Annotation of rpl/lapack/lapack/zpprfs.f, revision 1.1
1.1 ! bertrand 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
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