How to get the polynomial from fractions scilab F(s)=4x^2-2x-1/2x^3+2x^2-3x+1. I am working with the next quotient of polynomial in scilab software.
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I have built an xgboost model in R and exported it in PMML (with r2pmml).
I have tested the same dataset with R and PMML (with Java), the probabilities output are very close but they all have a small difference between 1e-8 and 1e-10.
These differences are too small to be caused by a issue with the input data.
Is it a classic behaviour of rounding between different language/software or I did a mistake somewhere.
the probabilities output are very close but they all have a small difference between 1e-8 and 1e-10.
The XGBoost library uses float32 data type (single-precision floating-point), which has a "natural precision" of around 1e-7 .. 1e-8 in this range (probability values, between 0 and 1).
So, your observed difference is less than the "natural precision", and should not be a cause for further concern.
The (J)PMML representation is carrying out exactly the same computations (summation of booster float values, applying a normalization function to it) as the native XGBoost representation.
I am trying to fit some kind of polynomical regression to a car engine performance curve. I know that the relationship between the studied two variables is not linear and should follow a quadratic function (performance v.s output power).
Power vs Performance
-14e-05{x^2}+0,009{x}+0,31545
Also I know that the derivative of the function that relates this two variables should be 0 (absolute maximum) when the engine is delivering the maximum power.
The problem comes when after fitting my curve I make the derivative of the function obtained through the polynomial regression I get has the maximum beyond the maximum real power output of the engine (under safety limits)
I have been looking for topics doing the same but I have found only issues related with the sum of the coefficients should be under certain value.
Any ideas to implement this in R?
I'm having a hard time understanding why it would be useful to use the Taylor series for a function in order to gain an approximation of a function, instead of just using the function itself when programming. If I can tell my computer to compute e^(.1) and it will give me an exact value, why would I take an approximation instead?
Taylor series are generally not used to approximate functions. Usually, some form of minimax polynomial is used.
Taylor series converge slowly (it takes many terms to get the accuracy desired) and are inefficient (they are more accurate near the point around which they are centered and less accurate away from it). The largest use of Taylor series is likely in mathematics classes and papers, where they are useful for examining the properties of functions and for learning about calculus.
To approximate functions, minimax polynomials are often used. A minimax polynomial has the minimum possible maximum error for a particular situation (interval over which a function is to be approximated, degree available for the polynomial). There is usually no analytical solution to finding a minimax polynomial. They are found numerically, using the Remez algorithm. Minimax polynomials can be tailored to suit particular needs, such as minimizing relative error or absolute error, approximating a function over a particular interval, and so on. Minimax polynomials need fewer terms than Taylor series to get acceptable results, and they “spread” the error over the interval instead of being better in the center and worse at the ends.
When you call the exp function to compute ex, you are likely using a minimax polynomial, because somebody has done the work for you and constructed a library routine that evaluates the polynomial. For the most part, the only arithmetic computer processors can do is addition, subtraction, multiplication, and division. So other functions have to be constructed from those operations. The first three give you polynomials, and polynomials are sufficient to approximate many functions, such as sine, cosine, logarithm, and exponentiation (with some additional operations of moving things into and out of the exponent field of floating-point values). Division adds rational functions, which is useful for functions like arctangent.
For two reasons. First and foremost - most processors do not have hardware implementations of complex operations like exponentials, logarithms, etc... In such cases the programming language may provide a library function for computing those - in other words, someone used a taylor series or other approximation for you.
Second, you may have a function that not even the language supports.
I recently wanted to use lookup tables with interpolation to get an angle and then compute the sin() and cos() of that angle. Trouble is that it's a DSP with no floating point and no trigonometric functions so those two functions are really slow (software implementation). Instead I put sin(x) in the table instead of x and then used the taylor series for y=sqrt(1-x*x) to compute the cos(x) from that. This taylor series is accurate over the range I needed with only 5 terms (denominators are all powers of two!) and can be implemented in fixed point using plain C and generates code that is faster than any other approach I could think of.
Mathematica has a four-parameter generalized inverse gamma distribution:
http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html
and gives its PDF on that page too. Has anyone implemented the density, distribution, quantile, and sampling-from functions for that in R?
I did make a quick start (the PDF is just the equations on that page translated into R) but if its done already I'll not bother with implementing the CDF and the quantile function.
Does a general function for computing the CDF (by integration of PDF) and the Quantile (by inversion of the CDF) of any distribution given the PDF exist?
[Note this is not the generalized inverse Gaussian]
Note also the 'Properties and Relations' dropdown on the Mathematica page, which seems to imply its not a special case or generalisation of anything (apart from the inverse gamma).
I started a package to implement this:
https://github.com/barryrowlingson/geninvgamma
Its only using simple inversion and integration of the density, so nothing clever. Currently random samples from the distribution are done by generating a U(0,1) and getting the quantile, which isn't very efficient or very accurate it seems..
Anyway, its a start.
According to this vignette (Appendix C2), the inverse gamma distribution is a special case of the generalized hyperbolic distribution which is implemented by the ghyp package.
I'm a new user of BLAS/Lapack, and I'm just wondering is there a routine which does Gaussian elimination or even Gaussian-Jordan elimination? I googled and looked at their documentations, but still couldn't find them.
Thanks a lot for helping me out!
Gaussian elimination is basically the same as LU factorization. The routine xGETRF computes the LU factorization (e.g., DGETRF for real double precision matrices). The U factor corresponds to the matrix after Gaussian elimination. The U factor is stored in the upper triangular part (including the diagonal) of the matrix A on exit.
LU factorization / Gaussian elimination is commonly used to solve linear systems of equations. You can use the xGETRS routine to solve a linear system once you have computed the LU factorization.