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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, 2: $ 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, LDB, LDX, N, NRHS 14: * .. 15: * .. Array Arguments .. 16: INTEGER IPIV( * ), IWORK( * ) 17: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ), 18: $ FERR( * ), WORK( * ), X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * DSPRFS improves the computed solution to a system of linear 25: * equations when the coefficient matrix is symmetric indefinite 26: * and packed, 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: * 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: * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) 44: * The upper or lower triangle of the symmetric matrix A, packed 45: * columnwise in a linear array. The j-th column of A is stored 46: * in the array AP as follows: 47: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 48: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 49: * 50: * AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) 51: * The factored form of the matrix A. AFP contains the block 52: * diagonal matrix D and the multipliers used to obtain the 53: * factor U or L from the factorization A = U*D*U**T or 54: * A = L*D*L**T as computed by DSPTRF, stored as a packed 55: * triangular matrix. 56: * 57: * IPIV (input) INTEGER array, dimension (N) 58: * Details of the interchanges and the block structure of D 59: * as determined by DSPTRF. 60: * 61: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) 62: * The right hand side matrix B. 63: * 64: * LDB (input) INTEGER 65: * The leading dimension of the array B. LDB >= max(1,N). 66: * 67: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) 68: * On entry, the solution matrix X, as computed by DSPTRS. 69: * On exit, the improved solution matrix X. 70: * 71: * LDX (input) INTEGER 72: * The leading dimension of the array X. LDX >= max(1,N). 73: * 74: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 75: * The estimated forward error bound for each solution vector 76: * X(j) (the j-th column of the solution matrix X). 77: * If XTRUE is the true solution corresponding to X(j), FERR(j) 78: * is an estimated upper bound for the magnitude of the largest 79: * element in (X(j) - XTRUE) divided by the magnitude of the 80: * largest element in X(j). The estimate is as reliable as 81: * the estimate for RCOND, and is almost always a slight 82: * overestimate of the true error. 83: * 84: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 85: * The componentwise relative backward error of each solution 86: * vector X(j) (i.e., the smallest relative change in 87: * any element of A or B that makes X(j) an exact solution). 88: * 89: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) 90: * 91: * IWORK (workspace) INTEGER array, dimension (N) 92: * 93: * INFO (output) INTEGER 94: * = 0: successful exit 95: * < 0: if INFO = -i, the i-th argument had an illegal value 96: * 97: * Internal Parameters 98: * =================== 99: * 100: * ITMAX is the maximum number of steps of iterative refinement. 101: * 102: * ===================================================================== 103: * 104: * .. Parameters .. 105: INTEGER ITMAX 106: PARAMETER ( ITMAX = 5 ) 107: DOUBLE PRECISION ZERO 108: PARAMETER ( ZERO = 0.0D+0 ) 109: DOUBLE PRECISION ONE 110: PARAMETER ( ONE = 1.0D+0 ) 111: DOUBLE PRECISION TWO 112: PARAMETER ( TWO = 2.0D+0 ) 113: DOUBLE PRECISION THREE 114: PARAMETER ( THREE = 3.0D+0 ) 115: * .. 116: * .. Local Scalars .. 117: LOGICAL UPPER 118: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ 119: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 120: * .. 121: * .. Local Arrays .. 122: INTEGER ISAVE( 3 ) 123: * .. 124: * .. External Subroutines .. 125: EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA 126: * .. 127: * .. Intrinsic Functions .. 128: INTRINSIC ABS, MAX 129: * .. 130: * .. External Functions .. 131: LOGICAL LSAME 132: DOUBLE PRECISION DLAMCH 133: EXTERNAL LSAME, DLAMCH 134: * .. 135: * .. Executable Statements .. 136: * 137: * Test the input parameters. 138: * 139: INFO = 0 140: UPPER = LSAME( UPLO, 'U' ) 141: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 142: INFO = -1 143: ELSE IF( N.LT.0 ) THEN 144: INFO = -2 145: ELSE IF( NRHS.LT.0 ) THEN 146: INFO = -3 147: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 148: INFO = -8 149: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 150: INFO = -10 151: END IF 152: IF( INFO.NE.0 ) THEN 153: CALL XERBLA( 'DSPRFS', -INFO ) 154: RETURN 155: END IF 156: * 157: * Quick return if possible 158: * 159: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 160: DO 10 J = 1, NRHS 161: FERR( J ) = ZERO 162: BERR( J ) = ZERO 163: 10 CONTINUE 164: RETURN 165: END IF 166: * 167: * NZ = maximum number of nonzero elements in each row of A, plus 1 168: * 169: NZ = N + 1 170: EPS = DLAMCH( 'Epsilon' ) 171: SAFMIN = DLAMCH( 'Safe minimum' ) 172: SAFE1 = NZ*SAFMIN 173: SAFE2 = SAFE1 / EPS 174: * 175: * Do for each right hand side 176: * 177: DO 140 J = 1, NRHS 178: * 179: COUNT = 1 180: LSTRES = THREE 181: 20 CONTINUE 182: * 183: * Loop until stopping criterion is satisfied. 184: * 185: * Compute residual R = B - A * X 186: * 187: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 ) 188: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ), 189: $ 1 ) 190: * 191: * Compute componentwise relative backward error from formula 192: * 193: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 194: * 195: * where abs(Z) is the componentwise absolute value of the matrix 196: * or vector Z. If the i-th component of the denominator is less 197: * than SAFE2, then SAFE1 is added to the i-th components of the 198: * numerator and denominator before dividing. 199: * 200: DO 30 I = 1, N 201: WORK( I ) = ABS( B( I, J ) ) 202: 30 CONTINUE 203: * 204: * Compute abs(A)*abs(X) + abs(B). 205: * 206: KK = 1 207: IF( UPPER ) THEN 208: DO 50 K = 1, N 209: S = ZERO 210: XK = ABS( X( K, J ) ) 211: IK = KK 212: DO 40 I = 1, K - 1 213: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 214: S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 215: IK = IK + 1 216: 40 CONTINUE 217: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S 218: KK = KK + K 219: 50 CONTINUE 220: ELSE 221: DO 70 K = 1, N 222: S = ZERO 223: XK = ABS( X( K, J ) ) 224: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK 225: IK = KK + 1 226: DO 60 I = K + 1, N 227: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK 228: S = S + ABS( AP( IK ) )*ABS( X( I, J ) ) 229: IK = IK + 1 230: 60 CONTINUE 231: WORK( K ) = WORK( K ) + S 232: KK = KK + ( N-K+1 ) 233: 70 CONTINUE 234: END IF 235: S = ZERO 236: DO 80 I = 1, N 237: IF( WORK( I ).GT.SAFE2 ) THEN 238: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) ) 239: ELSE 240: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) / 241: $ ( WORK( I )+SAFE1 ) ) 242: END IF 243: 80 CONTINUE 244: BERR( J ) = S 245: * 246: * Test stopping criterion. Continue iterating if 247: * 1) The residual BERR(J) is larger than machine epsilon, and 248: * 2) BERR(J) decreased by at least a factor of 2 during the 249: * last iteration, and 250: * 3) At most ITMAX iterations tried. 251: * 252: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 253: $ COUNT.LE.ITMAX ) THEN 254: * 255: * Update solution and try again. 256: * 257: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO ) 258: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 ) 259: LSTRES = BERR( J ) 260: COUNT = COUNT + 1 261: GO TO 20 262: END IF 263: * 264: * Bound error from formula 265: * 266: * norm(X - XTRUE) / norm(X) .le. FERR = 267: * norm( abs(inv(A))* 268: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 269: * 270: * where 271: * norm(Z) is the magnitude of the largest component of Z 272: * inv(A) is the inverse of A 273: * abs(Z) is the componentwise absolute value of the matrix or 274: * vector Z 275: * NZ is the maximum number of nonzeros in any row of A, plus 1 276: * EPS is machine epsilon 277: * 278: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 279: * is incremented by SAFE1 if the i-th component of 280: * abs(A)*abs(X) + abs(B) is less than SAFE2. 281: * 282: * Use DLACN2 to estimate the infinity-norm of the matrix 283: * inv(A) * diag(W), 284: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 285: * 286: DO 90 I = 1, N 287: IF( WORK( I ).GT.SAFE2 ) THEN 288: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) 289: ELSE 290: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1 291: END IF 292: 90 CONTINUE 293: * 294: KASE = 0 295: 100 CONTINUE 296: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ), 297: $ KASE, ISAVE ) 298: IF( KASE.NE.0 ) THEN 299: IF( KASE.EQ.1 ) THEN 300: * 301: * Multiply by diag(W)*inv(A'). 302: * 303: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, 304: $ INFO ) 305: DO 110 I = 1, N 306: WORK( N+I ) = WORK( I )*WORK( N+I ) 307: 110 CONTINUE 308: ELSE IF( KASE.EQ.2 ) THEN 309: * 310: * Multiply by inv(A)*diag(W). 311: * 312: DO 120 I = 1, N 313: WORK( N+I ) = WORK( I )*WORK( N+I ) 314: 120 CONTINUE 315: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, 316: $ INFO ) 317: END IF 318: GO TO 100 319: END IF 320: * 321: * Normalize error. 322: * 323: LSTRES = ZERO 324: DO 130 I = 1, N 325: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 326: 130 CONTINUE 327: IF( LSTRES.NE.ZERO ) 328: $ FERR( J ) = FERR( J ) / LSTRES 329: * 330: 140 CONTINUE 331: * 332: RETURN 333: * 334: * End of DSPRFS 335: * 336: END