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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, 2: $ 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, LDB, LDX, N, NRHS 14: * .. 15: * .. Array Arguments .. 16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 17: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), 18: $ X( LDX, * ) 19: * .. 20: * 21: * Purpose 22: * ======= 23: * 24: * ZPPRFS improves the computed solution to a system of linear 25: * equations when the coefficient matrix is Hermitian positive definite 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) COMPLEX*16 array, dimension (N*(N+1)/2) 44: * The upper or lower triangle of the Hermitian 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)*(2n-j)/2) = A(i,j) for j<=i<=n. 49: * 50: * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2) 51: * The triangular factor U or L from the Cholesky factorization 52: * A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF, 53: * packed columnwise in a linear array in the same format as A 54: * (see AP). 55: * 56: * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 57: * The right hand side matrix B. 58: * 59: * LDB (input) INTEGER 60: * The leading dimension of the array B. LDB >= max(1,N). 61: * 62: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 63: * On entry, the solution matrix X, as computed by ZPPTRS. 64: * On exit, the improved solution matrix X. 65: * 66: * LDX (input) INTEGER 67: * The leading dimension of the array X. LDX >= max(1,N). 68: * 69: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 70: * The estimated forward error bound for each solution vector 71: * X(j) (the j-th column of the solution matrix X). 72: * If XTRUE is the true solution corresponding to X(j), FERR(j) 73: * is an estimated upper bound for the magnitude of the largest 74: * element in (X(j) - XTRUE) divided by the magnitude of the 75: * largest element in X(j). The estimate is as reliable as 76: * the estimate for RCOND, and is almost always a slight 77: * overestimate of the true error. 78: * 79: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 80: * The componentwise relative backward error of each solution 81: * vector X(j) (i.e., the smallest relative change in 82: * any element of A or B that makes X(j) an exact solution). 83: * 84: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 85: * 86: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 87: * 88: * INFO (output) INTEGER 89: * = 0: successful exit 90: * < 0: if INFO = -i, the i-th argument had an illegal value 91: * 92: * Internal Parameters 93: * =================== 94: * 95: * ITMAX is the maximum number of steps of iterative refinement. 96: * 97: * ==================================================================== 98: * 99: * .. Parameters .. 100: INTEGER ITMAX 101: PARAMETER ( ITMAX = 5 ) 102: DOUBLE PRECISION ZERO 103: PARAMETER ( ZERO = 0.0D+0 ) 104: COMPLEX*16 CONE 105: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 106: DOUBLE PRECISION TWO 107: PARAMETER ( TWO = 2.0D+0 ) 108: DOUBLE PRECISION THREE 109: PARAMETER ( THREE = 3.0D+0 ) 110: * .. 111: * .. Local Scalars .. 112: LOGICAL UPPER 113: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ 114: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 115: COMPLEX*16 ZDUM 116: * .. 117: * .. Local Arrays .. 118: INTEGER ISAVE( 3 ) 119: * .. 120: * .. External Subroutines .. 121: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS 122: * .. 123: * .. Intrinsic Functions .. 124: INTRINSIC ABS, DBLE, DIMAG, MAX 125: * .. 126: * .. External Functions .. 127: LOGICAL LSAME 128: DOUBLE PRECISION DLAMCH 129: EXTERNAL LSAME, DLAMCH 130: * .. 131: * .. Statement Functions .. 132: DOUBLE PRECISION CABS1 133: * .. 134: * .. Statement Function definitions .. 135: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 136: * .. 137: * .. Executable Statements .. 138: * 139: * Test the input parameters. 140: * 141: INFO = 0 142: UPPER = LSAME( UPLO, 'U' ) 143: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 144: INFO = -1 145: ELSE IF( N.LT.0 ) THEN 146: INFO = -2 147: ELSE IF( NRHS.LT.0 ) THEN 148: INFO = -3 149: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 150: INFO = -7 151: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 152: INFO = -9 153: END IF 154: IF( INFO.NE.0 ) THEN 155: CALL XERBLA( 'ZPPRFS', -INFO ) 156: RETURN 157: END IF 158: * 159: * Quick return if possible 160: * 161: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 162: DO 10 J = 1, NRHS 163: FERR( J ) = ZERO 164: BERR( J ) = ZERO 165: 10 CONTINUE 166: RETURN 167: END IF 168: * 169: * NZ = maximum number of nonzero elements in each row of A, plus 1 170: * 171: NZ = N + 1 172: EPS = DLAMCH( 'Epsilon' ) 173: SAFMIN = DLAMCH( 'Safe minimum' ) 174: SAFE1 = NZ*SAFMIN 175: SAFE2 = SAFE1 / EPS 176: * 177: * Do for each right hand side 178: * 179: DO 140 J = 1, NRHS 180: * 181: COUNT = 1 182: LSTRES = THREE 183: 20 CONTINUE 184: * 185: * Loop until stopping criterion is satisfied. 186: * 187: * Compute residual R = B - A * X 188: * 189: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 190: CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 ) 191: * 192: * Compute componentwise relative backward error from formula 193: * 194: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 195: * 196: * where abs(Z) is the componentwise absolute value of the matrix 197: * or vector Z. If the i-th component of the denominator is less 198: * than SAFE2, then SAFE1 is added to the i-th components of the 199: * numerator and denominator before dividing. 200: * 201: DO 30 I = 1, N 202: RWORK( I ) = CABS1( B( I, J ) ) 203: 30 CONTINUE 204: * 205: * Compute abs(A)*abs(X) + abs(B). 206: * 207: KK = 1 208: IF( UPPER ) THEN 209: DO 50 K = 1, N 210: S = ZERO 211: XK = CABS1( X( K, J ) ) 212: IK = KK 213: DO 40 I = 1, K - 1 214: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 215: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 216: IK = IK + 1 217: 40 CONTINUE 218: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )* 219: $ XK + S 220: KK = KK + K 221: 50 CONTINUE 222: ELSE 223: DO 70 K = 1, N 224: S = ZERO 225: XK = CABS1( X( K, J ) ) 226: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK 227: IK = KK + 1 228: DO 60 I = K + 1, N 229: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 230: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 231: IK = IK + 1 232: 60 CONTINUE 233: RWORK( K ) = RWORK( K ) + S 234: KK = KK + ( N-K+1 ) 235: 70 CONTINUE 236: END IF 237: S = ZERO 238: DO 80 I = 1, N 239: IF( RWORK( I ).GT.SAFE2 ) THEN 240: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 241: ELSE 242: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 243: $ ( RWORK( I )+SAFE1 ) ) 244: END IF 245: 80 CONTINUE 246: BERR( J ) = S 247: * 248: * Test stopping criterion. Continue iterating if 249: * 1) The residual BERR(J) is larger than machine epsilon, and 250: * 2) BERR(J) decreased by at least a factor of 2 during the 251: * last iteration, and 252: * 3) At most ITMAX iterations tried. 253: * 254: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 255: $ COUNT.LE.ITMAX ) THEN 256: * 257: * Update solution and try again. 258: * 259: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) 260: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 ) 261: LSTRES = BERR( J ) 262: COUNT = COUNT + 1 263: GO TO 20 264: END IF 265: * 266: * Bound error from formula 267: * 268: * norm(X - XTRUE) / norm(X) .le. FERR = 269: * norm( abs(inv(A))* 270: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 271: * 272: * where 273: * norm(Z) is the magnitude of the largest component of Z 274: * inv(A) is the inverse of A 275: * abs(Z) is the componentwise absolute value of the matrix or 276: * vector Z 277: * NZ is the maximum number of nonzeros in any row of A, plus 1 278: * EPS is machine epsilon 279: * 280: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 281: * is incremented by SAFE1 if the i-th component of 282: * abs(A)*abs(X) + abs(B) is less than SAFE2. 283: * 284: * Use ZLACN2 to estimate the infinity-norm of the matrix 285: * inv(A) * diag(W), 286: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 287: * 288: DO 90 I = 1, N 289: IF( RWORK( I ).GT.SAFE2 ) THEN 290: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 291: ELSE 292: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 293: $ SAFE1 294: END IF 295: 90 CONTINUE 296: * 297: KASE = 0 298: 100 CONTINUE 299: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 300: IF( KASE.NE.0 ) THEN 301: IF( KASE.EQ.1 ) THEN 302: * 303: * Multiply by diag(W)*inv(A'). 304: * 305: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO ) 306: DO 110 I = 1, N 307: WORK( I ) = RWORK( I )*WORK( I ) 308: 110 CONTINUE 309: ELSE IF( KASE.EQ.2 ) THEN 310: * 311: * Multiply by inv(A)*diag(W). 312: * 313: DO 120 I = 1, N 314: WORK( I ) = RWORK( I )*WORK( I ) 315: 120 CONTINUE 316: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, 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, CABS1( 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 ZPPRFS 335: * 336: END