Annotation of rpl/lapack/lapack/dptrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
! 2: $ BERR, WORK, 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: INTEGER INFO, LDB, LDX, N, NRHS
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
! 14: $ E( * ), EF( * ), FERR( * ), WORK( * ),
! 15: $ X( LDX, * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DPTRFS improves the computed solution to a system of linear
! 22: * equations when the coefficient matrix is symmetric positive definite
! 23: * and tridiagonal, and provides error bounds and backward error
! 24: * estimates for the solution.
! 25: *
! 26: * Arguments
! 27: * =========
! 28: *
! 29: * N (input) INTEGER
! 30: * The order of the matrix A. N >= 0.
! 31: *
! 32: * NRHS (input) INTEGER
! 33: * The number of right hand sides, i.e., the number of columns
! 34: * of the matrix B. NRHS >= 0.
! 35: *
! 36: * D (input) DOUBLE PRECISION array, dimension (N)
! 37: * The n diagonal elements of the tridiagonal matrix A.
! 38: *
! 39: * E (input) DOUBLE PRECISION array, dimension (N-1)
! 40: * The (n-1) subdiagonal elements of the tridiagonal matrix A.
! 41: *
! 42: * DF (input) DOUBLE PRECISION array, dimension (N)
! 43: * The n diagonal elements of the diagonal matrix D from the
! 44: * factorization computed by DPTTRF.
! 45: *
! 46: * EF (input) DOUBLE PRECISION array, dimension (N-1)
! 47: * The (n-1) subdiagonal elements of the unit bidiagonal factor
! 48: * L from the factorization computed by DPTTRF.
! 49: *
! 50: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 51: * The right hand side matrix B.
! 52: *
! 53: * LDB (input) INTEGER
! 54: * The leading dimension of the array B. LDB >= max(1,N).
! 55: *
! 56: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 57: * On entry, the solution matrix X, as computed by DPTTRS.
! 58: * On exit, the improved solution matrix X.
! 59: *
! 60: * LDX (input) INTEGER
! 61: * The leading dimension of the array X. LDX >= max(1,N).
! 62: *
! 63: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 64: * The forward error bound for each solution vector
! 65: * X(j) (the j-th column of the solution matrix X).
! 66: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 67: * is an estimated upper bound for the magnitude of the largest
! 68: * element in (X(j) - XTRUE) divided by the magnitude of the
! 69: * largest element in X(j).
! 70: *
! 71: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 72: * The componentwise relative backward error of each solution
! 73: * vector X(j) (i.e., the smallest relative change in
! 74: * any element of A or B that makes X(j) an exact solution).
! 75: *
! 76: * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
! 77: *
! 78: * INFO (output) INTEGER
! 79: * = 0: successful exit
! 80: * < 0: if INFO = -i, the i-th argument had an illegal value
! 81: *
! 82: * Internal Parameters
! 83: * ===================
! 84: *
! 85: * ITMAX is the maximum number of steps of iterative refinement.
! 86: *
! 87: * =====================================================================
! 88: *
! 89: * .. Parameters ..
! 90: INTEGER ITMAX
! 91: PARAMETER ( ITMAX = 5 )
! 92: DOUBLE PRECISION ZERO
! 93: PARAMETER ( ZERO = 0.0D+0 )
! 94: DOUBLE PRECISION ONE
! 95: PARAMETER ( ONE = 1.0D+0 )
! 96: DOUBLE PRECISION TWO
! 97: PARAMETER ( TWO = 2.0D+0 )
! 98: DOUBLE PRECISION THREE
! 99: PARAMETER ( THREE = 3.0D+0 )
! 100: * ..
! 101: * .. Local Scalars ..
! 102: INTEGER COUNT, I, IX, J, NZ
! 103: DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
! 104: $ SAFMIN
! 105: * ..
! 106: * .. External Subroutines ..
! 107: EXTERNAL DAXPY, DPTTRS, XERBLA
! 108: * ..
! 109: * .. Intrinsic Functions ..
! 110: INTRINSIC ABS, MAX
! 111: * ..
! 112: * .. External Functions ..
! 113: INTEGER IDAMAX
! 114: DOUBLE PRECISION DLAMCH
! 115: EXTERNAL IDAMAX, DLAMCH
! 116: * ..
! 117: * .. Executable Statements ..
! 118: *
! 119: * Test the input parameters.
! 120: *
! 121: INFO = 0
! 122: IF( N.LT.0 ) THEN
! 123: INFO = -1
! 124: ELSE IF( NRHS.LT.0 ) THEN
! 125: INFO = -2
! 126: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 127: INFO = -8
! 128: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 129: INFO = -10
! 130: END IF
! 131: IF( INFO.NE.0 ) THEN
! 132: CALL XERBLA( 'DPTRFS', -INFO )
! 133: RETURN
! 134: END IF
! 135: *
! 136: * Quick return if possible
! 137: *
! 138: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 139: DO 10 J = 1, NRHS
! 140: FERR( J ) = ZERO
! 141: BERR( J ) = ZERO
! 142: 10 CONTINUE
! 143: RETURN
! 144: END IF
! 145: *
! 146: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 147: *
! 148: NZ = 4
! 149: EPS = DLAMCH( 'Epsilon' )
! 150: SAFMIN = DLAMCH( 'Safe minimum' )
! 151: SAFE1 = NZ*SAFMIN
! 152: SAFE2 = SAFE1 / EPS
! 153: *
! 154: * Do for each right hand side
! 155: *
! 156: DO 90 J = 1, NRHS
! 157: *
! 158: COUNT = 1
! 159: LSTRES = THREE
! 160: 20 CONTINUE
! 161: *
! 162: * Loop until stopping criterion is satisfied.
! 163: *
! 164: * Compute residual R = B - A * X. Also compute
! 165: * abs(A)*abs(x) + abs(b) for use in the backward error bound.
! 166: *
! 167: IF( N.EQ.1 ) THEN
! 168: BI = B( 1, J )
! 169: DX = D( 1 )*X( 1, J )
! 170: WORK( N+1 ) = BI - DX
! 171: WORK( 1 ) = ABS( BI ) + ABS( DX )
! 172: ELSE
! 173: BI = B( 1, J )
! 174: DX = D( 1 )*X( 1, J )
! 175: EX = E( 1 )*X( 2, J )
! 176: WORK( N+1 ) = BI - DX - EX
! 177: WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
! 178: DO 30 I = 2, N - 1
! 179: BI = B( I, J )
! 180: CX = E( I-1 )*X( I-1, J )
! 181: DX = D( I )*X( I, J )
! 182: EX = E( I )*X( I+1, J )
! 183: WORK( N+I ) = BI - CX - DX - EX
! 184: WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
! 185: 30 CONTINUE
! 186: BI = B( N, J )
! 187: CX = E( N-1 )*X( N-1, J )
! 188: DX = D( N )*X( N, J )
! 189: WORK( N+N ) = BI - CX - DX
! 190: WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
! 191: END IF
! 192: *
! 193: * Compute componentwise relative backward error from formula
! 194: *
! 195: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
! 196: *
! 197: * where abs(Z) is the componentwise absolute value of the matrix
! 198: * or vector Z. If the i-th component of the denominator is less
! 199: * than SAFE2, then SAFE1 is added to the i-th components of the
! 200: * numerator and denominator before dividing.
! 201: *
! 202: S = ZERO
! 203: DO 40 I = 1, N
! 204: IF( WORK( I ).GT.SAFE2 ) THEN
! 205: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 206: ELSE
! 207: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 208: $ ( WORK( I )+SAFE1 ) )
! 209: END IF
! 210: 40 CONTINUE
! 211: BERR( J ) = S
! 212: *
! 213: * Test stopping criterion. Continue iterating if
! 214: * 1) The residual BERR(J) is larger than machine epsilon, and
! 215: * 2) BERR(J) decreased by at least a factor of 2 during the
! 216: * last iteration, and
! 217: * 3) At most ITMAX iterations tried.
! 218: *
! 219: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 220: $ COUNT.LE.ITMAX ) THEN
! 221: *
! 222: * Update solution and try again.
! 223: *
! 224: CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
! 225: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 226: LSTRES = BERR( J )
! 227: COUNT = COUNT + 1
! 228: GO TO 20
! 229: END IF
! 230: *
! 231: * Bound error from formula
! 232: *
! 233: * norm(X - XTRUE) / norm(X) .le. FERR =
! 234: * norm( abs(inv(A))*
! 235: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
! 236: *
! 237: * where
! 238: * norm(Z) is the magnitude of the largest component of Z
! 239: * inv(A) is the inverse of A
! 240: * abs(Z) is the componentwise absolute value of the matrix or
! 241: * vector Z
! 242: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 243: * EPS is machine epsilon
! 244: *
! 245: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
! 246: * is incremented by SAFE1 if the i-th component of
! 247: * abs(A)*abs(X) + abs(B) is less than SAFE2.
! 248: *
! 249: DO 50 I = 1, N
! 250: IF( WORK( I ).GT.SAFE2 ) THEN
! 251: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 252: ELSE
! 253: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 254: END IF
! 255: 50 CONTINUE
! 256: IX = IDAMAX( N, WORK, 1 )
! 257: FERR( J ) = WORK( IX )
! 258: *
! 259: * Estimate the norm of inv(A).
! 260: *
! 261: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
! 262: *
! 263: * m(i,j) = abs(A(i,j)), i = j,
! 264: * m(i,j) = -abs(A(i,j)), i .ne. j,
! 265: *
! 266: * and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'.
! 267: *
! 268: * Solve M(L) * x = e.
! 269: *
! 270: WORK( 1 ) = ONE
! 271: DO 60 I = 2, N
! 272: WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
! 273: 60 CONTINUE
! 274: *
! 275: * Solve D * M(L)' * x = b.
! 276: *
! 277: WORK( N ) = WORK( N ) / DF( N )
! 278: DO 70 I = N - 1, 1, -1
! 279: WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
! 280: 70 CONTINUE
! 281: *
! 282: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
! 283: *
! 284: IX = IDAMAX( N, WORK, 1 )
! 285: FERR( J ) = FERR( J )*ABS( WORK( IX ) )
! 286: *
! 287: * Normalize error.
! 288: *
! 289: LSTRES = ZERO
! 290: DO 80 I = 1, N
! 291: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 292: 80 CONTINUE
! 293: IF( LSTRES.NE.ZERO )
! 294: $ FERR( J ) = FERR( J ) / LSTRES
! 295: *
! 296: 90 CONTINUE
! 297: *
! 298: RETURN
! 299: *
! 300: * End of DPTRFS
! 301: *
! 302: END
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