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cosolver/FM/FMconvert/fmin.f95
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fmin.f95
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PROGRAM MINF
! Sample root-finding program.
! FM_FIND_MIN is a function minimization routine that uses Brent's method.
! The equation to be solved is F(X,NF) = 0.
! X is the argument to the function.
! NF is the function number in case roots to several functions are needed.
IMPLICIT NONE
! CHARACTER(80) :: ST1,ST2
! Declare the multiple precision variables.
REAL (KIND(3.1D1)) :: A,B,TOL,XVAL,FVAL
REAL (KIND(3.1D1)), EXTERNAL :: F
! Set the FM precision to 50 significant digits (plus a few more "guard digits")
! CALL FM_SET(50)
! Find a minimum of the first function, X**3 - 9*X + 17.
! A, B are two endpoints of an interval in which the search takes place.
A = 1
B = 2
! TOL is the error tolerance. For most functions, the best accuracy we can obtain
! corresponds to about half the digits being used for the arithmetic.
! EPSILON(A) gives the relative accuracy of full precision, so SQRT(EPSILON(A))
! gives the relative accuracy of half precision.
TOL = SQRT(EPSILON(A))
! For this call no trace output will be done (KPRT = 0).
! KW = 6 is used, so any error messages will go to the screen.
WRITE (*,*) ' '
WRITE (*,*) ' '
WRITE (*,*) ' Case 1. Call FM_FIND_MIN to find a relative minimum between 1 and 2'
WRITE (*,*) ' for f(x) = X**3 - 9*X + 17.'
WRITE (*,*) ' Use KPRT = 0, so no output will be done in the routine, then'
WRITE (*,*) ' write the results from the main program.'
CALL FM_FIND_MIN(1,A,B,TOL,XVAL,FVAL,F,1,0,6)
! Write the result.
! CALL FM_FORM('F32.30',XVAL,ST1)
! CALL FM_FORM('F32.30',FVAL,ST2)
! WRITE (*,"(/' A minimum for function 1 is'/' x = ',A/' f(x) = ',A)") &
! TRIM(ST1),TRIM(ST2)
WRITE (*,"(/' A minimum for function 1 is'/5X," // &
"' x = ',ES25.16,', f(x) = ',ES25.16)") XVAL,FVAL
! Find a maximum of the first function, X**3 - 9*X + 17.
! This time we use FM_FIND_MIN's built-in trace (KPRT = 1) to print the final
! approximation to the root. The output will appear on more than one line, to
! allow for the possibility that precision could be hundreds or thousands of digits,
! so the number might not fit on one line.
WRITE (*,*) ' '
WRITE (*,*) ' '
WRITE (*,*) ' Case 2. Find a relative maximum between -5 and 1.'
WRITE (*,*) ' Use KPRT = 1, so FM_FIND_MIN will print the results.'
CALL FM_FIND_MIN(2,-5.0D0,1.0D0,TOL,XVAL,FVAL,F,1,1,6)
! Find a maximum of the first function, X**3 - 9*X + 17.
! See what happens when the maximum value is at an endpoint of the search interval.
! The algorithm still finds a relative maximum in the interior of the interval,
! not the absolute maximum at x=5.
WRITE (*,*) ' '
WRITE (*,*) ' '
WRITE (*,*) ' Case 3. Find a relative maximum between -5 and 5.'
WRITE (*,*) ' Use KPRT = 2, so FM_FIND_MIN will print all iterations,'
WRITE (*,*) ' as well as the final results.'
CALL FM_FIND_MIN(2,-5.0D0,5.0D0,TOL,XVAL,FVAL,F,1,2,6)
WRITE (*,*) ' '
END PROGRAM MINF
! http://www.netlib.no/netlib/go/FVAL.f
! function FVAL(ax,bx,f,tol)
SUBROUTINE FM_FIND_MIN(MIN_OR_MAX,AX,BX,TOL,XVAL,FVAL,F,NF,KPRT,KW)
! MIN_OR_MAX having value 1 means minimize the function, otherwise maximize.
! AX, BX define the endpoints of an interval in which the search takes place.
! TOL is the tolerance for the minimum. Usually TOL should be no less than
! sqrt(epsilon(ax)), meaning the x-coordinate XVAL of the extreme point will
! be accurate to only about half the digits carried. The y-coordinate FVAL
! should be accurate to nearly full precision.
! This happens because the typical graph is nearly parabolic near the minimum,
! and within sqrt(epsilon(ax)) of the minimum all the function values are
! essentially identical at the current precision.
! XVAL is returned as the value of X that minimizes (or maximizes) function F(X,NF).
! It is a relative extreme point, and may not be the global extreme point if the
! function has more than one extremum on the interval.
! FVAL is returned as the function value at XVAL.
! F(X,NF) is the function to be minimized. X is the argument and NF is the function
! number, in case several functions are defined within F.
! KPRT controls printing within the routine:
! KPRT = 0 for no output
! KPRT = 1 for the approximation to the root and the function
! value to be printed once at the end of the routine.
! KPRT = 2 for the approximation to the root and the function
! value to be printed each iteration.
! KW is the unit number for output.
! The method used is a combination of golden section search and successive parabolic interpolation.
! Convergence is never much slower than that for a fibonacci search. If f has a continuous second
! derivative which is positive at the minimum (which is not at ax or bx), then convergence is
! superlinear, and usually of the order of about 1.324....
! The function f is never evaluated at two points closer together than eps*abs(FVAL)+(tol/3),
! where eps is approximately the square root of the relative machine precision. If f is a
! unimodal function and the computed values of f are always unimodal when separated by at least
! eps*abs(x)+(tol/3), then FVAL approximates the abscissa of the global minimum of f on the
! interval ax,bx with an error less than 3*eps*abs(FVAL)+tol. If f is not unimodal, then FVAL
! may approximate a local, but perhaps non-global, minimum to the same accuracy.
! This routine is a slightly modified translation of function FVAL from netlib, which was adapted
! from the algol 60 procedure localmin given by Richard Brent in Algorithms For Minimization
! Without Derivatives, Prentice-Hall (1973).
IMPLICIT NONE
! CHARACTER(80) :: ST1,ST2
INTEGER :: J,MINV,MIN_OR_MAX,NF,KPRT,KW
REAL (KIND(3.1D1)) :: AX,BX,TOL,XVAL,FVAL
REAL (KIND(3.1D1)), EXTERNAL :: F
REAL (KIND(3.1D1)) :: A,B,C,D,E,EPS,XM,P,Q,R,TOL1,T2,U,V,W,FU,FV,FW,FX,X,TOL3
MINV = 1
IF (MIN_OR_MAX /= 1) MINV = -1
! C is the squared inverse of the golden ratio.
C = (3-SQRT(5.0D0))/2
! EPS is approximately the square root of the relative machine precision.
EPS = EPSILON(AX)
TOL1 = EPS + 1
EPS = SQRT(EPS)
A = MIN(AX,BX)
B = MAX(AX,BX)
V = A + C*(B-A)
W = V
X = V
E = 0
FX = F(X,NF)*MINV
FV = FX
FW = FX
TOL3 = TOL/3
J = 1
IF (KPRT == 2) THEN
WRITE (KW,*) ' '
IF (MIN_OR_MAX == 1) THEN
WRITE (KW,*) ' FM_FIND_MIN. Begin trace of all iterations.'
WRITE (KW,*) ' Search for a relative minimum on the interval'
WRITE (KW,"(13X,ES20.10,' to ',ES20.10/)") AX,BX
ELSE
WRITE (KW,*) ' FM_FIND_MIN. Begin trace of all iterations.'
WRITE (KW,*) ' Search for a relative maximum on the interval'
WRITE (KW,"(13X,ES20.10,' to ',ES20.10/)") AX,BX
ENDIF
! ST1 = FM_FORMAT('ES35.25',X)
! ST2 = FM_FORMAT('ES35.25',FX*MINV)
! WRITE (KW,"(' J =',I3,4X,' x = ',A)") J,TRIM(ST1)
! WRITE (KW,"(' ',3X,4X,'f(x) = ',A/)") TRIM(ST2)
WRITE (KW,"(' J =',I3,4X,'X =',ES25.16,' F(X) =',ES25.16)") J,X,FX*MINV
ENDIF
! The main loop starts here.
110 XM = (A+B)/2
TOL1 = EPS*ABS(X) + TOL3
T2 = 2*TOL1
! Check the stopping criterion.
IF (ABS(X-XM) <= (T2-(B-A)/2)) GO TO 160
P = 0
Q = 0
R = 0
IF (ABS(E) > TOL1) THEN
R = (X-W)*(FX-FV) ! Fit a parabola.
Q = (X-V)*(FX-FW)
P = (X-V)*Q-(X-W)*R
Q = 2*(Q-R)
IF (Q > 0) THEN
P = -P
ELSE
Q = -Q
ENDIF
R = E
E = D
ENDIF
IF ((ABS(P) >= ABS(Q*R/2)) .OR. (P <= Q*(A-X)) .OR. (P >= Q*(B-X))) GO TO 120
! Make a parabolic-interpolation step.
D = P/Q
U = X + D
! f must not be evaluated too close to ax or bx.
IF (((U-A) >= T2) .AND. ((B-U) >= T2)) GO TO 130
D = TOL1
IF (X >= XM) D = -D
GO TO 130
! Make a golden-section step.
120 IF (X < XM) THEN
E = B - X
ELSE
E = A - X
ENDIF
D = C*E
! f must not be evaluated too close to x.
130 IF (ABS(D) >= TOL1) THEN
U = X + D
ELSE
IF (D > 0) THEN
U = X + TOL1
ELSE
U = X - TOL1
ENDIF
ENDIF
FU = F(U,NF)*MINV
J = J + 1
IF (KPRT == 2) THEN
! ST1 = FM_FORMAT('ES35.25',U)
! ST2 = FM_FORMAT('ES35.25',FU*MINV)
! WRITE (KW,"(' J =',I3,4X,' x = ',A)") J,TRIM(ST1)
! WRITE (KW,"(' ',3X,4X,'f(x) = ',A/)") TRIM(ST2)
WRITE (KW,"(' J =',I3,4X,'X =',ES25.16,' F(X) =',ES25.16)") J,U,FU*MINV
ENDIF
! update a, b, v, w, and x.
IF (FX <= FU) THEN
IF (U < X) THEN
A = U
ELSE
B = U
ENDIF
ENDIF
IF (FU > FX) GO TO 140
IF (U < X) THEN
B = X
ELSE
A = X
ENDIF
V = W
FV = FW
W = X
FW = FX
X = U
FX = FU
GO TO 110
140 IF ((FU > FW) .AND. (W /= X)) GO TO 150
V = W
FV = FW
W = U
FW = FU
GO TO 110
150 IF ((FU > FV) .AND. (V /= X) .AND. (V /= W)) GO TO 110
V = U
FV = FU
GO TO 110
! end of main loop
160 XVAL = X
FVAL = FX*MINV
IF (KPRT >= 1) THEN
WRITE (KW,*) ' '
IF (MIN_OR_MAX == 1) THEN
WRITE (KW,"(' FM_FIND_MIN. Function ',I3,I6,' iterations. A relative minimum"// &
" on interval'/13X,ES20.10,' to ',ES20.10,' is')") &
NF,J,AX,BX
! ST1 = FM_FORMAT('ES35.25',XVAL)
! ST2 = FM_FORMAT('ES35.25',FVAL)
! WRITE (KW,"(15X,' x = ',A)") TRIM(ST1)
! WRITE (KW,"(15X,' f(x) = ',A)") TRIM(ST2)
WRITE (KW,"(15X,' x =',ES24.16,', f(x) =',ES24.16)") XVAL,FVAL
ELSE
WRITE (KW,"(' FM_FIND_MIN. Function ',I3,I6,' iterations. A relative maximum"// &
" on interval'/13X,ES20.10,' to ',ES20.10,' is')") &
NF,J,AX,BX
! ST1 = FM_FORMAT('ES35.25',XVAL)
! ST2 = FM_FORMAT('ES35.25',FVAL)
! WRITE (KW,"(15X,' x = ',A)") TRIM(ST1)
! WRITE (KW,"(15X,' f(x) = ',A)") TRIM(ST2)
WRITE (KW,"(15X,' x =',ES24.16,', f(x) =',ES24.16)") XVAL,FVAL
ENDIF
WRITE (KW,*) ' '
ENDIF
END SUBROUTINE FM_FIND_MIN
FUNCTION F(X,NF)
! X is the argument to the function.
! NF is the function number.
IMPLICIT NONE
INTEGER :: NF
REAL (KIND(3.1D1)) :: F,X
IF (NF == 1) THEN
F = X**3 - 9*X + 17
ELSE IF (NF == 2) THEN
F = X*TAN(X) - 1
ELSE
F = 3*X**2 + X - 2
ENDIF
END FUNCTION F
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