MaplePrimes - answers and comments on Question, Generate (an FFT) matrix from equation and have it simplify as I wish

simplify with side-relations?

acer — Sun, 30 Sep 2012 06:21:36 Z

Do you mean a reduction something like this?

simplify(F, {omega^(N/2)=-1});

For example,

restart; 

N:=6:

F:=Matrix(N,N,(i,j)->omega^((i-1)*(j-1)));

                 [1    1        1        1        1        1   ]
                 [                                             ]
                 [                2        3        4        5 ]
                 [1  omega   omega    omega    omega    omega  ]
                 [                                             ]
                 [        2       4        6        8        10]
                 [1  omega   omega    omega    omega    omega  ]
                 [                                             ]
            F := [        3       6        9        12       15]
                 [1  omega   omega    omega    omega    omega  ]
                 [                                             ]
                 [        4       8        12       16       20]
                 [1  omega   omega    omega    omega    omega  ]
                 [                                             ]
                 [        5       10       15       20       25]
                 [1  omega   omega    omega    omega    omega  ]

simplify(F,{omega^(N/2)=-1});

                  [1     1       1     1     1        1   ]
                  [                                       ]
                  [                 2                    2]
                  [1   omega   omega   -1  -omega  -omega ]
                  [                                       ]
                  [        2                    2         ]
                  [1  omega    -omega  1   omega   -omega ]
                  [                                       ]
                  [1    -1       1     -1    1       -1   ]
                  [                                       ]
                  [                 2                   2 ]
                  [1  -omega   omega   1   -omega  omega  ]
                  [                                       ]
                  [         2                   2         ]
                  [1  -omega   -omega  -1  omega    omega ]

acer

Yup, I meant a reduction something sorta

Jeff — Mon, 01 Oct 2012 06:04:42 Z

Yup, I meant a reduction something sorta like that! :D.

I tried all the "Simplification" choices on the menu. I didn't know you could type something like that. Good to go. Thanks.

efficiency

Alejandro Jakubi — Mon, 01 Oct 2012 10:25:43 Z

As acer has an inclination for efficiency issues, I add a brief note. For the original 8x8 matrix, N=8, I have compared simplify + side relation with the applyrule approach for the "full" simplification of the matrix terms to the form p or p*omega, where p is a 4th root of the unity. I have got these results:

> CodeTools:-Usage(simplify(F,{omega^(N/4)=I}));
memory used=2.19MiB, alloc change=8.00MiB, cpu time=37.00ms, real time=34.00ms

> CodeTools:-Usage(subs(`&*`=`*`,map2(applyrule1,r,F,2)));
memory used=1.03MiB, alloc change=0 bytes, cpu time=20.00ms, real time=19.00ms

Just a comment here on the second statement. I have used a custom variation of applyrule that applies the rules just once by default or any other given number of times, in this case 2. And the neutral operator `&*` is being used in a rule for avoiding premature automatic simplification. Certainly, the rules r (not shown here) are somewhat long (3 lines say), a fact that stresses the need for a mathematical pattern matcher. But yet, I was somewhat surprised at seeing the rule method operating much more efficiently both in time and memory.