'*******************************************************
'* Program to demonstrate the use of multi-dimensional *
'*     Steepest Descent Optimization subroutine        *
'* --------------------------------------------------- *
'*  Reference: BASIC Scientific Subroutines, Vol. II   *
'*  By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].  *
'* --------------------------------------------------- *
'* Example:   Find a local maximum of:                 *
'*            F(x,y,z) = sin(x)+2*cos(y)-sin(z)
'*                                                     *
'* SAMPLE RUN:                                         *
'*                                                     *
'* How many dimensions: 3                              *
'*                                                     *
'* Convergence criterion: 1e-9                         *
'*                                                     *
'* Maximum number of iterations: 50                    *
'*                                                     *
'* Starting constant: 1                                *
'*                                                     *
'* Input the starting point:                           *
'*                                                     *
'*     X( 1) = 1                                       *
'*     X( 2) = 1                                       *
'*     X( 3) = 1                                       *
'*                                                     *
'* The results are:                                    *
'*                                                     *
'*     X( 1) = 1.5707963                               *
'*     X( 2) = -0.0000435                              *
'*     X( 3) = -1.5707963                              *
'*                                                     *
'* Local maximum found = 4.0000000                     *
'*                                                     *
'* The number of steps was: 30                         *
'*                                                     *
'*******************************************************
defint i-n
defdbl a-h,o-z
cls
print
input " How many dimensions: ", l
print
dim D(4),X(l),X1(l),Y(4)
input " Convergence criterion: ", e
print
input " Maximum number of iterations: ", m
print
input " Starting constant: ", xk
print
print " Input the starting points:"
print
for i=1 to l
  print "    X(";i;") = "; : input X(i)
next

gosub 2000   'Call steepest descent optimization subroutine

print : print
print " The results are:"
print
for i=1 to l
  print "    X(";i;") = ";
  print using "##.#######"; X(i)
next
print
print " Local maximum found = ";
print using "##.#######"; yy
print
print " The number of iterations was ";n
print
end
'******************************************************
1000 ' Function subroutine
  yy = SIN(X(1))+2#*COS(X(2))-SIN(X(3))
  D(1)=COS(X(1)) : D(2)=-2#*SIN(X(2)) : D(3)=-COS(X(3))
return
'******************************************************

'***************************************************
'*    Steepest descent optimization subroutine     *
'* ----------------------------------------------- *
'* This routine finds the local maximum or minimum *
'* of an L-dimensional function using the method   *
'* of steepest decent, or the gradient.            *
'* The function and the L derivatives must be      *
'* available in subroutine 1000.                   *
'* ----------------------------------------------- *
'* INPUTS:                                         *
'*   l - The dimension of function to study        *
'*   e - The convergence criteria                  *
'*   m - The maximum number of iterations          *
'*   xk - A starting constant                      *
'*   X(i) - Initial values of variables            *
'* OUTPUTS:                                        *
'*   X(i) - The locally optimum set                *
'*   yy - The local maximum found                  *
'*   n - The number of iterations performed,       *                                  *
'***************************************************
2000 n=0
  'Start initial probe
  for j=1 to 3
    'Obtain yy and D(i)
    gosub 1000
    Y(j)=yy
    'Update X(i)
    gosub 3000
  next j
  'We now have a history to base the subsequent search on
  'Accelerate search if approach is monotonic
2050 if Y(2)-Y(1)=0 then goto 2051
  if (Y(3)-Y(2))/(Y(2)-Y(1))>0 then xk=xk*1.2
  'Decelerate if heading the wrong way
2051 if Y(3)<Y(2) then xk=xk/2#
  'Update the Y(i) if value has increased
  if Y(3)>Y(2) then goto 2100
  'Restore the X(i)
  for i=1 to l
    X(i)=X1(i)
  next
  goto 2200
2100 Y(1)=Y(2) : Y(2)=Y(3)
  'Obtain new values
2200 gosub 1000 : Y(3)=yy
  'Update X(i)
  gosub 3000
  'Check for maximum iterations and convergence
  n=n+1
  if n>=m then return
  if ABS(Y(3)-Y(2))<e then return
  'Try another iteration
  goto 2050
'end of subroutine 2000

3000 'Find the magnitude of the gradient
  dd=0
  for i=1 to l
    dd=dd+D(i)*D(i)
  next
  dd=SQR(dd)
  'Update the X(i)
  for i=1 to l
    'Save old values
    X1(i)=X(i)
    X(i)=X(i)+xk*D(i)/dd
  next
  gosub 1000 : Y(3)=yy
return

'End of file Steepds.bas