Annotation of rpl/lapack/lapack/dsprfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
! 2: $ 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, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IPIV( * ), IWORK( * )
! 17: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
! 18: $ FERR( * ), WORK( * ), X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DSPRFS improves the computed solution to a system of linear
! 25: * equations when the coefficient matrix is symmetric indefinite
! 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) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 44: * The upper or lower triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 49: *
! 50: * AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 51: * The factored form of the matrix A. AFP contains the block
! 52: * diagonal matrix D and the multipliers used to obtain the
! 53: * factor U or L from the factorization A = U*D*U**T or
! 54: * A = L*D*L**T as computed by DSPTRF, stored as a packed
! 55: * triangular matrix.
! 56: *
! 57: * IPIV (input) INTEGER array, dimension (N)
! 58: * Details of the interchanges and the block structure of D
! 59: * as determined by DSPTRF.
! 60: *
! 61: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 62: * The right hand side matrix B.
! 63: *
! 64: * LDB (input) INTEGER
! 65: * The leading dimension of the array B. LDB >= max(1,N).
! 66: *
! 67: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 68: * On entry, the solution matrix X, as computed by DSPTRS.
! 69: * On exit, the improved solution matrix X.
! 70: *
! 71: * LDX (input) INTEGER
! 72: * The leading dimension of the array X. LDX >= max(1,N).
! 73: *
! 74: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 75: * The estimated forward error bound for each solution vector
! 76: * X(j) (the j-th column of the solution matrix X).
! 77: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 78: * is an estimated upper bound for the magnitude of the largest
! 79: * element in (X(j) - XTRUE) divided by the magnitude of the
! 80: * largest element in X(j). The estimate is as reliable as
! 81: * the estimate for RCOND, and is almost always a slight
! 82: * overestimate of the true error.
! 83: *
! 84: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 85: * The componentwise relative backward error of each solution
! 86: * vector X(j) (i.e., the smallest relative change in
! 87: * any element of A or B that makes X(j) an exact solution).
! 88: *
! 89: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 90: *
! 91: * IWORK (workspace) INTEGER array, dimension (N)
! 92: *
! 93: * INFO (output) INTEGER
! 94: * = 0: successful exit
! 95: * < 0: if INFO = -i, the i-th argument had an illegal value
! 96: *
! 97: * Internal Parameters
! 98: * ===================
! 99: *
! 100: * ITMAX is the maximum number of steps of iterative refinement.
! 101: *
! 102: * =====================================================================
! 103: *
! 104: * .. Parameters ..
! 105: INTEGER ITMAX
! 106: PARAMETER ( ITMAX = 5 )
! 107: DOUBLE PRECISION ZERO
! 108: PARAMETER ( ZERO = 0.0D+0 )
! 109: DOUBLE PRECISION ONE
! 110: PARAMETER ( ONE = 1.0D+0 )
! 111: DOUBLE PRECISION TWO
! 112: PARAMETER ( TWO = 2.0D+0 )
! 113: DOUBLE PRECISION THREE
! 114: PARAMETER ( THREE = 3.0D+0 )
! 115: * ..
! 116: * .. Local Scalars ..
! 117: LOGICAL UPPER
! 118: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
! 119: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 120: * ..
! 121: * .. Local Arrays ..
! 122: INTEGER ISAVE( 3 )
! 123: * ..
! 124: * .. External Subroutines ..
! 125: EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
! 126: * ..
! 127: * .. Intrinsic Functions ..
! 128: INTRINSIC ABS, MAX
! 129: * ..
! 130: * .. External Functions ..
! 131: LOGICAL LSAME
! 132: DOUBLE PRECISION DLAMCH
! 133: EXTERNAL LSAME, DLAMCH
! 134: * ..
! 135: * .. Executable Statements ..
! 136: *
! 137: * Test the input parameters.
! 138: *
! 139: INFO = 0
! 140: UPPER = LSAME( UPLO, 'U' )
! 141: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 142: INFO = -1
! 143: ELSE IF( N.LT.0 ) THEN
! 144: INFO = -2
! 145: ELSE IF( NRHS.LT.0 ) THEN
! 146: INFO = -3
! 147: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 148: INFO = -8
! 149: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 150: INFO = -10
! 151: END IF
! 152: IF( INFO.NE.0 ) THEN
! 153: CALL XERBLA( 'DSPRFS', -INFO )
! 154: RETURN
! 155: END IF
! 156: *
! 157: * Quick return if possible
! 158: *
! 159: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 160: DO 10 J = 1, NRHS
! 161: FERR( J ) = ZERO
! 162: BERR( J ) = ZERO
! 163: 10 CONTINUE
! 164: RETURN
! 165: END IF
! 166: *
! 167: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 168: *
! 169: NZ = N + 1
! 170: EPS = DLAMCH( 'Epsilon' )
! 171: SAFMIN = DLAMCH( 'Safe minimum' )
! 172: SAFE1 = NZ*SAFMIN
! 173: SAFE2 = SAFE1 / EPS
! 174: *
! 175: * Do for each right hand side
! 176: *
! 177: DO 140 J = 1, NRHS
! 178: *
! 179: COUNT = 1
! 180: LSTRES = THREE
! 181: 20 CONTINUE
! 182: *
! 183: * Loop until stopping criterion is satisfied.
! 184: *
! 185: * Compute residual R = B - A * X
! 186: *
! 187: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 188: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
! 189: $ 1 )
! 190: *
! 191: * Compute componentwise relative backward error from formula
! 192: *
! 193: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
! 194: *
! 195: * where abs(Z) is the componentwise absolute value of the matrix
! 196: * or vector Z. If the i-th component of the denominator is less
! 197: * than SAFE2, then SAFE1 is added to the i-th components of the
! 198: * numerator and denominator before dividing.
! 199: *
! 200: DO 30 I = 1, N
! 201: WORK( I ) = ABS( B( I, J ) )
! 202: 30 CONTINUE
! 203: *
! 204: * Compute abs(A)*abs(X) + abs(B).
! 205: *
! 206: KK = 1
! 207: IF( UPPER ) THEN
! 208: DO 50 K = 1, N
! 209: S = ZERO
! 210: XK = ABS( X( K, J ) )
! 211: IK = KK
! 212: DO 40 I = 1, K - 1
! 213: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
! 214: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
! 215: IK = IK + 1
! 216: 40 CONTINUE
! 217: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
! 218: KK = KK + K
! 219: 50 CONTINUE
! 220: ELSE
! 221: DO 70 K = 1, N
! 222: S = ZERO
! 223: XK = ABS( X( K, J ) )
! 224: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
! 225: IK = KK + 1
! 226: DO 60 I = K + 1, N
! 227: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
! 228: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
! 229: IK = IK + 1
! 230: 60 CONTINUE
! 231: WORK( K ) = WORK( K ) + S
! 232: KK = KK + ( N-K+1 )
! 233: 70 CONTINUE
! 234: END IF
! 235: S = ZERO
! 236: DO 80 I = 1, N
! 237: IF( WORK( I ).GT.SAFE2 ) THEN
! 238: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 239: ELSE
! 240: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 241: $ ( WORK( I )+SAFE1 ) )
! 242: END IF
! 243: 80 CONTINUE
! 244: BERR( J ) = S
! 245: *
! 246: * Test stopping criterion. Continue iterating if
! 247: * 1) The residual BERR(J) is larger than machine epsilon, and
! 248: * 2) BERR(J) decreased by at least a factor of 2 during the
! 249: * last iteration, and
! 250: * 3) At most ITMAX iterations tried.
! 251: *
! 252: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 253: $ COUNT.LE.ITMAX ) THEN
! 254: *
! 255: * Update solution and try again.
! 256: *
! 257: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
! 258: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 259: LSTRES = BERR( J )
! 260: COUNT = COUNT + 1
! 261: GO TO 20
! 262: END IF
! 263: *
! 264: * Bound error from formula
! 265: *
! 266: * norm(X - XTRUE) / norm(X) .le. FERR =
! 267: * norm( abs(inv(A))*
! 268: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
! 269: *
! 270: * where
! 271: * norm(Z) is the magnitude of the largest component of Z
! 272: * inv(A) is the inverse of A
! 273: * abs(Z) is the componentwise absolute value of the matrix or
! 274: * vector Z
! 275: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 276: * EPS is machine epsilon
! 277: *
! 278: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
! 279: * is incremented by SAFE1 if the i-th component of
! 280: * abs(A)*abs(X) + abs(B) is less than SAFE2.
! 281: *
! 282: * Use DLACN2 to estimate the infinity-norm of the matrix
! 283: * inv(A) * diag(W),
! 284: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
! 285: *
! 286: DO 90 I = 1, N
! 287: IF( WORK( I ).GT.SAFE2 ) THEN
! 288: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 289: ELSE
! 290: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 291: END IF
! 292: 90 CONTINUE
! 293: *
! 294: KASE = 0
! 295: 100 CONTINUE
! 296: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 297: $ KASE, ISAVE )
! 298: IF( KASE.NE.0 ) THEN
! 299: IF( KASE.EQ.1 ) THEN
! 300: *
! 301: * Multiply by diag(W)*inv(A').
! 302: *
! 303: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
! 304: $ INFO )
! 305: DO 110 I = 1, N
! 306: WORK( N+I ) = WORK( I )*WORK( N+I )
! 307: 110 CONTINUE
! 308: ELSE IF( KASE.EQ.2 ) THEN
! 309: *
! 310: * Multiply by inv(A)*diag(W).
! 311: *
! 312: DO 120 I = 1, N
! 313: WORK( N+I ) = WORK( I )*WORK( N+I )
! 314: 120 CONTINUE
! 315: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
! 316: $ 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, ABS( 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 DSPRFS
! 335: *
! 336: END
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