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change_of_basis [2017/04/19 14:15] (current)
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 +====== Change of Basis ======
 +Change of basis is a useful transformation in a lot of aspect. This page shows how to use command CoB to apply change of basis on a program.
 +===== Usage =====
 +Here is the usage for CoB:
 +<sxh cs; gutter:true>
 +CoB(Program program, String systemName, String varName, String function)
 +</sxh>
 +The meaning for each parameters are:
 +  * program: the program you are processing.
 +  * systemName: the name for the system you want to solve.
 +  * varName: the name for the variable you want to apply change of basis on.
 +  * function: change of basis function.
 +===== Example =====
 +The following example does a matrix multiplication, C = A*B.
 +<sxh ab; gutter:true>
 +affine matrix_product {P, Q, R|P>0 && Q>0 && R>0}
 +given  float A {i,k| 0<=i<P && 0<=k<Q};
 +       float B {k,j| 0<=k<Q && 0<=j<R}; 
 +returns  float C {i,j,k| 0<=i<P && 0<=j<R && k==Q};
 +using // Using an accumulator locally
 +   float temp_C {i,j,k|0<=i<P && 0<=j<R && 0<=k<=Q};
 +through
 +temp_C = case
 +         {i,j,k|k-1>= 0} : ((i,j,k->i,j,k-1)@temp_C + ((i,j,k->i,k-1)@A * (i,j,k->k-1,j)@B));
 +         {i,j,k|k== 0} : (i,j,k->)@0;
 +         esac;
 +C = (i,j,k->i,j,k)@temp_C;
 +.
 +</sxh>
 +Now we are going to apply change of basis on variable "temp_C" using function (i,j,k->i,j,k+1). We can apply the transformation with the following code:
 +<sxh cs; gutter:true>
 +prog = ReadAlphabets("matrix_product.ab");
 +system = "matrix_product";
 +CoB(prog, system, "temp_C", "(i,j,k->i,j,k+1)");
 +</sxh> 
 +
 +The result for this transformation is shown below:
 +<sxh ab; gutter:true>
 +affine matrix_product {P, Q, R|P>0 && Q>0 && R>0}
 +given  float A {i,k| 0<=i<P && 0<=k<Q};
 +       float B {k,j| 0<=k<Q && 0<=j<R}; 
 +returns  float C {i,j,k| 0<=i<P && 0<=j<R && k==Q};
 +using // Using an accumulator locally
 +   float temp_C {i,j,k|0<=i<P && 0<=j<R && 1<=k<=Q+1};
 +through
 +temp_C = (i,j,k->i,j,k-1)@case
 +                          {i,j,k|k-1>= 0} : ((i,j,k->i,j,k-1)@(i,j,k->i,j,k+1)@temp_C + ((i,j,k->i,k-1)@A * (i,j,k->k-1,j)@B));
 +                          {i,j,k|k== 0} : (i,j,k->)@0;
 +                          esac;
 +C = (i,j,k->i,j,k)@(i,j,k->i,j,k+1)@temp_C;
 +.
 +</sxh>
 +We applied the change of basis function (i,j,k->i,j,k+1) on variable "temp_C", which means we shifted the domain of "temp_C" along k direction by one unit. We can see that the domain for "temp_C" is changed from:
 +<sxh ab; gutter:true>
 +float temp_C {i,j,k|0<=i<P && 0<=j<R && 0<=k<=Q};
 +</sxh>
 +to 
 +<sxh ab; gutter:true>
 +float temp_C {i,j,k|0<=i<P && 0<=j<R && 1<=k<=Q+1};
 +</sxh>
 +And all the access functions for temp_C is composed with function (i,j,k->i,j,k+1). For example, the access function in the computation of C is changed from
 +<sxh ab; gutter:true>
 +C = (i,j,k->i,j,k)@temp_C;
 +</sxh>
 +to
 +<sxh ab; gutter:true>
 +C = (i,j,k->i,j,k)@(i,j,k->i,j,k+1)@temp_C;
 +</sxh>
  
change_of_basis.txt ยท Last modified: 2017/04/19 14:15 (external edit)