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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, 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: * .. Scalar Arguments .. 10: CHARACTER UPLO 11: INTEGER INFO, LDB, LDX, N, NRHS 12: * .. 13: * .. Array Arguments .. 14: DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ), 15: $ RWORK( * ) 16: COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ), 17: $ X( LDX, * ) 18: * .. 19: * 20: * Purpose 21: * ======= 22: * 23: * ZPTRFS improves the computed solution to a system of linear 24: * equations when the coefficient matrix is Hermitian positive definite 25: * and tridiagonal, and provides error bounds and backward error 26: * estimates for the solution. 27: * 28: * Arguments 29: * ========= 30: * 31: * UPLO (input) CHARACTER*1 32: * Specifies whether the superdiagonal or the subdiagonal of the 33: * tridiagonal matrix A is stored and the form of the 34: * factorization: 35: * = 'U': E is the superdiagonal of A, and A = U**H*D*U; 36: * = 'L': E is the subdiagonal of A, and A = L*D*L**H. 37: * (The two forms are equivalent if A is real.) 38: * 39: * N (input) INTEGER 40: * The order of the matrix A. N >= 0. 41: * 42: * NRHS (input) INTEGER 43: * The number of right hand sides, i.e., the number of columns 44: * of the matrix B. NRHS >= 0. 45: * 46: * D (input) DOUBLE PRECISION array, dimension (N) 47: * The n real diagonal elements of the tridiagonal matrix A. 48: * 49: * E (input) COMPLEX*16 array, dimension (N-1) 50: * The (n-1) off-diagonal elements of the tridiagonal matrix A 51: * (see UPLO). 52: * 53: * DF (input) DOUBLE PRECISION array, dimension (N) 54: * The n diagonal elements of the diagonal matrix D from 55: * the factorization computed by ZPTTRF. 56: * 57: * EF (input) COMPLEX*16 array, dimension (N-1) 58: * The (n-1) off-diagonal elements of the unit bidiagonal 59: * factor U or L from the factorization computed by ZPTTRF 60: * (see UPLO). 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 ZPTTRS. 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 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). 82: * 83: * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 84: * The componentwise relative backward error of each solution 85: * vector X(j) (i.e., the smallest relative change in 86: * any element of A or B that makes X(j) an exact solution). 87: * 88: * WORK (workspace) COMPLEX*16 array, dimension (N) 89: * 90: * RWORK (workspace) DOUBLE PRECISION array, dimension (N) 91: * 92: * INFO (output) INTEGER 93: * = 0: successful exit 94: * < 0: if INFO = -i, the i-th argument had an illegal value 95: * 96: * Internal Parameters 97: * =================== 98: * 99: * ITMAX is the maximum number of steps of iterative refinement. 100: * 101: * ===================================================================== 102: * 103: * .. Parameters .. 104: INTEGER ITMAX 105: PARAMETER ( ITMAX = 5 ) 106: DOUBLE PRECISION ZERO 107: PARAMETER ( ZERO = 0.0D+0 ) 108: DOUBLE PRECISION ONE 109: PARAMETER ( ONE = 1.0D+0 ) 110: DOUBLE PRECISION TWO 111: PARAMETER ( TWO = 2.0D+0 ) 112: DOUBLE PRECISION THREE 113: PARAMETER ( THREE = 3.0D+0 ) 114: * .. 115: * .. Local Scalars .. 116: LOGICAL UPPER 117: INTEGER COUNT, I, IX, J, NZ 118: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN 119: COMPLEX*16 BI, CX, DX, EX, ZDUM 120: * .. 121: * .. External Functions .. 122: LOGICAL LSAME 123: INTEGER IDAMAX 124: DOUBLE PRECISION DLAMCH 125: EXTERNAL LSAME, IDAMAX, DLAMCH 126: * .. 127: * .. External Subroutines .. 128: EXTERNAL XERBLA, ZAXPY, ZPTTRS 129: * .. 130: * .. Intrinsic Functions .. 131: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX 132: * .. 133: * .. Statement Functions .. 134: DOUBLE PRECISION CABS1 135: * .. 136: * .. Statement Function definitions .. 137: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 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( NRHS.LT.0 ) THEN 150: INFO = -3 151: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 152: INFO = -9 153: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 154: INFO = -11 155: END IF 156: IF( INFO.NE.0 ) THEN 157: CALL XERBLA( 'ZPTRFS', -INFO ) 158: RETURN 159: END IF 160: * 161: * Quick return if possible 162: * 163: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN 164: DO 10 J = 1, NRHS 165: FERR( J ) = ZERO 166: BERR( J ) = ZERO 167: 10 CONTINUE 168: RETURN 169: END IF 170: * 171: * NZ = maximum number of nonzero elements in each row of A, plus 1 172: * 173: NZ = 4 174: EPS = DLAMCH( 'Epsilon' ) 175: SAFMIN = DLAMCH( 'Safe minimum' ) 176: SAFE1 = NZ*SAFMIN 177: SAFE2 = SAFE1 / EPS 178: * 179: * Do for each right hand side 180: * 181: DO 100 J = 1, NRHS 182: * 183: COUNT = 1 184: LSTRES = THREE 185: 20 CONTINUE 186: * 187: * Loop until stopping criterion is satisfied. 188: * 189: * Compute residual R = B - A * X. Also compute 190: * abs(A)*abs(x) + abs(b) for use in the backward error bound. 191: * 192: IF( UPPER ) THEN 193: IF( N.EQ.1 ) THEN 194: BI = B( 1, J ) 195: DX = D( 1 )*X( 1, J ) 196: WORK( 1 ) = BI - DX 197: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) 198: ELSE 199: BI = B( 1, J ) 200: DX = D( 1 )*X( 1, J ) 201: EX = E( 1 )*X( 2, J ) 202: WORK( 1 ) = BI - DX - EX 203: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) + 204: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) ) 205: DO 30 I = 2, N - 1 206: BI = B( I, J ) 207: CX = DCONJG( E( I-1 ) )*X( I-1, J ) 208: DX = D( I )*X( I, J ) 209: EX = E( I )*X( I+1, J ) 210: WORK( I ) = BI - CX - DX - EX 211: RWORK( I ) = CABS1( BI ) + 212: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) + 213: $ CABS1( DX ) + CABS1( E( I ) )* 214: $ CABS1( X( I+1, J ) ) 215: 30 CONTINUE 216: BI = B( N, J ) 217: CX = DCONJG( E( N-1 ) )*X( N-1, J ) 218: DX = D( N )*X( N, J ) 219: WORK( N ) = BI - CX - DX 220: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )* 221: $ CABS1( X( N-1, J ) ) + CABS1( DX ) 222: END IF 223: ELSE 224: IF( N.EQ.1 ) THEN 225: BI = B( 1, J ) 226: DX = D( 1 )*X( 1, J ) 227: WORK( 1 ) = BI - DX 228: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) 229: ELSE 230: BI = B( 1, J ) 231: DX = D( 1 )*X( 1, J ) 232: EX = DCONJG( E( 1 ) )*X( 2, J ) 233: WORK( 1 ) = BI - DX - EX 234: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) + 235: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) ) 236: DO 40 I = 2, N - 1 237: BI = B( I, J ) 238: CX = E( I-1 )*X( I-1, J ) 239: DX = D( I )*X( I, J ) 240: EX = DCONJG( E( I ) )*X( I+1, J ) 241: WORK( I ) = BI - CX - DX - EX 242: RWORK( I ) = CABS1( BI ) + 243: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) + 244: $ CABS1( DX ) + CABS1( E( I ) )* 245: $ CABS1( X( I+1, J ) ) 246: 40 CONTINUE 247: BI = B( N, J ) 248: CX = E( N-1 )*X( N-1, J ) 249: DX = D( N )*X( N, J ) 250: WORK( N ) = BI - CX - DX 251: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )* 252: $ CABS1( X( N-1, J ) ) + CABS1( DX ) 253: END IF 254: END IF 255: * 256: * Compute componentwise relative backward error from formula 257: * 258: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) 259: * 260: * where abs(Z) is the componentwise absolute value of the matrix 261: * or vector Z. If the i-th component of the denominator is less 262: * than SAFE2, then SAFE1 is added to the i-th components of the 263: * numerator and denominator before dividing. 264: * 265: S = ZERO 266: DO 50 I = 1, N 267: IF( RWORK( I ).GT.SAFE2 ) THEN 268: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) ) 269: ELSE 270: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) / 271: $ ( RWORK( I )+SAFE1 ) ) 272: END IF 273: 50 CONTINUE 274: BERR( J ) = S 275: * 276: * Test stopping criterion. Continue iterating if 277: * 1) The residual BERR(J) is larger than machine epsilon, and 278: * 2) BERR(J) decreased by at least a factor of 2 during the 279: * last iteration, and 280: * 3) At most ITMAX iterations tried. 281: * 282: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND. 283: $ COUNT.LE.ITMAX ) THEN 284: * 285: * Update solution and try again. 286: * 287: CALL ZPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO ) 288: CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 ) 289: LSTRES = BERR( J ) 290: COUNT = COUNT + 1 291: GO TO 20 292: END IF 293: * 294: * Bound error from formula 295: * 296: * norm(X - XTRUE) / norm(X) .le. FERR = 297: * norm( abs(inv(A))* 298: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) 299: * 300: * where 301: * norm(Z) is the magnitude of the largest component of Z 302: * inv(A) is the inverse of A 303: * abs(Z) is the componentwise absolute value of the matrix or 304: * vector Z 305: * NZ is the maximum number of nonzeros in any row of A, plus 1 306: * EPS is machine epsilon 307: * 308: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) 309: * is incremented by SAFE1 if the i-th component of 310: * abs(A)*abs(X) + abs(B) is less than SAFE2. 311: * 312: DO 60 I = 1, N 313: IF( RWORK( I ).GT.SAFE2 ) THEN 314: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) 315: ELSE 316: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) + 317: $ SAFE1 318: END IF 319: 60 CONTINUE 320: IX = IDAMAX( N, RWORK, 1 ) 321: FERR( J ) = RWORK( IX ) 322: * 323: * Estimate the norm of inv(A). 324: * 325: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by 326: * 327: * m(i,j) = abs(A(i,j)), i = j, 328: * m(i,j) = -abs(A(i,j)), i .ne. j, 329: * 330: * and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. 331: * 332: * Solve M(L) * x = e. 333: * 334: RWORK( 1 ) = ONE 335: DO 70 I = 2, N 336: RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) ) 337: 70 CONTINUE 338: * 339: * Solve D * M(L)' * x = b. 340: * 341: RWORK( N ) = RWORK( N ) / DF( N ) 342: DO 80 I = N - 1, 1, -1 343: RWORK( I ) = RWORK( I ) / DF( I ) + 344: $ RWORK( I+1 )*ABS( EF( I ) ) 345: 80 CONTINUE 346: * 347: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n. 348: * 349: IX = IDAMAX( N, RWORK, 1 ) 350: FERR( J ) = FERR( J )*ABS( RWORK( IX ) ) 351: * 352: * Normalize error. 353: * 354: LSTRES = ZERO 355: DO 90 I = 1, N 356: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) ) 357: 90 CONTINUE 358: IF( LSTRES.NE.ZERO ) 359: $ FERR( J ) = FERR( J ) / LSTRES 360: * 361: 100 CONTINUE 362: * 363: RETURN 364: * 365: * End of ZPTRFS 366: * 367: END