Computer program to calculate bernoulli numbers

This leads to potential problems with mismatched sizes of arrays, and calculation. Also, I tried to work out a method whereby all calculations were performed on the stack. I gave up sooner than I should have. Again, because getting the address of the storage and holding it a convenient variable was problematic. I'm sure there's a way of doing this. On the plus side, the result is brilliantly fast, and the cells allow for quite high precision - so much so, that I don't mind showing off the first 30 results.

I also impressed myself by inventing re-inventing? I was quite happy that I managed to utilize types and recursive functions with the intent keyword. I wasn't so happy in wasting so long in not understanding the array shape. In Fortran arrays start from Consequently, my allocate a n needed to be changed to allocate a 0:n. This is intentional, since I wanted Golang's ability to return multiple values. I also liked the method to determine the sign of the fraction, even though its full functionality won't get used in this case.

I was and am still very disappointed by the ability for Go to maintain accuracy when the numbers get large. It was tempting to move it to bigints, but it was late, so I postponed it. System; import java. Getting the development environment work was the biggest pain, here, as the Linux version of the compiler and run-time for some reason decided to not match : And the use of the -target 1.

So I reluctantly re-installed Eclipse for the first time in 5 years. On the plus side, my machine is now fast enough to run it! I also decided to use the 3-line version of the subtract method, which doesn't affect A[j], to gain precision. The first one I wrote, since I was doing a lot of Javascript work at the time. The idea to optimize the algorithm into the smallest possible space happened one Saturday morning because I was curious to see if I could get the code into characters to share in a tweet.

I spent about 3 hours trying. Ultimately, I couldn't, and had to resort to the image. One of the holy grails, and I finally managed it. It's not pretty, and it's certainly not idiomatic, but I now know why people love and hate LISP, according to temprament. Perhaps strangely, the reverse order i. No doubt to my love and minor experience with Forth. What did, however, was getting a loop to count down and terminate correctly!

I had to switch my approach from setq-loop to loop-for. This stopped the first iteration from being evaluated. A simple conversion this one. By the time I got to this one, I'd given up on the 'full' version of the subtraction, and just used the 3 line code which doesn't affect the a[j] element. Even so, I was surprised by the need to simplify at each step.

How I've missed you I remember the "declare before use" rule, and that for loops need downto for decrementing steps, but managed to forget almost all of the other gotchas. Also, there's a separate keyword for functions which return a value and procedures which do not.

Oh, there's so many semicolons that I ended putting them in too many places and short-circuited a for block. It took 15 minutes or more to find that bug, with writeln, all of things. As for the Bernoulli algorithm itself, it wasn't a difficult port per se. I switch to Int64 upon realized the limit of integers. Otherwise, it was strange going back to a language with such distinct definition and declaration sections but, given the age and the "declare before use" rules, it's a necessary evil.

And no worse than C. The sum in our first expression is actually the Basel problem, which was initially proved by Euler a long time ago. Until the double integral solution came out not too long ago. Sign up to join this community.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What is the simplest way to get Bernoulli numbers? Ask Question. Asked 3 years, 6 months ago. Active 13 days ago. Viewed 4k times. Tristian Tristian 4 4 silver badges 12 12 bronze badges.

Her famous -- and poetically wild -- father went to Greece, and she never knew him. Ada was 15 when she met the Cambridge mathematics professor Babbage years ago today. Babbage had already received funding from Parliament to build a " difference engine " that could do mathematical calculations.

While that project was still unfinished, he conceived in a new and broader idea: an "analytical engine" that "could not only foresee but could act on that foresight. They had three children, but Ada's family and social responsibilities did not keep her from continuing her study of advanced mathematics.

Babbage, meanwhile, gave a seminar on the analytic engine in Turin, Italy, in He persuaded Menabrea to write an outline of what the Analytical Engine could accomplish. In , Menabrea published a paper on the topic in French.

Lovelace, then known as Ada Byron, first met Babbage at a party in , when she was 17 and he was She could also understand how it worked, because she had been extensively tutored in mathematics throughout her childhood. After meeting in , Lovelace and Babbage remained a part of the same social circle and wrote to each other frequently. Ada Byron married William King in All of them were trivial in the sense that they did not require any kind of branching or looping.

It was here that she made her major contributions to computing. The other famous note is Note G. To illustrate its ability to handle even more complex problems, Lovelace provides her program calculating the Bernoulli numbers.

The program is essentially a list of operations, specified using the usual mathematical symbols. Though Lovelace was describing a method for computing the entire sequence of Bernoulli numbers up to some limit, the program she provided only illustrated one step of that process.

Her program calculated a number that she called B7, which modern mathematicians know as the eighth Bernoulli number. Her program thus sought to solve the following equation:. In the above, each term represents a coefficient in the polynomial formula for the sum of integers to a particular power. Here that power is eight, since the eighth Bernoulli number first appears in the formula for the sum of positive integers to the eighth power.

The B and A numbers represent the two kinds of factors that Bernoulli discovered. The values of A0, A1, A3, and A5 appear below. Here n represents the index of the Bernoulli number in the sequence of odd-numbered Bernoulli numbers starting with the first. It then enters a loop that repeats twice to calculate B3A3 and B5A5, since those are formed according to an identical pattern. After each product is calculated, it is added with all the previous products, so that by the end of the program the full sum has been obtained.

The C program contains two while loops, one nested inside the other. The variable v10 , in the original program and in the C translation, functions as a counter variable that decrements with each loop, a construct any programmer would be familiar with. She adds a superscript index here to each variable to indicate the successive values they hold. A superscript of two, for example, means that the value being used here is the second value that has been assigned to the variable since the beginning of the program.

To my frustration, I kept getting the wrong result. The bug was in the original! This may well have been a typesetting error and not an error in the program that Lovelace devised.