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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, 2: $ 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, LDB, LDX, N, NRHS 14: * .. 15: * .. Array Arguments .. 16: INTEGER IPIV( * ) 17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ) 18: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ), 19: $ X( LDX, * ) 20: * .. 21: * 22: * Purpose 23: * ======= 24: * 25: * ZSPRFS improves the computed solution to a system of linear 26: * equations when the coefficient matrix is symmetric indefinite 27: * and packed, and provides error bounds and backward error estimates 28: * for the solution. 29: * 30: * Arguments 31: * ========= 32: * 33: * UPLO (input) CHARACTER*1 34: * = 'U': Upper triangle of A is stored; 35: * = 'L': Lower triangle of A is stored. 36: * 37: * N (input) INTEGER 38: * The order of the matrix A. N >= 0. 39: * 40: * NRHS (input) INTEGER 41: * The number of right hand sides, i.e., the number of columns 42: * of the matrices B and X. NRHS >= 0. 43: * 44: * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) 45: * The upper or lower triangle of the symmetric matrix A, packed 46: * columnwise in a linear array. The j-th column of A is stored 47: * in the array AP as follows: 48: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 49: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. 50: * 51: * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2) 52: * The factored form of the matrix A. AFP contains the block 53: * diagonal matrix D and the multipliers used to obtain the 54: * factor U or L from the factorization A = U*D*U**T or 55: * A = L*D*L**T as computed by ZSPTRF, stored as a packed 56: * triangular matrix. 57: * 58: * IPIV (input) INTEGER array, dimension (N) 59: * Details of the interchanges and the block structure of D 60: * as determined by ZSPTRF. 61: * 62: * B (input) COMPLEX*16 array, dimension (LDB,NRHS) 63: * The right hand side matrix B. 64: * 65: * LDB (input) INTEGER 66: * The leading dimension of the array B. LDB >= max(1,N). 67: * 68: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) 69: * On entry, the solution matrix X, as computed by ZSPTRS. 70: * On exit, the improved solution matrix X. 71: * 72: * LDX (input) INTEGER 73: * The leading dimension of the array X. LDX >= max(1,N). 74: * 75: * FERR (output) DOUBLE PRECISION array, dimension (NRHS) 76: * The estimated forward error bound for each solution vector 77: * X(j) (the j-th column of the solution matrix X). 78: * If XTRUE is the true solution corresponding to X(j), FERR(j) 79: * is an estimated upper bound for the magnitude of the largest 80: * element in (X(j) - XTRUE) divided by the magnitude of the 81: * largest element in X(j). The estimate is as reliable as 82: * the estimate for RCOND, and is almost always a slight 83: * overestimate of the true error. 84: * 85: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 86: * The componentwise relative backward error of each solution 87: * vector X(j) (i.e., the smallest relative change in 88: * any element of A or B that makes X(j) an exact solution). 89: * 90: * WORK (workspace) COMPLEX*16 array, dimension (2*N) 91: * 92: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 93: * 94: * INFO (output) INTEGER 95: * = 0: successful exit 96: * < 0: if INFO = -i, the i-th argument had an illegal value 97: * 98: * Internal Parameters 99: * =================== 100: * 101: * ITMAX is the maximum number of steps of iterative refinement. 102: * 103: * ===================================================================== 104: * 105: * .. Parameters .. 106: INTEGER ITMAX 107: PARAMETER ( ITMAX = 5 ) 108: DOUBLE PRECISION ZERO 109: PARAMETER ( ZERO = 0.0D+0 ) 110: COMPLEX*16 ONE 111: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 112: DOUBLE PRECISION TWO 113: PARAMETER ( TWO = 2.0D+0 ) 114: DOUBLE PRECISION THREE 115: PARAMETER ( THREE = 3.0D+0 ) 116: * .. 117: * .. Local Scalars .. 118: LOGICAL UPPER 119: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ 120: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK 121: COMPLEX*16 ZDUM 122: * .. 123: * .. Local Arrays .. 124: INTEGER ISAVE( 3 ) 125: * .. 126: * .. External Subroutines .. 127: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSPMV, ZSPTRS 128: * .. 129: * .. Intrinsic Functions .. 130: INTRINSIC ABS, DBLE, DIMAG, MAX 131: * .. 132: * .. External Functions .. 133: LOGICAL LSAME 134: DOUBLE PRECISION DLAMCH 135: EXTERNAL LSAME, DLAMCH 136: * .. 137: * .. Statement Functions .. 138: DOUBLE PRECISION CABS1 139: * .. 140: * .. Statement Function definitions .. 141: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 142: * .. 143: * .. Executable Statements .. 144: * 145: * Test the input parameters. 146: * 147: INFO = 0 148: UPPER = LSAME( UPLO, 'U' ) 149: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 150: INFO = -1 151: ELSE IF( N.LT.0 ) THEN 152: INFO = -2 153: ELSE IF( NRHS.LT.0 ) THEN 154: INFO = -3 155: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 156: INFO = -8 157: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 158: INFO = -10 159: END IF 160: IF( INFO.NE.0 ) THEN 161: CALL XERBLA( 'ZSPRFS', -INFO ) 162: RETURN 163: END IF 164: * 165: * Quick return if possible 166: * 167: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 168: DO 10 J = 1, NRHS 169: FERR( J ) = ZERO 170: BERR( J ) = ZERO 171: 10 CONTINUE 172: RETURN 173: END IF 174: * 175: * NZ = maximum number of nonzero elements in each row of A, plus 1 176: * 177: NZ = N + 1 178: EPS = DLAMCH( 'Epsilon' ) 179: SAFMIN = DLAMCH( 'Safe minimum' ) 180: SAFE1 = NZ*SAFMIN 181: SAFE2 = SAFE1 / EPS 182: * 183: * Do for each right hand side 184: * 185: DO 140 J = 1, NRHS 186: * 187: COUNT = 1 188: LSTRES = THREE 189: 20 CONTINUE 190: * 191: * Loop until stopping criterion is satisfied. 192: * 193: * Compute residual R = B - A * X 194: * 195: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 ) 196: CALL ZSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 ) 197: * 198: * Compute componentwise relative backward error from formula 199: * 200: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 201: * 202: * where abs(Z) is the componentwise absolute value of the matrix 203: * or vector Z. If the i-th component of the denominator is less 204: * than SAFE2, then SAFE1 is added to the i-th components of the 205: * numerator and denominator before dividing. 206: * 207: DO 30 I = 1, N 208: RWORK( I ) = CABS1( B( I, J ) ) 209: 30 CONTINUE 210: * 211: * Compute abs(A)*abs(X) + abs(B). 212: * 213: KK = 1 214: IF( UPPER ) THEN 215: DO 50 K = 1, N 216: S = ZERO 217: XK = CABS1( X( K, J ) ) 218: IK = KK 219: DO 40 I = 1, K - 1 220: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 221: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 222: IK = IK + 1 223: 40 CONTINUE 224: RWORK( K ) = RWORK( K ) + CABS1( AP( KK+K-1 ) )*XK + S 225: KK = KK + K 226: 50 CONTINUE 227: ELSE 228: DO 70 K = 1, N 229: S = ZERO 230: XK = CABS1( X( K, J ) ) 231: RWORK( K ) = RWORK( K ) + CABS1( AP( KK ) )*XK 232: IK = KK + 1 233: DO 60 I = K + 1, N 234: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK 235: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) ) 236: IK = IK + 1 237: 60 CONTINUE 238: RWORK( K ) = RWORK( K ) + S 239: KK = KK + ( N-K+1 ) 240: 70 CONTINUE 241: END IF 242: S = ZERO 243: DO 80 I = 1, N 244: IF( RWORK( I ).GT.SAFE2 ) THEN 245: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 246: ELSE 247: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 248: $ ( RWORK( I )+SAFE1 ) ) 249: END IF 250: 80 CONTINUE 251: BERR( J ) = S 252: * 253: * Test stopping criterion. Continue iterating if 254: * 1) The residual BERR(J) is larger than machine epsilon, and 255: * 2) BERR(J) decreased by at least a factor of 2 during the 256: * last iteration, and 257: * 3) At most ITMAX iterations tried. 258: * 259: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 260: $ COUNT.LE.ITMAX ) THEN 261: * 262: * Update solution and try again. 263: * 264: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) 265: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 ) 266: LSTRES = BERR( J ) 267: COUNT = COUNT + 1 268: GO TO 20 269: END IF 270: * 271: * Bound error from formula 272: * 273: * norm(X - XTRUE) / norm(X) .le. FERR = 274: * norm( abs(inv(A))* 275: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 276: * 277: * where 278: * norm(Z) is the magnitude of the largest component of Z 279: * inv(A) is the inverse of A 280: * abs(Z) is the componentwise absolute value of the matrix or 281: * vector Z 282: * NZ is the maximum number of nonzeros in any row of A, plus 1 283: * EPS is machine epsilon 284: * 285: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 286: * is incremented by SAFE1 if the i-th component of 287: * abs(A)*abs(X) + abs(B) is less than SAFE2. 288: * 289: * Use ZLACN2 to estimate the infinity-norm of the matrix 290: * inv(A) * diag(W), 291: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) 292: * 293: DO 90 I = 1, N 294: IF( RWORK( I ).GT.SAFE2 ) THEN 295: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 296: ELSE 297: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 298: $ SAFE1 299: END IF 300: 90 CONTINUE 301: * 302: KASE = 0 303: 100 CONTINUE 304: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE ) 305: IF( KASE.NE.0 ) THEN 306: IF( KASE.EQ.1 ) THEN 307: * 308: * Multiply by diag(W)*inv(A'). 309: * 310: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) 311: DO 110 I = 1, N 312: WORK( I ) = RWORK( 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( I ) = RWORK( I )*WORK( I ) 320: 120 CONTINUE 321: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO ) 322: END IF 323: GO TO 100 324: END IF 325: * 326: * Normalize error. 327: * 328: LSTRES = ZERO 329: DO 130 I = 1, N 330: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) ) 331: 130 CONTINUE 332: IF( LSTRES.NE.ZERO ) 333: $ FERR( J ) = FERR( J ) / LSTRES 334: * 335: 140 CONTINUE 336: * 337: RETURN 338: * 339: * End of ZSPRFS 340: * 341: END