Use with caution -- the set-up can be right and the answer wrong as in: What is the norm of vector (1/2, 1, 1)?
The norm of a vector is defined as the length or magnitude of the vector. The norm of a vector v = (v1, v2, ..., vn) is given by the formula: ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2) So, for the vector (1/2, 1, 1), the norm is: || (1/2, 1, 1) || = sqrt( (1/2)^2 + 1^2 + 1^2 ) = sqrt( 1/4 + 1 + 1 ) = sqrt( 6.25 ) = 2.5 Therefore, the norm of the vector (1/2, 1, 1) is 2.5. The answer: `assert sqrt(sum([i**2 for i in (1/2,1,1)])) == 1.5` On Wednesday, March 29, 2023 at 7:48:48 AM UTC-5 e...@tssfl.co wrote: > ChatGPT is quite impressive! For those who do not have access to ChatGPT > and/or Python, can use it with SageCell at > https://tssfl.com/viewtopic.php?t=6651 > > SageCell supports countless Python libraries including SymPy. > > I tested ChatGPT with a number of examples, it tries solve even complex > maths problems, e.g., : > > Example 1: > > Find a 4x4 matrix which is a Jordan canonical form of the matrix A = > matrix([[5, -1, 0, 0],[9, -1, 0, 0],[0, 0, 7, -2], [0, 0, 12, -3]]). > > assistant: > > To find the Jordan canonical form of a matrix, we need to find the > eigenvalues and eigenvectors, and then use them to construct the Jordan > blocks. > > First, we find the eigenvalues of A by solving for the determinant of (A - > lambda*I), where I is the identity matrix: > > |5 - lambda -1 0 0 | > |9 -1 - lambda 0 0 | > |0 0 7 - lambda -2 | > |0 0 12 -3 - lambda| > > = (5 - lambda)(-1 - lambda)(7 - lambda)(-3 - lambda) - 9*(-1)*(7 - > lambda)*(12) > = lambda^4 - 8*lambda^3 + 11*lambda^2 + 38*lambda - 360 > = (lambda - 10)(lambda - 5)(lambda + 3)(lambda - 4) > > Therefore, the eigenvalues are: lambda1 = 10, lambda2 = 5, lambda3 = -3, > lambda4 = 4. > > Next, we find the eigenvectors corresponding to each eigenvalue. For lambda1 > = 10, we solve (A - 10*I)x = 0: > > | -5 -1 0 0 | > | 9 -11 0 0 | > | 0 0 -3 -2 | > | 0 0 12 -13 | > > By performing elementary row operations to reduce this matrix, we obtain: > > | 1 0 0 1 | > | 0 1 0 3/2 | > | 0 0 1 2 | > | 0 0 0 0 | > > So the eigenvector corresponding to lambda1 is x1 = [1, -3/2, -2, 1]. > > Similarly, for lambda2 = 5, we solve (A - 5*I)x = 0: > > | 0 -1 0 0 | > | 9 -6 0 0 | > | 0 0 2 -2 | > | 0 0 12 -8 | > > By performing elementary row operations to reduce this matrix, we obtain: > > | 1 -3 0 0 | > | 0 0 1 -1 | > | 0 0 0 0 | > | 0 0 0 0 | > > So the eigenvector corresponding to lambda2 is x2 = [3, 1, 0, 0]. > > For lambda3 = -3, we solve (A + 3*I)x = 0: > > | 8 -1 0 0 | > | 9 -2 0 0 | > | 0 0 4 -2 | > | 0 0 12 -6 | > > By performing elementary row operations to reduce this matrix, we obtain: > > | 1 -1/2 0 0 | > | 0 0 1 -1/2 | > | 0 0 0 0 | > | 0 0 0 0 | > > So the eigenvector corresponding to lambda3 is x3 = [1/2, 1, 1/2, 0]. > > Finally, for lambda4 = 4, we solve (A - 4*I)x = 0: > > | 1 -1 0 0 | > | 9 -5 0 0 | > | 0 0 3 -2 | > | 0 0 12 -7 | > > By performing elementary row operations to reduce this matrix, we obtain: > > | 1 -1 0 0 | > | 0 0 1 -2/3 | > | 0 0 0 0 | > | 0 0 0 0 | > > So the eigenvector corresponding to lambda4 is x4 = [1, 1/3, 0, 0]. > > We can now construct the Jordan canonical form of the matrix A by arranging > these eigenvectors in blocks, as follows: > > J = [10 1 0 0 ] > [0 10 0 0 ] > [0 0 -3 1 ] > [0 0 0 4 ] > > This is the desired 4x4 Jordan canonical form of A. > > Example 2: > > Find an invertible 3x3 real matrix P which puts the matrix A = > matrix([[3,4,0],[-1, -3, -2],[1, 2, 1]]) into rational form. > > assistant: > > To put a matrix in rational form, we want to reduce it to row echelon form > and then perform row operations to get all leading coefficients to be 1 and > all other entries in the same columns to be 0. Here are the steps to do it > for matrix A: > > Step 1: Choose the first element in the first row to be the pivot and > eliminate all entries in the column below it. To do this, subtract the first > row multiplied by (-1/3) from the second row, and add the first row to the > third row. > > [3 4 0] > [0 1 2] > [4 6 1] > > Step 2: Choose the second element in the second row to be the pivot and > eliminate all entries in the column below it. To do this, subtract the second > row multiplied by 2 from the third row. > > [3 4 0] > [0 1 2] > [0 -2 -7] > > Step 3: Choose the third element in the third row to be the pivot and > eliminate all entries in the column above it. To do this, add the second row > multiplied by 2 to the third row. > > [3 4 0] > [0 1 2] > [0 0 -3] > > Now we have the row echelon form of matrix A. To get it in rational form, we > can perform further row operations to get all leading coefficients to be 1. > We can achieve this by multiplying the third row by -1/3. > > [3 4 0] > [0 1 2] > [0 0 1] > > Now we have the rational form of matrix A. To find the invertible matrix P > that puts A into this form, we need to keep track of the row operations we > performed. > > First, we subtracted the first row multiplied by (-1/3) from the second row. > This is equivalent to multiplying by the matrix: > > [1 0 0] > [-1/3 1 0] > [0 0 1] > > Next, we added the first row to the third row. This is equivalent to > multiplying by the matrix: > > [1 0 0] > [0 1 0] > [1 0 1] > > Then, we subtracted the second row multiplied by 2 from the third row. This > is equivalent to multiplying by the matrix: > > [1 0 0] > [0 1 0] > [0 -2 1] > > Finally, we multiplied the third row by -1/3. This is equivalent to > multiplying by the matrix: > > [1 0 0] > [0 1 0] > [0 0 -1/3] > > To find P, we multiply these matrices in order: > > P = [1 0 0] @ [-1/3 1 0] @ [1 0 0] @ [0 1 0] @ [0 -2 1] @ [1 0 0] > @ [0 1 0] @ [0 0 -1/3] > > Simplifying this expression gives: > > P = matrix([[-1/3, -8/3, -2/3],[1/3, 5/3, 2/3],[-4/3, -2/3, 1/3]]) > > We can check that matrix A can be transformed to its rational form by > multiplying it by P and verifying that the result is indeed the rational form: > > A_rational = P.inv() @ A @ P > > Output: > A_rational = matrix([[1, 0, 0],[0, 1, 0],[0, 0, 1]]) > > > Elimboto > > > On Wed, Mar 29, 2023 at 3:32 AM Aaron Meurer <asme...@gmail.com> wrote: > >> If you don't have access to the ChatGPT beta, someone made a Streamlit >> app that does the same thing. You can give it a math problem and it uses >> ChatGPT to generate and run SymPy code to solve it >> https://mathgpt.streamlit.app/ >> >> Aaron Meurer >> >> On Thu, Mar 23, 2023 at 3:28 PM Aaron Meurer <asme...@gmail.com> wrote: >> >>> Actually I didn't notice it but there's literally an example of ChatGPT >>> generating SymPy code and executing on >>> https://openai.com/blog/chatgpt-plugins (scroll down to where it says >>> "code interpreter"). It's their main example of calling out to Python. >>> >>> Aaron Meurer >>> >>> On Thu, Mar 23, 2023 at 2:42 PM Aaron Meurer <asme...@gmail.com> wrote: >>> >>>> >>>> >>>> On Thu, Mar 23, 2023 at 12:24 PM S.Y. Lee <syle...@gmail.com> wrote: >>>> >>>>> Wolfram had recently announced the collaboration of chatGPT and >>>>> wolfram alpha >>>>> ChatGPT Gets Its “Wolfram Superpowers”!—Stephen Wolfram Writings >>>>> <https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/> >>>> >>>> >>>> I wouldn't call this a "collaboration". OpenAI is adding a plugin >>>> system to ChatGPT and Wolfram is one of the first plugins >>>> https://openai.com/blog/chatgpt-plugins. >>>> >>>> If you scroll down on that page, there is also a tool that lets it >>>> execute Python code. I don't know if it has access to SymPy, but it likely >>>> does, since it seems to have access to other popular libraries like pandas >>>> and matplotlib. If anyone has access to ChatGPT Plus, could you check? >>>> >>>> >>>>> >>>>> They start to use chatgpt to generate the Wolfram code. >>>>> And it is likely to help the issues with correctness about math or >>>>> science facts >>>>> because once it translates to the wolfram functions, and the wolfram >>>>> function runs without error, the answer is correct. >>>>> >>>>> However, the argument I'd give is that making a combination of code >>>>> with >>>>> *simplify, solve, integrate *are not still informative because even >>>>> though they do things logically correct, >>>>> they have problem that they can't inform people how to solve the >>>>> problems in details. >>>>> >>>>> So I'm thinking about an idea whether language models should really >>>>> generate a code that assembles the *rules* used in simplify, solve. >>>>> And then it can be more human readable (or can get more human readable >>>>> logic from it) >>>>> if it is like compose(solve_trig, simplify_cos, ...) >>>>> (in some pseudo sympy code) >>>>> >>>> >>>> A language model can do pretty much anything, so long as it's seen >>>> enough examples of it before. You can often just give it examples of what >>>> you want in the prompt and it will figure it out. >>>> >>>> ChatGPT (and other LLMs) are basically just huge pattern matching >>>> machines. That's all they are, which is why they break down whenever they >>>> have to do actual logic or reason about something they haven't seen >>>> before. >>>> So I suspect they could be used quite successfully for symbolic pattern >>>> matching, especially when that pattern matching is "fuzzy", like trying to >>>> find the best heuristic function to apply to an expression. >>>> >>>> Aaron Meurer >>>> >>>> >>>>> On Saturday, December 17, 2022 at 5:47:39 PM UTC+9 smi...@gmail.com >>>>> wrote: >>>>> >>>>>> In reviewing a PR related to units, I found ChatGPT to get correct >>>>>> the idea that a foot is bigger than an inch, but it said that a volt is >>>>>> bigger than a statvolt (see quoted GPT response [here]( >>>>>> https://github.com/sympy/sympy/pull/24325#issuecomment-1354343306)). >>>>>> >>>>>> /c >>>>>> >>>>>> On Thursday, December 15, 2022 at 1:58:33 PM UTC-6 Aaron Meurer wrote: >>>>>> >>>>>>> The trend with LLMs is much less structured. It doesn't use any >>>>>>> formalism. It just guesses the next character of the input based on >>>>>>> training on billions of examples. >>>>>>> >>>>>>> That's why I think that tools like SymPy that are more structured >>>>>>> can be useful. GPT can already write SymPy code pretty well, much >>>>>>> better >>>>>>> than it can do the actual mathematics. It may be as simple as >>>>>>> automatically >>>>>>> appending "and write SymPy code to verify this" to the end of a prompt >>>>>>> whenever it involves mathematics. This sort of approach has already >>>>>>> been >>>>>>> proven to be able to solve university math problems (see >>>>>>> https://www.pnas.org/doi/pdf/10.1073/pnas.2123433119, where they >>>>>>> literally just take the input problem and prepend "use sympy" and the >>>>>>> neural network model does the rest). >>>>>>> >>>>>>> Aaron Meurer >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Thu, Dec 15, 2022 at 2:21 AM S.Y. Lee <syle...@gmail.com> wrote: >>>>>>> >>>>>>>> > My hope is that tools like SymPy can be used as oracles for tools >>>>>>>> like GPT to help them verify their mathematics. >>>>>>>> >>>>>>>> In the most general context, "correct mathematics" can also be >>>>>>>> considered some "grammar". >>>>>>>> So there should be some grammar between Type-0 grammar to Type-1 >>>>>>>> grammar in Chomsky hierarchy >>>>>>>> <https://en.wikipedia.org/wiki/Chomsky_hierarchy>. >>>>>>>> In this context, a parser, or a parser with sympy oracle is the >>>>>>>> solution for such problem, >>>>>>>> such that any other ideas to solve such problem can be isomorphic >>>>>>>> to. >>>>>>>> >>>>>>>> However, building up such parser is off-direction for the >>>>>>>> researches of deep learning itself, >>>>>>>> because it would need a lot of efforts by experts, to interpret the >>>>>>>> sentence generated by GPT, >>>>>>>> and design a phrase structure grammar for it. >>>>>>>> >>>>>>>> I also thought about an idea that they can just tag arithmetics >>>>>>>> using SKI combinator calculus >>>>>>>> <https://en.wikipedia.org/wiki/SKI_combinator_calculus>. >>>>>>>> In this way, there is no wrong arithmetics for every random >>>>>>>> sequence of alphabets. >>>>>>>> >>>>>>>> However, I'm not sure that if this idea is already refuted by such >>>>>>>> contemporary researchers >>>>>>>> because it should be pretty much well-known. >>>>>>>> >>>>>>>> On Thursday, December 15, 2022 at 12:45:53 AM UTC+2 >>>>>>>> asme...@gmail.com wrote: >>>>>>>> >>>>>>>>> I've seen many examples on Twitter and elsewhere of chatgpt doing >>>>>>>>> terribly on actual mathematics. I instead asked it to "Find the >>>>>>>>> extreme >>>>>>>>> values of the function x^3 + 5x^2 + 3x - 9". It got the first part >>>>>>>>> with the >>>>>>>>> derivative and the quadratic equation correct, but then did faulty >>>>>>>>> arithmetic when plugging the values back into the expression (the >>>>>>>>> correct >>>>>>>>> values are 0 and -256/27). It makes a sign error and also gets the >>>>>>>>> addition >>>>>>>>> wrong even with the wrong sign. >>>>>>>>> >>>>>>>>> My hope is that tools like SymPy can be used as oracles for tools >>>>>>>>> like GPT to help them verify their mathematics. Obviously in this >>>>>>>>> case you >>>>>>>>> don't need SymPy to just verify arithmetic, but I've also seen it do >>>>>>>>> just >>>>>>>>> as poorly on symbolic algebra problems. >>>>>>>>> >>>>>>>>> I've also pasted the full log below: >>>>>>>>> >>>>>>>>> [image: Screen Shot 2022-12-14 at 3.41.32 PM.png] >>>>>>>>> [image: Screen Shot 2022-12-14 at 3.41.42 PM.png] >>>>>>>>> >>>>>>>>> To find the extreme values of a function, we need to find the >>>>>>>>> critical points of the function, which are the points at which the >>>>>>>>> derivative of the function is equal to 0 or does not exist. >>>>>>>>> >>>>>>>>> The derivative of the function x^3 + 5x^2 + 3x - 9 is 3x^2 + 10x + >>>>>>>>> 3. To find the critical points, we need to solve the equation 3x^2 + >>>>>>>>> 10x + >>>>>>>>> 3 = 0. >>>>>>>>> >>>>>>>>> To solve this equation, we can use the quadratic formula: >>>>>>>>> >>>>>>>>> x = (-b ± √(b^2 - 4ac)) / 2a >>>>>>>>> >>>>>>>>> where a, b, and c are the coefficients of the quadratic equation >>>>>>>>> ax^2 + bx + c = 0. In our case, a = 3, b = 10, and c = 3, so the >>>>>>>>> formula >>>>>>>>> becomes: >>>>>>>>> >>>>>>>>> x = (-10 ± √(10^2 - 4 * 3 * 3)) / 2 * 3 >>>>>>>>> >>>>>>>>> = (-10 ± √(100 - 36)) / 6 >>>>>>>>> >>>>>>>>> = (-10 ± √(64)) / 6 >>>>>>>>> >>>>>>>>> = (-10 ± 8) / 6 >>>>>>>>> >>>>>>>>> = (-18, -2) / 6 >>>>>>>>> >>>>>>>>> = (-3, -1/3) >>>>>>>>> >>>>>>>>> These are the critical points of the function x^3 + 5x^2 + 3x - 9. >>>>>>>>> >>>>>>>>> To find the extreme values of the function, we need to evaluate >>>>>>>>> the function at these critical points and at any points where the >>>>>>>>> derivative does not exist. In this case, the derivative exists for >>>>>>>>> all >>>>>>>>> values of x, so we only need to evaluate the function at the critical >>>>>>>>> points. >>>>>>>>> >>>>>>>>> If we plug in the critical points into the function, we get: >>>>>>>>> >>>>>>>>> x^3 + 5x^2 + 3x - 9 = (-3)^3 + 5(-3)^2 + 3(-3) - 9 = -27 + 45 + 9 >>>>>>>>> - 9 = 8 >>>>>>>>> >>>>>>>>> and >>>>>>>>> >>>>>>>>> x^3 + 5x^2 + 3x - 9 = (-1/3)^3 + 5(-1/3)^2 + 3(-1/3) - 9 = -1/27 + >>>>>>>>> 5/9 + 1 - 9 = 5/9 - 1/27 - 8/9 = -64/27 >>>>>>>>> >>>>>>>>> The extreme values of the function x^3 + 5x^2 + 3x - 9 are 8 and >>>>>>>>> -64/27. >>>>>>>>> >>>>>>>>> On Wed, Dec 14, 2022 at 3:35 PM gu...@uwosh.edu <gu...@uwosh.edu> >>>>>>>>> wrote: >>>>>>>>> >>>>>>>>>> Just for some additional perspective. I have also tried this on >>>>>>>>>> some general chemistry word problems. In general, I see it getting >>>>>>>>>> basic >>>>>>>>>> one logical step processes correct (e.g. a single step dilution or >>>>>>>>>> grams -> >>>>>>>>>> moles). Things with multiple steps or requiring understanding the >>>>>>>>>> physical >>>>>>>>>> situation it does poorly on. That said, I think it does better than >>>>>>>>>> some of >>>>>>>>>> my weakest students. It does not seem to be able to use significant >>>>>>>>>> figures >>>>>>>>>> in computations (also a problem for my weaker students). >>>>>>>>>> >>>>>>>>>> It seems to be improving rapidly. If it can get to reliably >>>>>>>>>> differentiating between correct (workable) solutions and erroneous >>>>>>>>>> ones, it >>>>>>>>>> will be more useful to most people (including my students) than >>>>>>>>>> searches of >>>>>>>>>> the internet or a cheating sight such as Chegg. >>>>>>>>>> >>>>>>>>>> My two cents worth of opinion. >>>>>>>>>> >>>>>>>>>> Jonathan >>>>>>>>>> >>>>>>>>>> On Wednesday, December 14, 2022 at 4:28:05 PM UTC-6 Francesco >>>>>>>>>> Bonazzi wrote: >>>>>>>>>> >>>>>>>>>>> [image: chatgpt.sympy.matrix_diag.png] >>>>>>>>>>> >>>>>>>>>>> On Wednesday, December 14, 2022 at 11:26:37 p.m. UTC+1 Francesco >>>>>>>>>>> Bonazzi wrote: >>>>>>>>>>> >>>>>>>>>>>> Not everything is perfect... ChatGPT misses the *convert_to( >>>>>>>>>>>> ... ) *function in *sympy.physics.units*, furthermore, the >>>>>>>>>>>> given code does not work: >>>>>>>>>>>> >>>>>>>>>>>> [image: chatgpt.sympy.unit_conv.png] >>>>>>>>>>>> >>>>>>>>>>>> On Wednesday, December 14, 2022 at 11:24:29 p.m. UTC+1 >>>>>>>>>>>> Francesco Bonazzi wrote: >>>>>>>>>>>> >>>>>>>>>>>>> [image: chatgpt.sympy.logical_inference.png] >>>>>>>>>>>>> >>>>>>>>>>>>> On Wednesday, December 14, 2022 at 11:23:43 p.m. UTC+1 >>>>>>>>>>>>> Francesco Bonazzi wrote: >>>>>>>>>>>>> >>>>>>>>>>>>>> https://en.wikipedia.org/wiki/ChatGPT >>>>>>>>>>>>>> >>>>>>>>>>>>>> Some tested examples attached as pictures to this post. Quite >>>>>>>>>>>>>> impressive... >>>>>>>>>>>>>> >>>>>>>>>>>>>> -- >>>>>>>>>> You received this message because you are subscribed to the >>>>>>>>>> Google Groups "sympy" group. >>>>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>>>> send an email to sympy+un...@googlegroups.com. >>>>>>>>>> >>>>>>>>> To view this discussion on the web visit >>>>>>>>>> https://groups.google.com/d/msgid/sympy/6af62b19-1fb0-4681-9fd2-5e5fccfcb46fn%40googlegroups.com >>>>>>>>>> >>>>>>>>>> <https://groups.google.com/d/msgid/sympy/6af62b19-1fb0-4681-9fd2-5e5fccfcb46fn%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>>>>>>> . >>>>>>>>>> >>>>>>>>> -- >>>>>>>> You received this message because you are subscribed to the Google >>>>>>>> Groups "sympy" group. >>>>>>>> To unsubscribe from this group and stop receiving emails from it, >>>>>>>> send an email to sympy+un...@googlegroups.com. >>>>>>>> >>>>>>> To view this discussion on the web visit >>>>>>>> https://groups.google.com/d/msgid/sympy/74847ca3-124b-414d-aa36-01eb91096310n%40googlegroups.com >>>>>>>> >>>>>>>> <https://groups.google.com/d/msgid/sympy/74847ca3-124b-414d-aa36-01eb91096310n%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>>>>> . >>>>>>>> >>>>>>> -- >>>>> You received this message because you are subscribed to the Google >>>>> Groups "sympy" group. >>>>> To unsubscribe from this group and stop receiving emails from it, send >>>>> an email to sympy+un...@googlegroups.com. >>>>> To view this discussion on the web visit >>>>> https://groups.google.com/d/msgid/sympy/1fbd88f8-3513-4e02-a576-266352f3531fn%40googlegroups.com >>>>> >>>>> <https://groups.google.com/d/msgid/sympy/1fbd88f8-3513-4e02-a576-266352f3531fn%40googlegroups.com?utm_medium=email&utm_source=footer> >>>>> . >>>>> >>>> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to sympy+un...@googlegroups.com. >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAKgW%3D6L3qy-h94G8OrTqtjrqTu9woZQpXZxaZx7mjcD1L7q32g%40mail.gmail.com >> >> <https://groups.google.com/d/msgid/sympy/CAKgW%3D6L3qy-h94G8OrTqtjrqTu9woZQpXZxaZx7mjcD1L7q32g%40mail.gmail.com?utm_medium=email&utm_source=footer> >> . >> > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/9fd4fc7a-32d4-4ab3-b561-f12cbb6ab331n%40googlegroups.com.
