Optimization of the function and Jacobian matrix

Structural knowledge and functional properties should be used by the symbolic environment in order to improve the efficiency of the resulting compiled code [21]. Optimisation can be used for more efficient evaluation of the function and its Jacobian matrix in Newton's method, in much the same way as in the Runge-Kutta example of section . For example, the optimization process can resolve matrix symmetries.

Illustration of these issues is given by a simple example. Consider a Jacobian matrix derived from a function of two variables .

In[1]:= f[x_,y_] := {x + y + 2 x Sin[y]^2, x + y + x^2 Sin[2 y]};

In[2]:= jac[f_List,vars_List] := Outer[D,f,vars];

In[3]:= matrix = jac[f[x,y],{x,y}];

In[4]:= matrix //MatrixForm

Out[4]//MatrixForm=
            2
1 + 2 Sin[y]            1 + 4 x Cos[y] Sin[y]

                               2
1 + 2 x Sin[2 y]        1 + 2 x  Cos[2 y]

In fact this matrix turns out to be symmetric:

In[5]:= simpmat = Simplify[matrix];
In[6]:= simpmat //MatrixForm

Out[6]//MatrixForm=
2 - Cos[2 y]        1 + 2 x Sin[2 y]

                           2
1 + 2 x Sin[2 y]    1 + 2 x  Cos[2 y]

In[7]:= FortranAssign[m, simpmat, AssignOptimize->True,
          OptimizePower->True,OptimizeTimes->False]
Out[7]//OutputForm=
        o1=cos(2.d0*y)
        o2=sin(2.d0*y)
        o3=1.d0+2.d0*o2*x
        m(1,1)=2.d0-o1
        m(1,2)=o3
        m(2,1)=o3
        m(2,2)=1.d0+2.d0*o1*x**2