Threaded Mode | Linear Mode

peacecalc · (This post was last modified: 11-18-2018 11:14 PM by peacecalc.)

Hello friends,

like Eddie Shore showed us HERE an algorithm for integration a function with two variables with the simpson rule and a matrix. He implemented this for the HP 71B.

I do the same thing for the HP 50G but not so elegant, it is brute force:

I implementated this formular:

\[ F = \int_a^b\int_c^d f(t,s)dtds \\ \sim \frac{ha}{3} \left( \frac{hi}{3}\left(
f(a,c) + f(b,c) + \sum_{j=1}^{k-1}\left( 2f(t_j,c) + 4f(t_j,c)\right) +
f(a,d) + f(b,d) + \sum_{j=1}^{k-1}\left( 2f(t_j,d) + 4f(t_j,d)\right) + \\
\sum_{i=1}^{m-1}\left(2\left(f(a,s_i)+f(b,s_i) +
\sum_{j=1}^{k-1}\left( 2f(t_j,s_i) + 4f(t_j,s_i)\right)\right) + \\ 4\left(f(a,s_i)+f(b,s_i) +
\sum_{j=1}^{k-1}\left( 2f(t_j,s_i) + 4f(t_j,s_i)\right)\right)\right)\right)\right) \]

That looks horrible, but I used the stack to sum up all function values and multiplied them afterwards with 2 or 4. And the indices in formular above has to be disdinguish between even or odd numbers (only values with odd indices has to be multiplied be 4 and even with 2). I used in the FOR loops no integer values but the values for the variables (the hp 50g is very happy to use a real variable in the FOR loop.

For instance I used my little program for estimate antiderivatives with harmonic sphere function multiplied with a light function to geht the coeffecients.The one angle goes from 0 to pi, the other one from 0 to 2pi. With N = 15 the hp 50g has to calculate 30*60 = 180 function values and it takes 2 minutes at average. That seems to be very long, but it is faster as you take the built-in function \[ \int \]. I have the impression that the built in function works then (when you have more variables) with recursion.

Code:

%%HP: T(2)A(R)F(.);

« \-> UA OA UI OI N    @@lower bound outer antiderivative

                @@higher bound outer antiderivative

                @@lower bound inner antiderivative

                @@higher bound inner antiderivative

                @@counts of partitial areas (N >= 2)

                @@otherwise the formular here is wrong 

                @@in between the bounds (same for both)

   «

    « DUP 2. * \-> TU TO SI H H2 @@ inner loop defined as a subprogram

      « TO SI FCSIM              @@load the function with two variables

                             @@with two input (numeric) values, each for                          @@every variable

    TU SI FCSIM              @@same again 

    0. DUP                  @@reserve two stackplaces for summation

                             @@the function values

    TU H + TO H2 - FOR K          @@init the for-loop

    K SI FCSIM + SWAP          @@save the value in the one stack place

                             @@and sum it up with the former ones

    K H + SI FCSIM + SWAP     @@save the value in the other stack place

                             @@and sum it up with the former ones

    H2 STEP                      @@increment the loop variable K with the 

                             @@double value 

    TO H - SI FCSIM +          @@calulate the last value which is multi-

                             @@plied later on with 4

    4. * SWAP 2. * + + +         @@multiply the sum of values in their 

                             @@stackplaces and sum it up

      »

    » 

    OI UI - N 2. * / DUP 2. *   @@calculate the delta H for the inner loop

                            @@and the double value of that

    OA UA - N 2. * / DUP 2. * @@calculate the delta H for the outer loop

                            @@and the double value of that

    \-> SFU HF HF2 HT HT2    @@local variables for the subprogram and the

                                @@delta values    

    « TICKS -2. SF -3. SF -105. SF @@ticks for time mesure (can be left out)

                               @@switch to numerical necessary for speed  

    UI OI UA HF SFU EVAL           @@load the subprogram with the boundaries

                               @@of the inner antiderivative, 

                                @@the value of outer antiderivative and the 

                                      @@delta h of the inner antiderivative

    UI OI OA HF SFU EVAL           @@same one line before 

    0. DUP                    @@reserve two stackplaces for summation

                               @@the function values

    UA HT + OA HT2 - FOR I      @@init the for-loop

    UI OI I HF SFU EVAL + SWAP    @@save the value in the one stack place

                             @@and sum it up with the former ones 

    UI OI I HT + HF SFU EVAL + SWAP @@save the value in the other stackplace                                                     

                                                   @@and sum it up with 

        HT2 STEP                            @@increment the loop variable K with the 

                                    @@double value 

    UI OI OA HT - HF SFU EVAL +       @@calulate the last value which is                                        

                                                       @@multiplied later on with 4

    4. * SWAP 2. * + + + HF * HT * 9. /   @@multiply the sum of values in 

                                                            @@their stackplaces and

                                        @@multiply it the delta h 

                                        @@for both antiderivatives

          TICKS ROT - 8192. /    @@can be left out only for time mesure 

    -2. CF -3. CF -105. CF    @@switch back to symbolic mode 

    »

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