?? dlatrs.f
字號:
JLAST = N
JINC = 1
ELSE
JFIRST = N
JLAST = 1
JINC = -1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 80
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, M(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 60 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
XJ = ONE + CNORM( J )
GROW = MIN( GROW, XBND / XJ )
*
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
TJJ = ABS( A( J, J ) )
IF( XJ.GT.TJJ )
$ XBND = XBND*( TJJ / XJ )
60 CONTINUE
GROW = MIN( GROW, XBND )
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 70 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = ( 1 + CNORM(j) )*G(j-1)
*
XJ = ONE + CNORM( J )
GROW = GROW / XJ
70 CONTINUE
END IF
80 CONTINUE
END IF
*
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
* Use the Level 2 BLAS solve if the reciprocal of the bound on
* elements of X is not too small.
*
CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
ELSE
*
* Use a Level 1 BLAS solve, scaling intermediate results.
*
IF( XMAX.GT.BIGNUM ) THEN
*
* Scale X so that its components are less than or equal to
* BIGNUM in absolute value.
*
SCALE = BIGNUM / XMAX
CALL DSCAL( N, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( NOTRAN ) THEN
*
* Solve A * x = b
*
DO 110 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 100
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by 1/b(j).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
* to avoid overflow when dividing by A(j,j).
*
REC = ( TJJ*BIGNUM ) / XJ
IF( CNORM( J ).GT.ONE ) THEN
*
* Scale by 1/CNORM(j) to avoid overflow when
* multiplying x(j) times column j.
*
REC = REC / CNORM( J )
END IF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A*x = 0.
*
DO 90 I = 1, N
X( I ) = ZERO
90 CONTINUE
X( J ) = ONE
XJ = ONE
SCALE = ZERO
XMAX = ZERO
END IF
100 CONTINUE
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j of A.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
* Scale x by 1/(2*abs(x(j))).
*
REC = REC*HALF
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
* Scale x by 1/2.
*
CALL DSCAL( N, HALF, X, 1 )
SCALE = SCALE*HALF
END IF
*
IF( UPPER ) THEN
IF( J.GT.1 ) THEN
*
* Compute the update
* x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
$ 1 )
I = IDAMAX( J-1, X, 1 )
XMAX = ABS( X( I ) )
END IF
ELSE
IF( J.LT.N ) THEN
*
* Compute the update
* x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
$ X( J+1 ), 1 )
I = J + IDAMAX( N-J, X( J+1 ), 1 )
XMAX = ABS( X( I ) )
END IF
END IF
110 CONTINUE
*
ELSE
*
* Solve A' * x = b
*
DO 160 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) - sum A(k,j)*x(k).
* k<>j
*
XJ = ABS( X( J ) )
USCAL = TSCAL
REC = ONE / MAX( XMAX, ONE )
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
* If x(j) could overflow, scale x by 1/(2*XMAX).
*
REC = REC*HALF
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.ONE ) THEN
*
* Divide by A(j,j) when scaling x if A(j,j) > 1.
*
REC = MIN( ONE, REC*TJJ )
USCAL = USCAL / TJJS
END IF
IF( REC.LT.ONE ) THEN
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
SUMJ = ZERO
IF( USCAL.EQ.ONE ) THEN
*
* If the scaling needed for A in the dot product is 1,
* call DDOT to perform the dot product.
*
IF( UPPER ) THEN
SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 )
ELSE IF( J.LT.N ) THEN
SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
END IF
ELSE
*
* Otherwise, use in-line code for the dot product.
*
IF( UPPER ) THEN
DO 120 I = 1, J - 1
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
120 CONTINUE
ELSE IF( J.LT.N ) THEN
DO 130 I = J + 1, N
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
130 CONTINUE
END IF
END IF
*
IF( USCAL.EQ.TSCAL ) THEN
*
* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
* was not used to scale the dotproduct.
*
X( J ) = X( J ) - SUMJ
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 150
END IF
*
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale X by 1/abs(x(j)).
*
REC = ONE / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
REC = ( TJJ*BIGNUM ) / XJ
CALL DSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A'*x = 0.
*
DO 140 I = 1, N
X( I ) = ZERO
140 CONTINUE
X( J ) = ONE
SCALE = ZERO
XMAX = ZERO
END IF
150 CONTINUE
ELSE
*
* Compute x(j) := x(j) / A(j,j) - sumj if the dot
* product has already been divided by 1/A(j,j).
*
X( J ) = X( J ) / TJJS - SUMJ
END IF
XMAX = MAX( XMAX, ABS( X( J ) ) )
160 CONTINUE
END IF
SCALE = SCALE / TSCAL
END IF
*
* Scale the column norms by 1/TSCAL for return.
*
IF( TSCAL.NE.ONE ) THEN
CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
END IF
*
RETURN
*
* End of DLATRS
*
END
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