Annotation of rpl/lapack/lapack/dporfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
! 2: $ LDX, FERR, BERR, WORK, IWORK, 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 DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER UPLO
! 13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DPORFS improves the computed solution to a system of linear
! 25: * equations when the coefficient matrix is symmetric positive definite,
! 26: * and provides error bounds and backward error estimates for the
! 27: * 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: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 44: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
! 45: * upper triangular part of A contains the upper triangular part
! 46: * of the matrix A, and the strictly lower triangular part of A
! 47: * is not referenced. If UPLO = 'L', the leading N-by-N lower
! 48: * triangular part of A contains the lower triangular part of
! 49: * the matrix A, and the strictly upper triangular part of A is
! 50: * not referenced.
! 51: *
! 52: * LDA (input) INTEGER
! 53: * The leading dimension of the array A. LDA >= max(1,N).
! 54: *
! 55: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
! 56: * The triangular factor U or L from the Cholesky factorization
! 57: * A = U**T*U or A = L*L**T, as computed by DPOTRF.
! 58: *
! 59: * LDAF (input) INTEGER
! 60: * The leading dimension of the array AF. LDAF >= max(1,N).
! 61: *
! 62: * B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 69: * On entry, the solution matrix X, as computed by DPOTRS.
! 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) DOUBLE PRECISION array, dimension (3*N)
! 91: *
! 92: * IWORK (workspace) INTEGER 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: DOUBLE PRECISION ONE
! 111: PARAMETER ( ONE = 1.0D+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, J, K, KASE, NZ
! 120: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 121: * ..
! 122: * .. Local Arrays ..
! 123: INTEGER ISAVE( 3 )
! 124: * ..
! 125: * .. External Subroutines ..
! 126: EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA
! 127: * ..
! 128: * .. Intrinsic Functions ..
! 129: INTRINSIC ABS, MAX
! 130: * ..
! 131: * .. External Functions ..
! 132: LOGICAL LSAME
! 133: DOUBLE PRECISION DLAMCH
! 134: EXTERNAL LSAME, DLAMCH
! 135: * ..
! 136: * .. Executable Statements ..
! 137: *
! 138: * Test the input parameters.
! 139: *
! 140: INFO = 0
! 141: UPPER = LSAME( UPLO, 'U' )
! 142: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 143: INFO = -1
! 144: ELSE IF( N.LT.0 ) THEN
! 145: INFO = -2
! 146: ELSE IF( NRHS.LT.0 ) THEN
! 147: INFO = -3
! 148: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 149: INFO = -5
! 150: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
! 151: INFO = -7
! 152: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 153: INFO = -9
! 154: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 155: INFO = -11
! 156: END IF
! 157: IF( INFO.NE.0 ) THEN
! 158: CALL XERBLA( 'DPORFS', -INFO )
! 159: RETURN
! 160: END IF
! 161: *
! 162: * Quick return if possible
! 163: *
! 164: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 165: DO 10 J = 1, NRHS
! 166: FERR( J ) = ZERO
! 167: BERR( J ) = ZERO
! 168: 10 CONTINUE
! 169: RETURN
! 170: END IF
! 171: *
! 172: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 173: *
! 174: NZ = N + 1
! 175: EPS = DLAMCH( 'Epsilon' )
! 176: SAFMIN = DLAMCH( 'Safe minimum' )
! 177: SAFE1 = NZ*SAFMIN
! 178: SAFE2 = SAFE1 / EPS
! 179: *
! 180: * Do for each right hand side
! 181: *
! 182: DO 140 J = 1, NRHS
! 183: *
! 184: COUNT = 1
! 185: LSTRES = THREE
! 186: 20 CONTINUE
! 187: *
! 188: * Loop until stopping criterion is satisfied.
! 189: *
! 190: * Compute residual R = B - A * X
! 191: *
! 192: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 193: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
! 194: $ WORK( N+1 ), 1 )
! 195: *
! 196: * Compute componentwise relative backward error from formula
! 197: *
! 198: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
! 199: *
! 200: * where abs(Z) is the componentwise absolute value of the matrix
! 201: * or vector Z. If the i-th component of the denominator is less
! 202: * than SAFE2, then SAFE1 is added to the i-th components of the
! 203: * numerator and denominator before dividing.
! 204: *
! 205: DO 30 I = 1, N
! 206: WORK( I ) = ABS( B( I, J ) )
! 207: 30 CONTINUE
! 208: *
! 209: * Compute abs(A)*abs(X) + abs(B).
! 210: *
! 211: IF( UPPER ) THEN
! 212: DO 50 K = 1, N
! 213: S = ZERO
! 214: XK = ABS( X( K, J ) )
! 215: DO 40 I = 1, K - 1
! 216: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
! 217: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
! 218: 40 CONTINUE
! 219: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
! 220: 50 CONTINUE
! 221: ELSE
! 222: DO 70 K = 1, N
! 223: S = ZERO
! 224: XK = ABS( X( K, J ) )
! 225: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
! 226: DO 60 I = K + 1, N
! 227: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
! 228: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
! 229: 60 CONTINUE
! 230: WORK( K ) = WORK( K ) + S
! 231: 70 CONTINUE
! 232: END IF
! 233: S = ZERO
! 234: DO 80 I = 1, N
! 235: IF( WORK( I ).GT.SAFE2 ) THEN
! 236: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 237: ELSE
! 238: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 239: $ ( WORK( I )+SAFE1 ) )
! 240: END IF
! 241: 80 CONTINUE
! 242: BERR( J ) = S
! 243: *
! 244: * Test stopping criterion. Continue iterating if
! 245: * 1) The residual BERR(J) is larger than machine epsilon, and
! 246: * 2) BERR(J) decreased by at least a factor of 2 during the
! 247: * last iteration, and
! 248: * 3) At most ITMAX iterations tried.
! 249: *
! 250: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 251: $ COUNT.LE.ITMAX ) THEN
! 252: *
! 253: * Update solution and try again.
! 254: *
! 255: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
! 256: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 257: LSTRES = BERR( J )
! 258: COUNT = COUNT + 1
! 259: GO TO 20
! 260: END IF
! 261: *
! 262: * Bound error from formula
! 263: *
! 264: * norm(X - XTRUE) / norm(X) .le. FERR =
! 265: * norm( abs(inv(A))*
! 266: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
! 267: *
! 268: * where
! 269: * norm(Z) is the magnitude of the largest component of Z
! 270: * inv(A) is the inverse of A
! 271: * abs(Z) is the componentwise absolute value of the matrix or
! 272: * vector Z
! 273: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 274: * EPS is machine epsilon
! 275: *
! 276: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
! 277: * is incremented by SAFE1 if the i-th component of
! 278: * abs(A)*abs(X) + abs(B) is less than SAFE2.
! 279: *
! 280: * Use DLACN2 to estimate the infinity-norm of the matrix
! 281: * inv(A) * diag(W),
! 282: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
! 283: *
! 284: DO 90 I = 1, N
! 285: IF( WORK( I ).GT.SAFE2 ) THEN
! 286: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 287: ELSE
! 288: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 289: END IF
! 290: 90 CONTINUE
! 291: *
! 292: KASE = 0
! 293: 100 CONTINUE
! 294: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 295: $ KASE, ISAVE )
! 296: IF( KASE.NE.0 ) THEN
! 297: IF( KASE.EQ.1 ) THEN
! 298: *
! 299: * Multiply by diag(W)*inv(A').
! 300: *
! 301: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
! 302: DO 110 I = 1, N
! 303: WORK( N+I ) = WORK( I )*WORK( N+I )
! 304: 110 CONTINUE
! 305: ELSE IF( KASE.EQ.2 ) THEN
! 306: *
! 307: * Multiply by inv(A)*diag(W).
! 308: *
! 309: DO 120 I = 1, N
! 310: WORK( N+I ) = WORK( I )*WORK( N+I )
! 311: 120 CONTINUE
! 312: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
! 313: END IF
! 314: GO TO 100
! 315: END IF
! 316: *
! 317: * Normalize error.
! 318: *
! 319: LSTRES = ZERO
! 320: DO 130 I = 1, N
! 321: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 322: 130 CONTINUE
! 323: IF( LSTRES.NE.ZERO )
! 324: $ FERR( J ) = FERR( J ) / LSTRES
! 325: *
! 326: 140 CONTINUE
! 327: *
! 328: RETURN
! 329: *
! 330: * End of DPORFS
! 331: *
! 332: END
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