Re: numpy (matrix solver) - python vs. matlab
On 05/04/2012 05:52 AM, Steven D'Aprano wrote: On Thu, 03 May 2012 19:30:35 +0200, someone wrote: So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would counter your hand-waving (correlates really good, within approx 5% of *what*?) with hand-waving of my own: Within 5% of experiments of course. There is not much else to compare with. Sure, that's exactly what I would expect! *wink* By the way, if I didn't say so earlier, I'll say so now: the interpretation of how bad the condition number is will depend on the underlying physics and/or mathematics of the situation. The interpretation of loss of digits of precision is a general rule of thumb that holds in many diverse situations, not a rule of physics that cannot be broken in this universe. If you have found a scenario where another interpretation of condition number applies, good for you. That doesn't change the fact that, under normal circumstances when trying to solve systems of linear equations, a condition number of 1e6 is likely to blow away *all* the accuracy in your measured data. (Very few physical measurements are accurate to more than six digits.) Not true, IMHO. Eigenfrequencies (I think that is a very typical physical measurement and I cannot think of something that is more typical) don't need to be accurate with 6 digits. I'm happy with below 5% error. So if an eigenfrequency is measured to 100 Hz, I'm happy if the numerical model gives a result in the 5%-range of 95-105 Hz. This I got with a condition number of approx. 1e6 and it's good enough for me. I don't think anyone expects 6-digit accuracy with eigenfrequncies. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/04/2012 06:15 AM, Russ P. wrote: On May 3, 4:59 pm, someonenewsbo...@gmail.com wrote: On 05/04/2012 12:58 AM, Russ P. wrote: Ok, but I just don't understand what's in the empirical category, sorry... I didn't look it up, but as far as I know, empirical just means based on experiment, which means based on measured data. Unless I am FEM based on measurement data? Still, I don't understand it, sorry. mistaken , a finite element analysis is not based on measured data. I'm probably a bit narrow-thinking because I just worked with this small FEM-program (in Matlab), but can you please give an example of a matrix-problem that is based on measurement data? Yes, the results can be *compared* with measured data and perhaps calibrated with measured data, but those are not the same thing. Exactly. That's why I don't understand what solving a matrix system using measurement/empirical data, could typically be an example of...? I agree with Steven D's comment above, and I will reiterate that a condition number of 1e6 would not inspire confidence in me. If I had a condition number like that, I would look for a better model. But that's just a gut reaction, not a hard scientific rule. I don't have any better model and don't know anything better. I still think that 5% accuracy is good enough and that nobody needs 6-digits precision for practical/engineering/empirical work... Maybe quantum physicists needs more than 6 digits of accuracy, but most practical/engineering problems are ok with an accuracy of 5%, I think, IMHO... Please tell me if I'm wrong. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 11:45 PM, Russ P. wrote: On May 2, 1:29 pm, someonenewsbo...@gmail.com wrote: If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. I just solved a FEM eigenvalue problem where the condition number of the mass and stiffness matrices was something like 1e6... Result looked good to me... So I don't understand what you're saying about 10 = 1 or 2 digits. I think my problem was accurate enough, though I don't know what error with 1e6 in condition number, I should expect. How did you arrive at 1 or 2 digits for cond(A)=10, if I may ask ? As Steven pointed out earlier, it all depends on the precision you are dealing with. If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of 1e6 may be fine. But you had better be using double precision (64- bit) floating point numbers (which are the default in Python, of course). Those have approximately 12 digits of precision, so you are in good shape. Single-precision floats only have 6 or 7 digits of precision, so you'd be in trouble there. For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would say that the problem lies with the highest natural frequencies, they for sure cannot be verified - there's too little energy in them. But the lowest frequencies (the most important ones) are good, I think - even for high cond number. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 3, 10:30 am, someone newsbo...@gmail.com wrote: On 05/02/2012 11:45 PM, Russ P. wrote: On May 2, 1:29 pm, someonenewsbo...@gmail.com wrote: If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. I just solved a FEM eigenvalue problem where the condition number of the mass and stiffness matrices was something like 1e6... Result looked good to me... So I don't understand what you're saying about 10 = 1 or 2 digits. I think my problem was accurate enough, though I don't know what error with 1e6 in condition number, I should expect. How did you arrive at 1 or 2 digits for cond(A)=10, if I may ask ? As Steven pointed out earlier, it all depends on the precision you are dealing with. If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of 1e6 may be fine. But you had better be using double precision (64- bit) floating point numbers (which are the default in Python, of course). Those have approximately 12 digits of precision, so you are in good shape. Single-precision floats only have 6 or 7 digits of precision, so you'd be in trouble there. For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would say that the problem lies with the highest natural frequencies, they for sure cannot be verified - there's too little energy in them. But the lowest frequencies (the most important ones) are good, I think - even for high cond number. Did you mention earlier what FEM stands for? If so, I missed it. Is it finite-element modeling? Whatever the case, note that I said, If you are just doing pure mathematical or numerical work with no real- world measurement error, then a condition number of 1e6 may be fine. I forgot much more than I know about finite-element modeling, but isn't it a purely numerical method of analysis? If that is the case, then my comment above is relevant. By the way, I didn't mean to patronize you with my earlier explanation of orthogonal transformations. They are fundamental to understanding the SVD, and I thought it might be interesting to anyone who is not familiar with the concept. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/03/2012 07:55 PM, Russ P. wrote: On May 3, 10:30 am, someonenewsbo...@gmail.com wrote: On 05/02/2012 11:45 PM, Russ P. wrote: For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would say that the problem lies with the highest natural frequencies, they for sure cannot be verified - there's too little energy in them. But the lowest frequencies (the most important ones) are good, I think - even for high cond number. Did you mention earlier what FEM stands for? If so, I missed it. Is it finite-element modeling? Whatever the case, note that I said, If Sorry, yes: Finite Element Model. you are just doing pure mathematical or numerical work with no real- world measurement error, then a condition number of 1e6 may be fine. I forgot much more than I know about finite-element modeling, but isn't it a purely numerical method of analysis? If that I'm not sure exactly, what is the definition of a purely numerical method of analysis? I would guess that the answer is yes, it's a purely numerical method? But I also thing it's a practical engineering or scientific work... is the case, then my comment above is relevant. Uh, I just don't understand the difference: 1) For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. vs. 2) If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of, 1e6 may be fine. I would think that FEM is a practical engineering work and also pure numerical work... Or something... By the way, I didn't mean to patronize you with my earlier explanation of orthogonal transformations. They are fundamental to understanding the SVD, and I thought it might be interesting to anyone who is not familiar with the concept. Don't worry, I think it was really good and I don't think anyone patronized me, on the contrary, people was/is very helpful. SVD isn't my strongest side and maybe I should've thought a bit more about this singular matrix and perhaps realized what some people here already explained, a bit earlier (maybe before I asked). Anyway, it's been good to hear/read what you've (and others) have written. Yesterday and earlier today I was at work during the day so answering/replying took a bit longer than I like, considering the huge flow of posts in the matlab group. But now I'm home most of the time, for the next 3 days and will check for followup posts quite frequent, I think... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
Yeah, I realized that I should rephrase my previous statement to something like this: For any *empirical* engineering or scientific work, I'd say that a condition number of 1e6 is likely to be unacceptable. I'd put finite elements into the category of theoretical and numerical rather than empirical. Still, a condition number of 1e6 would bother me, but maybe that's just me. --Russ P. On May 3, 3:21 pm, someone newsbo...@gmail.com wrote: On 05/03/2012 07:55 PM, Russ P. wrote: On May 3, 10:30 am, someonenewsbo...@gmail.com wrote: On 05/02/2012 11:45 PM, Russ P. wrote: For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would say that the problem lies with the highest natural frequencies, they for sure cannot be verified - there's too little energy in them. But the lowest frequencies (the most important ones) are good, I think - even for high cond number. Did you mention earlier what FEM stands for? If so, I missed it. Is it finite-element modeling? Whatever the case, note that I said, If Sorry, yes: Finite Element Model. you are just doing pure mathematical or numerical work with no real- world measurement error, then a condition number of 1e6 may be fine. I forgot much more than I know about finite-element modeling, but isn't it a purely numerical method of analysis? If that I'm not sure exactly, what is the definition of a purely numerical method of analysis? I would guess that the answer is yes, it's a purely numerical method? But I also thing it's a practical engineering or scientific work... is the case, then my comment above is relevant. Uh, I just don't understand the difference: 1) For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. vs. 2) If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of, 1e6 may be fine. I would think that FEM is a practical engineering work and also pure numerical work... Or something... By the way, I didn't mean to patronize you with my earlier explanation of orthogonal transformations. They are fundamental to understanding the SVD, and I thought it might be interesting to anyone who is not familiar with the concept. Don't worry, I think it was really good and I don't think anyone patronized me, on the contrary, people was/is very helpful. SVD isn't my strongest side and maybe I should've thought a bit more about this singular matrix and perhaps realized what some people here already explained, a bit earlier (maybe before I asked). Anyway, it's been good to hear/read what you've (and others) have written. Yesterday and earlier today I was at work during the day so answering/replying took a bit longer than I like, considering the huge flow of posts in the matlab group. But now I'm home most of the time, for the next 3 days and will check for followup posts quite frequent, I think... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/04/2012 12:58 AM, Russ P. wrote: Yeah, I realized that I should rephrase my previous statement to something like this: For any *empirical* engineering or scientific work, I'd say that a condition number of 1e6 is likely to be unacceptable. Still, I don't understand it. Do you have an example of this kind of work, if it's not FEM? I'd put finite elements into the category of theoretical and numerical rather than empirical. Still, a condition number of 1e6 would bother me, but maybe that's just me. Ok, but I just don't understand what's in the empirical category, sorry... Maybe the conclusion is just that if cond(A) 1e15 or 1e16, then that problem shouldn't be solved and maybe this is also approx. where matlab has it's warning-threshold (maybe, I'm just guessing here)... So, maybe I could perhaps use that limit in my future python program (when I find out how to get the condition number etc, but I assume this can be googled for with no problems)... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On Thu, 03 May 2012 19:30:35 +0200, someone wrote: On 05/02/2012 11:45 PM, Russ P. wrote: On May 2, 1:29 pm, someonenewsbo...@gmail.com wrote: If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. I just solved a FEM eigenvalue problem where the condition number of the mass and stiffness matrices was something like 1e6... Result looked good to me... So I don't understand what you're saying about 10 = 1 or 2 digits. I think my problem was accurate enough, though I don't know what error with 1e6 in condition number, I should expect. How did you arrive at 1 or 2 digits for cond(A)=10, if I may ask ? As Steven pointed out earlier, it all depends on the precision you are dealing with. If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of 1e6 may be fine. But you had better be using double precision (64- bit) floating point numbers (which are the default in Python, of course). Those have approximately 12 digits of precision, so you are in good shape. Single-precision floats only have 6 or 7 digits of precision, so you'd be in trouble there. For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. So how do you explain that the natural frequencies from FEM (with condition number ~1e6) generally correlates really good with real measurements (within approx. 5%), at least for the first 3-4 natural frequencies? I would counter your hand-waving (correlates really good, within approx 5% of *what*?) with hand-waving of my own: Sure, that's exactly what I would expect! *wink* By the way, if I didn't say so earlier, I'll say so now: the interpretation of how bad the condition number is will depend on the underlying physics and/or mathematics of the situation. The interpretation of loss of digits of precision is a general rule of thumb that holds in many diverse situations, not a rule of physics that cannot be broken in this universe. If you have found a scenario where another interpretation of condition number applies, good for you. That doesn't change the fact that, under normal circumstances when trying to solve systems of linear equations, a condition number of 1e6 is likely to blow away *all* the accuracy in your measured data. (Very few physical measurements are accurate to more than six digits.) -- Steven -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 3, 4:59 pm, someone newsbo...@gmail.com wrote: On 05/04/2012 12:58 AM, Russ P. wrote: Yeah, I realized that I should rephrase my previous statement to something like this: For any *empirical* engineering or scientific work, I'd say that a condition number of 1e6 is likely to be unacceptable. Still, I don't understand it. Do you have an example of this kind of work, if it's not FEM? I'd put finite elements into the category of theoretical and numerical rather than empirical. Still, a condition number of 1e6 would bother me, but maybe that's just me. Ok, but I just don't understand what's in the empirical category, sorry... I didn't look it up, but as far as I know, empirical just means based on experiment, which means based on measured data. Unless I am mistaken , a finite element analysis is not based on measured data. Yes, the results can be *compared* with measured data and perhaps calibrated with measured data, but those are not the same thing. I agree with Steven D's comment above, and I will reiterate that a condition number of 1e6 would not inspire confidence in me. If I had a condition number like that, I would look for a better model. But that's just a gut reaction, not a hard scientific rule. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 01:05 AM, Paul Rubin wrote: someonenewsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD I don't understand what else I should write. I gave the singular matrix and that's it. Nothing more is to say about this problem, except it would be nice to learn some things for future use (for instance understanding SVD more - perhaps someone geometrically can explain SVD, that'll be really nice, I hope)... just separates out the orthogonal and scaling parts of the transformation induced by a matrix. Whether that is of any use to you is unclear since you don't say what you're trying to do. Still, I dont think I completely understand SVD. SVD (at least in Matlab) returns 3 matrices, one is a diagonal matrix I think. I think I would better understand it with geometric examples, if one would be so kind to maybe write something about that... I can plot 3D vectors in matlab, very easily so maybe I better understand SVD if I hear/read the geometric explanation (references to textbook/similar is also appreciated). -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 01:38 AM, Russ P. wrote: On May 1, 4:05 pm, Paul Rubinno.em...@nospam.invalid wrote: someonenewsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD just separates out the orthogonal and scaling parts of the transformation induced by a matrix. Whether that is of any use to you is unclear since you don't say what you're trying to do. I agree with the first sentence, but I take slight issue with the word just in the second. The orthogonal part of the transformation is non-distorting, but the scaling part essentially distorts the space. At least that's how I think about it. The larger the ratio between the largest and smallest singular value, the more distortion there is. SVD may or may not be the best choice for the final algorithm, but it is useful for visualizing the transformation you are applying. It can provide clues about the quality of the selection of independent variables, state variables, or inputs. Me would like to hear more! :-) It would really appreciate if anyone could maybe post a simple SVD example and tell what the vectors from the SVD represents geometrically / visually, because I don't understand it good enough and I'm sure it's very important, when it comes to solving matrix systems... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 1, 11:03 pm, someone newsbo...@gmail.com wrote: On 05/02/2012 01:38 AM, Russ P. wrote: On May 1, 4:05 pm, Paul Rubinno.em...@nospam.invalid wrote: someonenewsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD just separates out the orthogonal and scaling parts of the transformation induced by a matrix. Whether that is of any use to you is unclear since you don't say what you're trying to do. I agree with the first sentence, but I take slight issue with the word just in the second. The orthogonal part of the transformation is non-distorting, but the scaling part essentially distorts the space. At least that's how I think about it. The larger the ratio between the largest and smallest singular value, the more distortion there is. SVD may or may not be the best choice for the final algorithm, but it is useful for visualizing the transformation you are applying. It can provide clues about the quality of the selection of independent variables, state variables, or inputs. Me would like to hear more! :-) It would really appreciate if anyone could maybe post a simple SVD example and tell what the vectors from the SVD represents geometrically / visually, because I don't understand it good enough and I'm sure it's very important, when it comes to solving matrix systems... SVD is perhaps the ultimate matrix decomposition and the ultimate tool for linear analysis. Google it and take a look at the excellent Wikipedia page on it. I would be wasting my time if I tried to compete with that. To really appreciate the SVD, you need some background in linear algebra. In particular, you need to understand orthogonal transformations. Think about a standard 3D Cartesian coordinate frame. A rotation of the coordinate frame is an orthogonal transformation of coordinates. The original frame and the new frame are both orthogonal. A vector in one frame is converted to the other frame by multiplying by an orthogonal matrix. The main feature of an orthogonal matrix is that its transpose is its inverse (hence the inverse is trivial to compute). The SVD can be thought of as factoring any linear transformation into a rotation, then a scaling, followed by another rotation. The scaling is represented by the middle matrix of the transformation, which is a diagonal matrix of the same dimensions as the original matrix. The singular values can be read off of the diagonal. If any of them are zero, then the original matrix is singular. If the ratio of the largest to smallest singular value is large, then the original matrix is said to be poorly conditioned. Standard Cartesian coordinate frames are orthogonal. Imagine an x-y coordinate frame in which the axes are not orthogonal. Such a coordinate frame is possible, but they are rarely used. If the axes are parallel, the coordinate frame will be singular and will basically reduce to one-dimensional. If the x and y axes are nearly parallel, the coordinate frame could still be used in theory, but it will be poorly conditioned. You will need large numbers to represent points fairly close to the origin, and small deviations will translate into large changes in coordinate values. That can lead to problems due to numerical roundoff errors and other kinds of errors. --Russ P. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
someone writes: except it would be nice to learn some things for future use (for instance understanding SVD more - perhaps someone geometrically can explain SVD, that'll be really nice, I hope)... The Wikipedia article looks promising to me: http://en.wikipedia.org/wiki/Singular_value_decomposition Lots in it, with annotated links to more, including a song. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
Russ P. russ.paie...@gmail.com writes: The SVD can be thought of as factoring any linear transformation into a rotation, then a scaling, followed by another rotation. Ah yes, my description was backwards, sorry. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
someone newsbo...@gmail.com writes: You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD I don't understand what else I should write. I gave the singular matrix and that's it. Nothing more is to say about this problem, You could write what your application is. What do the numbers in that matrix actually represent? except it would be nice to learn some things for future use (for instance understanding SVD more The Wikipedia article about SVD is pretty good. Still, I dont think I completely understand SVD. SVD (at least in Matlab) returns 3 matrices, one is a diagonal matrix I think. Yes. Two diagonal matrices and a unitary matrix. The diagonal matrices have the singular values which are the magnitudes of the matrix's eigenvalues. A matrix (for simplicity consider it to have nonzero determinant) represents a linear transformation in space. Think of a bunch of vectors as arrows sticking out of the origin in different directions, and consider what happens to them if you use the matrix to act on all of them at once. You could break the transformation into 3 steps: 1) scale the axes by suitable amounts, so all the vectors stretch or shrink depending on their directions. 2) Rotate all the vectors without changing their lengths, through some combination of angles; 3) undo the scaling transformation so that the vectors get their original lengths back. Each of these 3 steps can be described by a matrix and the 3 matrices are what SVD calculates. I still don't have any idea whether this info is going to do you any good. There are some quite knowledgeable numerics folks here (I'm not one) who can probably be of more help, but frankly your questions don't make a whole lot of sense. You might get the book Numerical Recipes which is old but instructive. Maybe someone else here has other recommendations. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 5/2/2012 8:00, someone wrote: Still, I dont think I completely understand SVD. SVD (at least in Matlab) returns 3 matrices, one is a diagonal matrix I think. I think I would better understand it with geometric examples, if one would be so kind to maybe write something about that... I can plot 3D vectors in matlab, very easily so maybe I better understand SVD if I hear/read the geometric explanation (references to textbook/similar is also appreciated). Russ's post is a very good starting point. I hope you read it. Kiuhnm -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On Wed, 02 May 2012 08:00:44 +0200, someone wrote: On 05/02/2012 01:05 AM, Paul Rubin wrote: someonenewsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD I don't understand what else I should write. I gave the singular matrix and that's it. You can't judge what an acceptable condition number is unless you know what your data is. http://mathworld.wolfram.com/ConditionNumber.html http://en.wikipedia.org/wiki/Condition_number If your condition number is ten, then you should expect to lose one digit of accuracy in your solution, over and above whatever loss of accuracy comes from the numeric algorithm. A condition number of 64 will lose six bits, or about 1.8 decimal digits, of accuracy. If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. -- Steven -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
Russ P. russ.paie...@gmail.com wrote in message news:2275231f-405f-4ee3-966a-40c821b7c...@2g2000yqp.googlegroups.com... On May 1, 11:52 am, someone newsbo...@gmail.com wrote: On 04/30/2012 03:35 AM, Nasser M. Abbasi wrote: On 04/29/2012 07:59 PM, someone wrote: I do not use python much myself, but a quick google showed that pyhton scipy has API for linalg, so use, which is from the documentation, the following code example X = scipy.linalg.solve(A, B) But you still need to check the cond(). If it is too large, not good. How large and all that, depends on the problem itself. But the rule of thumb, the lower the better. Less than 100 can be good in general, but I really can't give you a fixed number to use, as I am not an expert in this subjects, others who know more about it might have better recommendations. Ok, that's a number... Anyone wants to participate and do I hear something better than less than 100 can be good in general ? If I don't hear anything better, the limit is now 100... What's the limit in matlab (on the condition number of the matrices), by the way, before it comes up with a warning ??? I'm not going to answer, and the reason why is that saying the limit is X may lead you to believe that as long as my condition number is less than X, I'm safe. That's not the case. The threshold above which MATLAB warns is the fence that separates the tourists from the edge of the cliff of singularity -- just because you haven't crossed the fence doesn't mean you're not in danger of tripping and falling into the ravine. The threshold of acceptability really depends on the problem you are trying to solve. I agree with this statement, although you generally don't want it to be too big. The definition of too big is somewhat fluid, though. I haven't solved linear equations for a long time, but off hand, I would say that a condition number over 10 is questionable. That seems pretty low as a general bound IMO. A high condition number suggests that the selection of independent variables for the linear function you are trying to fit is not quite right. For a poorly conditioned matrix, your modeling function will be very sensitive to measurement noise and other sources of error, if applicable. If the condition number is 100, then any input on one particular axis gets magnified 100 times more than other inputs. Unless your inputs are very precise, that is probably not what you want. Or something like that. Russ, you and the OP (and others) may be interested in one of the books that Cleve Moler has written and made freely available on the MathWorks website: http://www.mathworks.com/moler/ The chapter Linear Equations in Numerical Computing with MATLAB includes a section (section 2.9, starting on page 14 if I remember correctly) that discusses norm and condition number and gives a more formal statement of what you described. The code included in the book is written in MATLAB, but even if you don't use MATLAB (since I know this has been cross-posted to comp.lang.python) there's plenty of good, crunchy mathematics in that section. -- Steve Lord sl...@mathworks.com To contact Technical Support use the Contact Us link on http://www.mathworks.com -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 01:03 PM, Kiuhnm wrote: On 5/2/2012 8:00, someone wrote: Still, I dont think I completely understand SVD. SVD (at least in Matlab) returns 3 matrices, one is a diagonal matrix I think. I think I would better understand it with geometric examples, if one would be so kind to maybe write something about that... I can plot 3D vectors in matlab, very easily so maybe I better understand SVD if I hear/read the geometric explanation (references to textbook/similar is also appreciated). Russ's post is a very good starting point. I hope you read it. Ofcourse I do. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 08:36 AM, Russ P. wrote: On May 1, 11:03 pm, someonenewsbo...@gmail.com wrote: On 05/02/2012 01:38 AM, Russ P. wrote: .. On May 1, 4:05 pm, Paul Rubinno.em...@nospam.invalidwrote: It would really appreciate if anyone could maybe post a simple SVD example and tell what the vectors from the SVD represents geometrically / visually, because I don't understand it good enough and I'm sure it's very important, when it comes to solving matrix systems... SVD is perhaps the ultimate matrix decomposition and the ultimate tool for linear analysis. Google it and take a look at the excellent Wikipedia page on it. I would be wasting my time if I tried to compete with that. Ok. To really appreciate the SVD, you need some background in linear algebra. In particular, you need to understand orthogonal transformations. Think about a standard 3D Cartesian coordinate frame. A rotation of the coordinate frame is an orthogonal transformation of coordinates. The original frame and the new frame are both orthogonal. Yep. A vector in one frame is converted to the other frame by multiplying by an orthogonal matrix. The main feature of an orthogonal matrix is that its transpose is its inverse (hence the inverse is trivial to compute). As far as i know, you have to replace orthogonal with: orthonormal. That much I at least think I know without even going to wikipedia first... The SVD can be thought of as factoring any linear transformation into a rotation, then a scaling, followed by another rotation. The scaling is represented by the middle matrix of the transformation, which is a diagonal matrix of the same dimensions as the original matrix. The singular values can be read off of the diagonal. If any of them are zero, then the original matrix is singular. If the ratio of the largest to smallest singular value is large, then the original matrix is said to be poorly conditioned. Aah, thank you very much. I can easily recognize some of this... Standard Cartesian coordinate frames are orthogonal. Imagine an x-y coordinate frame in which the axes are not orthogonal. Such a coordinate frame is possible, but they are rarely used. If the axes are parallel, the coordinate frame will be singular and will basically reduce to one-dimensional. If the x and y axes are nearly parallel, the coordinate frame could still be used in theory, but it will be poorly conditioned. You will need large numbers to represent points fairly close to the origin, and small deviations will translate into large changes in coordinate values. That can lead to problems due to numerical roundoff errors and other kinds of errors. Thank you very much for your time. It always helps to get the same explanation from different people with slightly different ways of explaining it. Thanks! -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 01:52 PM, Steven D'Aprano wrote: On Wed, 02 May 2012 08:00:44 +0200, someone wrote: On 05/02/2012 01:05 AM, Paul Rubin wrote: someonenewsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD I don't understand what else I should write. I gave the singular matrix and that's it. You can't judge what an acceptable condition number is unless you know what your data is. http://mathworld.wolfram.com/ConditionNumber.html http://en.wikipedia.org/wiki/Condition_number If your condition number is ten, then you should expect to lose one digit of accuracy in your solution, over and above whatever loss of accuracy comes from the numeric algorithm. A condition number of 64 will lose six bits, or about 1.8 decimal digits, of accuracy. If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. I just solved a FEM eigenvalue problem where the condition number of the mass and stiffness matrices was something like 1e6... Result looked good to me... So I don't understand what you're saying about 10 = 1 or 2 digits. I think my problem was accurate enough, though I don't know what error with 1e6 in condition number, I should expect. How did you arrive at 1 or 2 digits for cond(A)=10, if I may ask ? -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/02/2012 04:47 PM, Steven_Lord wrote: Russ, you and the OP (and others) may be interested in one of the books that Cleve Moler has written and made freely available on the MathWorks website: http://www.mathworks.com/moler/ The chapter Linear Equations in Numerical Computing with MATLAB includes a section (section 2.9, starting on page 14 if I remember correctly) that discusses norm and condition number and gives a more formal statement of what you described. The code included in the book is written in MATLAB, but even if you don't use MATLAB (since I know this has been cross-posted to comp.lang.python) there's plenty of good, crunchy mathematics in that section. I use matlab more than python. I just want to learn python, which seems extremely powerful and easy but yet, I'm a python beginner. Thank you very much for the reference, Mr. Lord. I'll look closely at that Moler-book, in about 1/2 hour or so Looking forward to learning more about this... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 2, 1:29 pm, someone newsbo...@gmail.com wrote: If your data starts off with only 1 or 2 digits of accuracy, as in your example, then the result is meaningless -- the accuracy will be 2-2 digits, or 0 -- *no* digits in the answer can be trusted to be accurate. I just solved a FEM eigenvalue problem where the condition number of the mass and stiffness matrices was something like 1e6... Result looked good to me... So I don't understand what you're saying about 10 = 1 or 2 digits. I think my problem was accurate enough, though I don't know what error with 1e6 in condition number, I should expect. How did you arrive at 1 or 2 digits for cond(A)=10, if I may ask ? As Steven pointed out earlier, it all depends on the precision you are dealing with. If you are just doing pure mathematical or numerical work with no real-world measurement error, then a condition number of 1e6 may be fine. But you had better be using double precision (64- bit) floating point numbers (which are the default in Python, of course). Those have approximately 12 digits of precision, so you are in good shape. Single-precision floats only have 6 or 7 digits of precision, so you'd be in trouble there. For any practical engineering or scientific work, I'd say that a condition number of 1e6 is very likely to be completely unacceptable. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On Apr 29, 5:17 pm, someone newsbo...@gmail.com wrote: On 04/30/2012 12:39 AM, Kiuhnm wrote: So Matlab at least warns about Matrix is close to singular or badly scaled, which python (and I guess most other languages) does not... A is not just close to singular: it's singular! Ok. When do you define it to be singular, btw? Which is the most accurate/best, even for such a bad matrix? Is it possible to say something about that? Looks like python has a lot more digits but maybe that's just a random result... I mean Element 1,1 = 2.81e14 in Python, but something like 3e14 in Matlab and so forth - there's a small difference in the results... Both results are *wrong*: no inverse exists. What's the best solution of the two wrong ones? Best least-squares solution or whatever? With python, I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? If cond(A) is high, you're trying to solve your problem the wrong way. So you're saying that in another language (python) I should check the condition number, before solving anything? You should try to avoid matrix inversion altogether if that's the case. For instance you shouldn't invert a matrix just to solve a linear system. What then? Cramer's rule? If you really want to know just about everything there is to know about a matrix, take a look at its Singular Value Decomposition (SVD). I've never used numpy, but I assume it can compute an SVD. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
There is linalg.pinv, which computes a pseudoinverse based on SVD that works on all matrices, regardless of the rank of the matrix. It merely approximates A*A.I = I as well as A permits though, rather than being a true inverse, which may not exist. Anyway, there are no general answers for this kind of thing. In all non-textbook problems I can think of, the properties of your matrix are highly constrained by the problem you are working on; which additional tests are required to check for corner cases thus depends on the problem. Often, if you have found an elegant solution to your problem, no such corner cases exist. In that case, MATLAB is just wasting your time with its automated checks. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/30/2012 02:57 AM, Paul Rubin wrote: someonenewsbo...@gmail.com writes: A is not just close to singular: it's singular! Ok. When do you define it to be singular, btw? Singular means the determinant is zero, i.e. the rows or columns are not linearly independent. Let's give names to the three rows: a = [1 2 3]; b = [11 12 13]; c = [21 22 23]. Then notice that c = 2*b - a. So c is linearly dependent on a and b. Geometrically this means the three vectors are in the same plane, so the matrix doesn't have an inverse. Oh, thak you very much for a good explanation. Which is the most accurate/best, even for such a bad matrix? What are you trying to do? If you are trying to calculate stuff with matrices, you really should know some basic linear algebra. Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, although I'm not an expert like others in this forum, I know for sure :-) -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/01/2012 08:56 AM, Russ P. wrote: On Apr 29, 5:17 pm, someonenewsbo...@gmail.com wrote: On 04/30/2012 12:39 AM, Kiuhnm wrote: You should try to avoid matrix inversion altogether if that's the case. For instance you shouldn't invert a matrix just to solve a linear system. What then? Cramer's rule? If you really want to know just about everything there is to know about a matrix, take a look at its Singular Value Decomposition (SVD). I know a bit about SVD - I used it for a short period of time in Matlab, though I'm definately not an expert in it and I don't understand the whole theory with orthogality behind making it work so elegant as it is/does work out. I've never used numpy, but I assume it can compute an SVD. I'm making my first steps now with numpy, so there's a lot I don't know and haven't tried with numpy... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/30/2012 03:35 AM, Nasser M. Abbasi wrote: On 04/29/2012 07:59 PM, someone wrote: I do not use python much myself, but a quick google showed that pyhton scipy has API for linalg, so use, which is from the documentation, the following code example X = scipy.linalg.solve(A, B) But you still need to check the cond(). If it is too large, not good. How large and all that, depends on the problem itself. But the rule of thumb, the lower the better. Less than 100 can be good in general, but I really can't give you a fixed number to use, as I am not an expert in this subjects, others who know more about it might have better recommendations. Ok, that's a number... Anyone wants to participate and do I hear something better than less than 100 can be good in general ? If I don't hear anything better, the limit is now 100... What's the limit in matlab (on the condition number of the matrices), by the way, before it comes up with a warning ??? -- http://mail.python.org/mailman/listinfo/python-list
RE: numpy (matrix solver) - python vs. matlab
I'm making my first steps now with numpy, so there's a lot I don't know and haven't tried with numpy... An excellent reason to subscribe to the numpy mailing list and talk on there :) Ramit Ramit Prasad | JPMorgan Chase Investment Bank | Currencies Technology 712 Main Street | Houston, TX 77002 work phone: 713 - 216 - 5423 -- This email is confidential and subject to important disclaimers and conditions including on offers for the purchase or sale of securities, accuracy and completeness of information, viruses, confidentiality, legal privilege, and legal entity disclaimers, available at http://www.jpmorgan.com/pages/disclosures/email. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 01/05/2012 2:43 PM, someone wrote: [snip] a = [1 2 3]; b = [11 12 13]; c = [21 22 23]. Then notice that c = 2*b - a. So c is linearly dependent on a and b. Geometrically this means the three vectors are in the same plane, so the matrix doesn't have an inverse. Does it not mean that there are three parallel planes? Consider the example in two dimensional space. Colin W. [snip] -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 1, 11:52 am, someone newsbo...@gmail.com wrote: On 04/30/2012 03:35 AM, Nasser M. Abbasi wrote: On 04/29/2012 07:59 PM, someone wrote: I do not use python much myself, but a quick google showed that pyhton scipy has API for linalg, so use, which is from the documentation, the following code example X = scipy.linalg.solve(A, B) But you still need to check the cond(). If it is too large, not good. How large and all that, depends on the problem itself. But the rule of thumb, the lower the better. Less than 100 can be good in general, but I really can't give you a fixed number to use, as I am not an expert in this subjects, others who know more about it might have better recommendations. Ok, that's a number... Anyone wants to participate and do I hear something better than less than 100 can be good in general ? If I don't hear anything better, the limit is now 100... What's the limit in matlab (on the condition number of the matrices), by the way, before it comes up with a warning ??? The threshold of acceptability really depends on the problem you are trying to solve. I haven't solved linear equations for a long time, but off hand, I would say that a condition number over 10 is questionable. A high condition number suggests that the selection of independent variables for the linear function you are trying to fit is not quite right. For a poorly conditioned matrix, your modeling function will be very sensitive to measurement noise and other sources of error, if applicable. If the condition number is 100, then any input on one particular axis gets magnified 100 times more than other inputs. Unless your inputs are very precise, that is probably not what you want. Or something like that. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/01/2012 09:59 PM, Colin J. Williams wrote: On 01/05/2012 2:43 PM, someone wrote: [snip] a = [1 2 3]; b = [11 12 13]; c = [21 22 23]. Then notice that c = 2*b - a. So c is linearly dependent on a and b. Geometrically this means the three vectors are in the same plane, so the matrix doesn't have an inverse. Does it not mean that there are three parallel planes? Consider the example in two dimensional space. I actually drawed it and saw it... It means that you can construct a 2D plane and all 3 vectors are in this 2D-plane... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 05/01/2012 10:54 PM, Russ P. wrote: On May 1, 11:52 am, someonenewsbo...@gmail.com wrote: On 04/30/2012 03:35 AM, Nasser M. Abbasi wrote: What's the limit in matlab (on the condition number of the matrices), by the way, before it comes up with a warning ??? The threshold of acceptability really depends on the problem you are trying to solve. I haven't solved linear equations for a long time, but off hand, I would say that a condition number over 10 is questionable. Anyone knows the threshold for Matlab for warning when solving x=A\b ? I tried edit slash but this seems to be internal so I cannot see what criteria the warning is based upon... A high condition number suggests that the selection of independent variables for the linear function you are trying to fit is not quite right. For a poorly conditioned matrix, your modeling function will be very sensitive to measurement noise and other sources of error, if applicable. If the condition number is 100, then any input on one particular axis gets magnified 100 times more than other inputs. Unless your inputs are very precise, that is probably not what you want. Or something like that. Ok. So it's like a frequency-response-function, output divided by input... -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 5/1/12 10:21 PM, someone wrote: On 05/01/2012 10:54 PM, Russ P. wrote: On May 1, 11:52 am, someonenewsbo...@gmail.com wrote: On 04/30/2012 03:35 AM, Nasser M. Abbasi wrote: What's the limit in matlab (on the condition number of the matrices), by the way, before it comes up with a warning ??? The threshold of acceptability really depends on the problem you are trying to solve. I haven't solved linear equations for a long time, but off hand, I would say that a condition number over 10 is questionable. Anyone knows the threshold for Matlab for warning when solving x=A\b ? I tried edit slash but this seems to be internal so I cannot see what criteria the warning is based upon... The documentation for that operator is here: http://www.mathworks.co.uk/help/techdoc/ref/mldivide.html -- Robert Kern I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth. -- Umberto Eco -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 5/1/2012 21:59, Colin J. Williams wrote: On 01/05/2012 2:43 PM, someone wrote: [snip] a = [1 2 3]; b = [11 12 13]; c = [21 22 23]. Then notice that c = 2*b - a. So c is linearly dependent on a and b. Geometrically this means the three vectors are in the same plane, so the matrix doesn't have an inverse. Does it not mean that there are three parallel planes? They're not parallel because our matrix has rank 2, not 1. Anyway, have a look at this: http://en.wikipedia.org/wiki/Parallelepiped#Volume It follows that our matrix A whose rows are a, b and c represents a parallelepiped. If our vectors are collinear or coplanar, the parallelepiped is degenerate, i.e. has volume 0. The converse is also true. Kiuhnm -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
someone newsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD just separates out the orthogonal and scaling parts of the transformation induced by a matrix. Whether that is of any use to you is unclear since you don't say what you're trying to do. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On May 1, 4:05 pm, Paul Rubin no.em...@nospam.invalid wrote: someone newsbo...@gmail.com writes: Actually I know some... I just didn't think so much about, before writing the question this as I should, I know theres also something like singular value decomposition that I think can help solve otherwise illposed problems, You will probably get better advice if you are able to describe what problem (ill-posed or otherwise) you are actually trying to solve. SVD just separates out the orthogonal and scaling parts of the transformation induced by a matrix. Whether that is of any use to you is unclear since you don't say what you're trying to do. I agree with the first sentence, but I take slight issue with the word just in the second. The orthogonal part of the transformation is non-distorting, but the scaling part essentially distorts the space. At least that's how I think about it. The larger the ratio between the largest and smallest singular value, the more distortion there is. SVD may or may not be the best choice for the final algorithm, but it is useful for visualizing the transformation you are applying. It can provide clues about the quality of the selection of independent variables, state variables, or inputs. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 4/30/2012 2:17, someone wrote: On 04/30/2012 12:39 AM, Kiuhnm wrote: So Matlab at least warns about Matrix is close to singular or badly scaled, which python (and I guess most other languages) does not... A is not just close to singular: it's singular! Ok. When do you define it to be singular, btw? Which is the most accurate/best, even for such a bad matrix? Is it possible to say something about that? Looks like python has a lot more digits but maybe that's just a random result... I mean Element 1,1 = 2.81e14 in Python, but something like 3e14 in Matlab and so forth - there's a small difference in the results... Both results are *wrong*: no inverse exists. What's the best solution of the two wrong ones? Best least-squares solution or whatever? Trust me. They're both so wrong that it doesn't matter. Have a look at A*inv(A) and inv(A)*A and you'll see by yourself. With python, I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? If cond(A) is high, you're trying to solve your problem the wrong way. So you're saying that in another language (python) I should check the condition number, before solving anything? Yes, unless you already know that it will always be low by design. You should try to avoid matrix inversion altogether if that's the case. For instance you shouldn't invert a matrix just to solve a linear system. What then? Look at the documentation of the library you're using. Cramer's rule? Surprisingly, yes. That's an option. See A condensation-based application of Cramerʼs rule for solving large-scale linear systems Popular linear codes are based on Gaussian elimination or some iterative method, though. Kiuhnm -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 4/30/2012 3:35, Nasser M. Abbasi wrote: But you still need to check the cond(). If it is too large, not good. How large and all that, depends on the problem itself. But the rule of thumb, the lower the better. Less than 100 can be good in general, but I really can't give you a fixed number to use, as I am not an expert in this subjects, others who know more about it might have better recommendations. Alas, there's no fixed number and as if that wasn't enough, there are many condition numbers, each one with different properties. For instance, the Skeel condition number is scale-invariant and it's useful when a matrix is ill-conditioned just because its rows are out of scale. Kiuhnm -- http://mail.python.org/mailman/listinfo/python-list
numpy (matrix solver) - python vs. matlab
Hi, Notice cross-post, I hope you bear over with me for doing that (and I imagine that some of you also like python in the matlab-group like myself)... -- Python vs. Matlab: -- Python: from numpy import matrix from numpy import linalg A = matrix( [[1,2,3],[11,12,13],[21,22,23]] ) print A= print A print A.I (inverse of A)= print A.I A.I (inverse of A)= [[ 2.81466387e+14 -5.62932774e+14 2.81466387e+14] [ -5.62932774e+14 1.12586555e+15 -5.62932774e+14] [ 2.81466387e+14 -5.62932774e+14 2.81466387e+14]] Matlab: A=[1 2 3; 11 12 13; 21 22 23] A = 1 2 3 111213 212223 inv(A) Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.067522e-17. ans = 1.0e+15 * 0.3002 -0.60050.3002 -0.60051.2010 -0.6005 0.3002 -0.60050.3002 -- Python vs. Matlab: -- So Matlab at least warns about Matrix is close to singular or badly scaled, which python (and I guess most other languages) does not... Which is the most accurate/best, even for such a bad matrix? Is it possible to say something about that? Looks like python has a lot more digits but maybe that's just a random result... I mean Element 1,1 = 2.81e14 in Python, but something like 3e14 in Matlab and so forth - there's a small difference in the results... With python, I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? I hope you matlabticians like this topic, at least I myself find it interesting and many of you probably also program in some other language and then maybe you'll find this worthwhile to read about. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 4/30/2012 0:17, someone wrote: Hi, Notice cross-post, I hope you bear over with me for doing that (and I imagine that some of you also like python in the matlab-group like myself)... -- Python vs. Matlab: -- Python: from numpy import matrix from numpy import linalg A = matrix( [[1,2,3],[11,12,13],[21,22,23]] ) print A= print A print A.I (inverse of A)= print A.I A.I (inverse of A)= [[ 2.81466387e+14 -5.62932774e+14 2.81466387e+14] [ -5.62932774e+14 1.12586555e+15 -5.62932774e+14] [ 2.81466387e+14 -5.62932774e+14 2.81466387e+14]] Matlab: A=[1 2 3; 11 12 13; 21 22 23] A = 1 2 3 11 12 13 21 22 23 inv(A) Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.067522e-17. ans = 1.0e+15 * 0.3002 -0.6005 0.3002 -0.6005 1.2010 -0.6005 0.3002 -0.6005 0.3002 -- Python vs. Matlab: -- So Matlab at least warns about Matrix is close to singular or badly scaled, which python (and I guess most other languages) does not... A is not just close to singular: it's singular! Which is the most accurate/best, even for such a bad matrix? Is it possible to say something about that? Looks like python has a lot more digits but maybe that's just a random result... I mean Element 1,1 = 2.81e14 in Python, but something like 3e14 in Matlab and so forth - there's a small difference in the results... Both results are *wrong*: no inverse exists. With python, I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? If cond(A) is high, you're trying to solve your problem the wrong way. You should try to avoid matrix inversion altogether if that's the case. For instance you shouldn't invert a matrix just to solve a linear system. Kiuhnm -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/30/2012 12:39 AM, Kiuhnm wrote: So Matlab at least warns about Matrix is close to singular or badly scaled, which python (and I guess most other languages) does not... A is not just close to singular: it's singular! Ok. When do you define it to be singular, btw? Which is the most accurate/best, even for such a bad matrix? Is it possible to say something about that? Looks like python has a lot more digits but maybe that's just a random result... I mean Element 1,1 = 2.81e14 in Python, but something like 3e14 in Matlab and so forth - there's a small difference in the results... Both results are *wrong*: no inverse exists. What's the best solution of the two wrong ones? Best least-squares solution or whatever? With python, I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? If cond(A) is high, you're trying to solve your problem the wrong way. So you're saying that in another language (python) I should check the condition number, before solving anything? You should try to avoid matrix inversion altogether if that's the case. For instance you shouldn't invert a matrix just to solve a linear system. What then? Cramer's rule? -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/29/2012 05:17 PM, someone wrote: I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? I hope you'll check the condition number all the time. You could be designing a building where people will live in it. If do not check the condition number, you'll end up with a building that will fall down when a small wind hits it and many people will die all because you did not bother to check the condition number when you solved the equations you used in your design. Also, as was said, do not use INV(A) directly to solve equations. --Nasser -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
someone newsbo...@gmail.com writes: A is not just close to singular: it's singular! Ok. When do you define it to be singular, btw? Singular means the determinant is zero, i.e. the rows or columns are not linearly independent. Let's give names to the three rows: a = [1 2 3]; b = [11 12 13]; c = [21 22 23]. Then notice that c = 2*b - a. So c is linearly dependent on a and b. Geometrically this means the three vectors are in the same plane, so the matrix doesn't have an inverse. Which is the most accurate/best, even for such a bad matrix? What are you trying to do? If you are trying to calculate stuff with matrices, you really should know some basic linear algebra. -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/30/2012 02:38 AM, Nasser M. Abbasi wrote: On 04/29/2012 05:17 PM, someone wrote: I would also kindly ask about how to avoid this problem in the future, I mean, this maybe means that I have to check the condition number at all times before doing anything at all ? How to do that? I hope you'll check the condition number all the time. So how big can it (cond-number) be before I should do something else? And what to do then? Cramers rule or pseudoinverse or something else? You could be designing a building where people will live in it. If do not check the condition number, you'll end up with a building that will fall down when a small wind hits it and many people will die all because you did not bother to check the condition number when you solved the equations you used in your design. Also, as was said, do not use INV(A) directly to solve equations. In Matlab I used x=A\b. I used inv(A) in python. Should I use some kind of pseudo-inverse or what do you suggest? -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/29/2012 07:59 PM, someone wrote: Also, as was said, do not use INV(A) directly to solve equations. In Matlab I used x=A\b. good. I used inv(A) in python. Should I use some kind of pseudo-inverse or what do you suggest? I do not use python much myself, but a quick google showed that pyhton scipy has API for linalg, so use, which is from the documentation, the following code example X = scipy.linalg.solve(A, B) But you still need to check the cond(). If it is too large, not good. How large and all that, depends on the problem itself. But the rule of thumb, the lower the better. Less than 100 can be good in general, but I really can't give you a fixed number to use, as I am not an expert in this subjects, others who know more about it might have better recommendations. --Nasser -- http://mail.python.org/mailman/listinfo/python-list
Re: numpy (matrix solver) - python vs. matlab
On 04/29/2012 07:17 PM, someone wrote: Ok. When do you define it to be singular, btw? There are things you can see right away about a matrix A being singular without doing any computation. By just looking at it. For example, If you see a column (or row) being a linear combination of other column(s) (or row(s)) then this is a no no. In your case you have 1 2 3 111213 212223 You can see right away that if you multiply the second row by 2, and subtract from that one times the first row, then you obtain the third row. Hence the third row is a linear combination of the first row and the second row. no good. When you get a row (or a column) being a linear combination of others rows (or columns), then this means the matrix is singular. --Nasser -- http://mail.python.org/mailman/listinfo/python-list
