# Optimizing the coefficients of basis functions to fit a curve

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Scott Kaiser on 29 Aug 2023
Commented: Matt J on 29 Aug 2023
I have 3 vectors F1, F2, and F3. I want to linearly combine these vectors such that I create a new function W(z)=a*F1+b*F2+c*F3, where a,b,c are just constant coefficients and the 3 vectors are essentially basis functions. These vectors are 1x70 doubles, and are shown plotted below (note that the horizontal axis 'Range' is the 'z'-variable in the equations below). The problem I am having is finding a way to calculate the coefficients a,b,c so that W=a*F1+b*F2+c*F3 most closely matches a function given by: Plotted together, W and r are shown below... What sort of optimization process can I use to build a function W=a*F1+b*F2+c*F3 that fits 'W' to 'r' and returns the coefficients a,b,c used to do so? Please note that F1,F2, and F3 are all 1x70 doubles, and are not explicitly functions of z anymore. They have each individually been evaluated. And so I tried using the curvefitting app with no success because of this. I really just need a systematic way to optimize the coefficients a,b, and c. Is there a built in way to do this?

Matt J on 29 Aug 2023
Edited: Matt J on 29 Aug 2023
You don't need any special curve fitting tool. Due to the linearity of the model, the coefficients can be found using mldivide, \
coefficients = [F1(:),F2(:),F3(:)]\r(:)
Image Analyst on 29 Aug 2023
Edited: Image Analyst on 29 Aug 2023
Good answer Matt. I wish this trick was better known.
Shouldn't W be in the denominator instead of r?
Also note there is a similar version mrdivide, / that uses forward slash. Since I can never remember which to use I always have to look at the documentation and see which one best matches up with my data and coefficients. If you have a good mnemonic to avoid having to check the documentation, I'd like to hear it.
help mldivide
\ Backslash or left matrix divide. A\B is the matrix division of A into B, which is roughly the same as INV(A)*B , except it is computed in a different way. If A is an N-by-N matrix and B is a column vector with N components, or a matrix with several such columns, then X = A\B is the solution to the equation A*X = B. A warning message is printed if A is badly scaled or nearly singular. A\EYE(SIZE(A)) produces the inverse of A. If A is an M-by-N rectangular matrix with M~=N and B is a column vector with M components, or a matrix with several such columns, then X = A\B is the solution in the least squares sense to the under- or overdetermined system of equations A*X = B. The effective rank, K, of A is determined from the QR decomposition with pivoting. A solution X is computed which has at most K nonzero components per column. If K < N this will usually not be the same solution as PINV(A)*B. A\EYE(SIZE(A)) produces a generalized inverse of A. X = MLDIVIDE(A,B) is called for the syntax 'A \ B' when A or B is an object. See MATLAB Operators and Special Characters for more details. See also LDIVIDE, RDIVIDE, MRDIVIDE, PAGEMLDIVIDE. Documentation for mldivide doc mldivide Other uses of mldivide codistributed/mldivide laurpoly/mldivide comm/mldivide StaticModel/mldivide distributed/mldivide sym/mldivide duration/mldivide symbolic/mldivide DynamicSystem/mldivide tall/mldivide gpuArray/mldivide timeseries/mldivide LagOp/mldivide tsdata.datametadata/mldivide
help mrdivide
/ Right matrix divide. B/A is the matrix division of A into B, which is roughly the same as B*INV(A) , except it is computed in a different way. More precisely, B/A = (A'\B')'. See MLDIVIDE for details. C = MRDIVIDE(B,A) is called for the syntax 'B / A' when B or A is an object. See MATLAB Operators and Special Characters for more details. See also MLDIVIDE, RDIVIDE, LDIVIDE, PAGEMRDIVIDE. Documentation for mrdivide doc mrdivide Other uses of mrdivide codistributed/mrdivide laurpoly/mrdivide distributed/mrdivide se2/mrdivide dlarray/mrdivide StaticModel/mrdivide duration/mrdivide sym/mrdivide DynamicSystem/mrdivide symbolic/mrdivide embedded.fi/mrdivide tall/mrdivide gpuArray/mrdivide timeseries/mrdivide LagOp/mrdivide tsdata.datametadata/mrdivide
Matt J on 29 Aug 2023
Shouldn't W be in the denominator instead of r?
W=[F1(:),F2(:),F3(:)] is in the denominator. I jut didn't create a W variable explicitly.

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