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