: * .. External Subroutines .. 107: EXTERNAL DAXPY, DPTTRS, XERBLA 108: * .. 109: * .. Intrinsic Functions .. 110: INTRINSIC ABS, MAX 111: * .. 112: * .. External Functions .. 113: INTEGER IDAMAX 114: DOUBLE PRECISION DLAMCH 115: EXTERNAL IDAMAX, DLAMCH 116: * .. 117: * .. Executable Statements .. 118: * 119: * Test the input parameters. 120: * 121: INFO = 0 122: IF( N.LT.0 ) THEN 123: INFO = -1 124: ELSE IF( NRHS.LT.0 ) THEN 125: INFO = -2 126: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 127: INFO = -8 128: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 129: INFO = -10 130: END IF 131: IF( INFO.NE.0 ) THEN 132: CALL XERBLA( 'DPTRFS', -INFO ) 133: RETURN 134: END IF 135: * 136: * Quick return if possible 137: * 138: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 139: DO 10 J = 1, NRHS 140: FERR( J ) = ZERO 141: BERR( J ) = ZERO 142: 10 CONTINUE 143: RETURN 144: END IF 145: * 146: * NZ = maximum number of nonzero elements in each row of A, plus 1 147: * 148: NZ = 4 149: EPS = DLAMCH( 'Epsilon' ) 150: SAFMIN = DLAMCH( 'Safe minimum' ) 151: SAFE1 = NZ*SAFMIN 152: SAFE2 = SAFE1 / EPS 153: * 154: * Do for each right hand side 155: * 156: DO 90 J = 1, NRHS 157: * 158: COUNT = 1 159: LSTRES = THREE 160: 20 CONTINUE 161: * 162: * Loop until stopping criterion is satisfied. 163: * 164: * Compute residual R = B - A * X. Also compute 165: * abs(A)*abs(x) + abs(b) for use in the backward error bound. 166: * 167: IF( N.EQ.1 ) THEN 168: BI = B( 1, J ) 169: DX = D( 1 )*X( 1, J ) 170: WORK( N+1 ) = BI - DX 171: WORK( 1 ) = ABS( BI ) + ABS( DX ) 172: ELSE 173: BI = B( 1, J ) 174: DX = D( 1 )*X( 1, J ) 175: EX = E( 1 )*X( 2, J ) 176: WORK( N+1 ) = BI - DX - EX 177: WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX ) 178: DO 30 I = 2, N - 1 179: BI = B( I, J ) 180: CX = E( I-1 )*X( I-1, J ) 181: DX = D( I )*X( I, J ) 182: EX = E( I )*X( I+1, J ) 183: WORK( N+I ) = BI - CX - DX - EX 184: WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX ) 185: 30 CONTINUE 186: BI = B( N, J ) 187: CX = E( N-1 )*X( N-1, J ) 188: DX = D( N )*X( N, J ) 189: WORK( N+N ) = BI - CX - DX 190: WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX ) 191: END IF 192: * 193: * Compute componentwise relative backward error from formula 194: * 195: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 196: * 197: * where abs(Z) is the componentwise absolute value of the matrix 198: * or vector Z. If the i-th component of the denominator is less 199: * than SAFE2, then SAFE1 is added to the i-th components of the 200: * numerator and denominator before dividing. 201: * 202: S = ZERO 203: DO 40 I = 1, N 204: IF( WORK( I ).GT.SAFE2 ) THEN 205: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 206: ELSE 207: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 208: $ ( WORK( I )+SAFE1 ) ) 209: END IF 210: 40 CONTINUE 211: BERR( J ) = S 212: * 213: * Test stopping criterion. Continue iterating if 214: * 1) The residual BERR(J) is larger than machine epsilon, and 215: * 2) BERR(J) decreased by at least a factor of 2 during the 216: * last iteration, and 217: * 3) At most ITMAX iterations tried. 218: * 219: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 220: $ COUNT.LE.ITMAX ) THEN 221: * 222: * Update solution and try again. 223: * 224: CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO ) 225: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 226: LSTRES = BERR( J ) 227: COUNT = COUNT + 1 228: GO TO 20 229: END IF 230: * 231: * Bound error from formula 232: * 233: * norm(X - XTRUE) / norm(X) .le. FERR = 234: * norm( abs(inv(A))* 235: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 236: * 237: * where 238: * norm(Z) is the magnitude of the largest component of Z 239: * inv(A) is the inverse of A 240: * abs(Z) is the componentwise absolute value of the matrix or 241: * vector Z 242: * NZ is the maximum number of nonzeros in any row of A, plus 1 243: * EPS is machine epsilon 244: * 245: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 246: * is incremented by SAFE1 if the i-th component of 247: * abs(A)*abs(X) + abs(B) is less than SAFE2. 248: * 249: DO 50 I = 1, N 250: IF( WORK( I ).GT.SAFE2 ) THEN 251: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 252: ELSE 253: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 254: END IF 255: 50 CONTINUE 256: IX = IDAMAX( N, WORK, 1 ) 257: FERR( J ) = WORK( IX ) 258: * 259: * Estimate the norm of inv(A). 260: * 261: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by 262: * 263: * m(i,j) = abs(A(i,j)), i = j, 264: * m(i,j) = -abs(A(i,j)), i .ne. j, 265: * 266: * and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. 267: * 268: * Solve M(L) * x = e. 269: * 270: WORK( 1 ) = ONE 271: DO 60 I = 2, N 272: WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) ) 273: 60 CONTINUE 274: * 275: * Solve D * M(L)' * x = b. 276: * 277: WORK( N ) = WORK( N ) / DF( N ) 278: DO 70 I = N - 1, 1, -1 279: WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) ) 280: 70 CONTINUE 281: * 282: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n. 283: * 284: IX = IDAMAX( N, WORK, 1 ) 285: FERR( J ) = FERR( J )*ABS( WORK( IX ) ) 286: * 287: * Normalize error. 288: * 289: LSTRES = ZERO 290: DO 80 I = 1, N 291: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 292: 80 CONTINUE 293: IF( LSTRES.NE.ZERO ) 294: $ FERR( J ) = FERR( J ) / LSTRES 295: * 296: 90 CONTINUE 297: * 298: RETURN 299: * 300: * End of DPTRFS 301: * 302: END