Matlab nonlinear least squares

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Nonlinear Optimization. Solve constrained or unconstrained nonlinear problems with one or more objectives, in serial or parallel. To set up a nonlinear optimization problem for solution, first decide between a problem-based approach and solver-based approach. See First Choose Problem-Based or Solver-Based Approach.We review Square POS, including features such as integrations, multiple ways to pay, inventory management and more. By clicking "TRY IT", I agree to receive newsletters and promoti...'trust-region-dogleg' is the only algorithm that is specially designed to solve nonlinear equations. The others attempt to minimize the sum of squares of the function. The 'trust-region' algorithm is effective on sparse problems. It can use special techniques such as a Jacobian multiply function for large-scale problems.$\begingroup$ I see from your comments on the answers that you're actually doing nonlinear least squares. You'd have had good answers more quickly if you'd started with that information. I have at least added a relevant tag. $\endgroup$ -The model equation for this problem is. y ( t) = A 1 exp ( r 1 t) + A 2 exp ( r 2 t), where A 1, A 2, r 1, and r 2 are the unknown parameters, y is the response, and t is time. The problem requires data for times tdata and (noisy) response measurements ydata. The goal is to find the best A and r, meaning those values that minimize.Wondering what it will cost to side your home? Click here to see a complete cost guide by siding type, home size and more, plus tips on choosing the right material. Expert Advice O...This problem comes actually from interrupted work two years ago. At that time, Prof. John C Nash, one of the authors of the package optimx has suggested that there has been quite a lot of work for Matlab for Automatic Differentiation, which might be the reason that the Matlab lsqnonlin performs better than the optimization functions/algorithms ...

Subtract the fit of the Theil regression off. Use LOESS to fit a smooth curve. Find the peak to get a rough estimate of A, and the x-value corresponding to the peak to get a rough estimate of B. Take the LOESS fits whose y-values are > 60% of the estimate of A as observations and fit a quadratic. This example shows how to perform nonlinear fitting of complex-valued data. While most Optimization Toolbox™ solvers and algorithms operate only on real-valued data, least-squares solvers and fsolve can work on both real-valued and complex-valued data for unconstrained problems. The objective function must be analytic in the complex function ... Improve Model Fit with Weights. This example shows how to fit a polynomial model to data using both the linear least-squares method and the weighted least-squares method for comparison. Generate sample data from different normal distributions by using the randn function. for k=1:20. r = k*randn([20,1]) + (1/20)*(k^3); rnorm = [rnorm;r];Constrained Optimization Definition. Constrained minimization is the problem of finding a vector x that is a local minimum to a scalar function f ( x ) subject to constraints on the allowable x: min x f ( x) such that one or more of the following holds: c(x) ≤ 0, ceq(x) = 0, A·x ≤ b, Aeq·x = beq, l ≤ x ≤ u. There are even more ...The natural logarithm function in MATLAB is log(). To calculate the natural logarithm of a scalar, vector or array, A, enter log(A). Log(A) calculates the natural logarithm of each...Points that are farther from the line than would be expected by random chance get zero weight. For most cases, the bisquare weight method is preferred over LAR because it simultaneously seeks to find a curve that fits the bulk of the data using the usual least-squares approach, and it minimizes the effect of outliers.This MATLAB function returns the 95% confidence intervals ci for the nonlinear least-squares parameter estimates beta. Skip to content. Toggle Main Navigation. ... returns the 95% confidence intervals ci for the nonlinear least-squares parameter estimates beta. Before calling nlparci, get the estimated coefficients beta, residuals r ...

Fit experimental data with linear piecewise continuos function with given x-axis break points. Generates 1-D look-up table (LUT) optimal (least-square sense with continuity constraint) y-axis points from experimental (x,y) data given a vector of x-axis break points. Note that x-axis break points should be chosen such that every bin has enough ...The objective function for this problem is the sum of squares of the differences between the ODE solution with parameters r and the solution with the true parameters yvals. To express this objective function, first write a MATLAB function that computes the ODE solution using parameters r. This function is the RtoODE function.Answers (1) If you have the Statistics Toolbox, you should be able to do this with the nlinfit () function. Sign in to comment. Sign in to answer this question. Non linear least squares regression. Learn more about non-linear least squares regression, alkalinity.Note that this function is not a linear model, but a nonlinear least squares (nls) fit, so not an lm fit. r; coefficient-of-determination; Share. Improve this question. Follow edited Aug 6, 2023 at 16:10. M--27.5k 8 8 gold badges 68 68 silver badges 100 100 bronze badges.The unstable camera path is one which gives the jittering or shake to the video. I have camera path specified using camera position which is a 3d-data. camera path - (cx,cy,cz); As i plot in matlab, i can visually see the shakiness of the camera motion. So now i require a least squares fitting to be done on the camera path specified by (cx,cy,cz);

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The Matlab back-slash operator computes a least squares solution to such a system. beta = X\y The basis functions might also involve some nonlinear parameters, α1,...,αp. The problem is separable if it involves both linear and nonlinear parameters: y(t) ≈ β1ϕ1(t,α)+ ··· +βnϕn(t,α). The elements of the design matrix depend upon both ...Mar 29, 2015 ... Wen Shen, Penn State University. Lectures are based on my book: "An Introduction to Numerical Computation", published by World Scientific, ...How to use Matlab for non linear least squares Michaelis-Menten parameters estimation. 1. Fitting data in least square sense to nonlinear equation. 0. Solving a system of nonlinear equations. 0. solve multidimensional equation using least square method in matlab. 0.x = lsqr(A,b) attempts to solve the system of linear equations A*x = b for x using the Least Squares Method . lsqr finds a least squares solution for x that minimizes norm(b-A*x). When A is consistent, the least squares solution is also a solution of the linear system. When the attempt is successful, lsqr displays a message to confirm convergence.Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. x = 5×1. -0.1899 -0.8174 7.8199 0.0026 -0.0388. resnorm = 0.1143.Feb 11, 2009 · The function LMFsolve.m serves for finding optimal solution of an overdetermined system of nonlinear equations in the least-squares sense. The standard Levenberg- Marquardt algorithm was modified by Fletcher and coded in FORTRAN many years ago.

Value Description Supported Fits "auto" Default value for all interpolant fit types. Set ExtrapolationMethod to "auto" to automatically assign an extrapolation method when you use the fit function.. All interpolant fit types and cubicspline curve fits "none" No extrapolation. When you use fitOptions with the fit function to evaluate query points outside of the convex hull, fit returns NaN.It can be applied to solve a nonlinear least square optimization problem. This function provides a way using the unscented Kalman filter to solve nonlinear least square optimization problems. Three examples are included: a general optimization problem, a problem to solve a set of nonlinear equations represented by a neural …The objective function for this problem is the sum of squares of the differences between the ODE solution with parameters r and the solution with the true parameters yvals. To express this objective function, first write a MATLAB function that computes the ODE solution using parameters r. This function is the RtoODE function.This code allows users to define new variable nodes and new factors/edges/cost functions. The framework is reorganized with necessary warnings for the extension of the new node and new edge. When the new node is defined, the information needs to be given in the “GetNodeTypeDimension”, “SetNodeDefaultValue” and “update_state”.Summary Assuming you have a weight matrix W (which can be a sparse diagonal matrix), and the nonlinear fitting function F, then the fitting function using lambda-expression and premultiply the measurement data ydata with the Cholesky factor R as given below.. R = chol( W, 'upper'); F_w = @(x, xdata) R * F( x, xdata); ydata_w = R * ydata; x_w = lsqcurvefit(F_w, x0, xdata, ydata_w)I'm wondering if anyone has thought about using lsqnonlin to solve non-linear least squares problems with relative constraints on parameter estimates. Whereas it's straightforward to limit parameter estimates in an absolute sense by specifying lower and/or upper bounds, I'm wondering if it's possible to specify parameter values relative to one another.Nonlinear least square regression. Learn more about regression i have (x , y) data the function between x and y is y = 0.392* (1 - (x / b1) .^ b2 i want to use nonlinear least square regression to obtain the values of b1 and b2 can any one help me wit...CONTENTS: A MATLAB implementation of CGLS, the Conjugate Gradient method for unsymmetric linear equations and least squares problems: Solve or minimize or solve Ax = b ∥Ax − b∥2 (ATA + sI)x = ATb, Solve A x = b or minimize ‖ A x − b ‖ 2 or solve ( A T A + s I) x = A T b, where the matrix A A may be square or rectangular (represented ...Nonlinear Least Squares So far we have looked at nding a \best t" solution to alinear system (linear least-squares) A more di cult situation is when we consider least-squares for nonlinearsystems Key point: We are referring to linearity in theparameters, not linearity of themodel (e.g. polynomial p n(x;b) = b 0 + b 1x + :::+ b nxn is nonlinear ...Regular nonlinear least squares algorithms are appropriate when measurement errors all have the same variance. When that assumption is not true, it is appropriate to used a weighted fit. ... You clicked a link that …• Nonlinear least squares problem • Linear least squares problem • Gradient descent • Cholesky solver • QR solver • Gauss-Newton Method A quick detour Next • Nonlinear …Looking for things to do in Times Square at night? Click this to discover the most fun activities and places to go at night in Times Square! AND GET FR Times Square is a world-famo...

This example shows how to fit a nonlinear function to data using several Optimization Toolbox™ algorithms. Problem Setup. Consider the following data: Data = ... [0.0000 …

The model equation for this problem is. y ( t) = A 1 exp ( r 1 t) + A 2 exp ( r 2 t), where A 1, A 2, r 1, and r 2 are the unknown parameters, y is the response, and t is time. The problem requires data for times tdata and (noisy) response measurements ydata. The goal is to find the best A and r, meaning those values that minimize.Least Squares. Solve least-squares (curve-fitting) problems. Least squares problems have two types. Linear least-squares solves min|| C * x - d || 2, possibly with bounds or linear constraints. See Linear Least Squares. Nonlinear least-squares solves min (∑|| F ( xi ) - yi || 2 ), where F ( xi ) is a nonlinear function and yi is data.The unstable camera path is one which gives the jittering or shake to the video. I have camera path specified using camera position which is a 3d-data. camera path - (cx,cy,cz); As i plot in matlab, i can visually see the shakiness of the camera motion. So now i require a least squares fitting to be done on the camera path specified by (cx,cy,cz);Obtain Residuals from Nonnegative Least Squares ... Call lsqnonneg with outputs to obtain the solution, residual norm, and residual vector. Prepare a C matrix and ...Sep 16, 2013 · If mu, Sigma, kappa, and y0 are your decision variables, then this is a nonlinear constraint, and the only solver that addresses problems with nonlinear constraints is fmincon. You would include the constraint as follows (I assume that the vector x is [mu, Sigma, kappa, y0]): Theme. Copy. function [c,ceq] = confun (x) Feasible Generalized Least Squares. Panel Corrected Standard Errors. Ordinary Least Squares. When you fit multivariate linear regression models using mvregress, you can use the optional name-value pair 'algorithm','cwls' to choose least squares estimation. In this case, by default, mvregress returns ordinary least squares (OLS) estimates using ...The function The LMFnlsq.m serves for finding optimal solution of an overdetermined system of nonlinear equations in the least-squares sense. The standard Levenberg- Marquardt algorithm was modified by Fletcher and coded in FORTRAN many years ago (see the Reference). This version of LMFnlsq is its complete MATLAB implementation complemented by ...A nonlinear function in math creates a graph that is not a straight line, according to Columbia University. Three nonlinear functions commonly used in business applications include...For non-linear least squares, an approximation can be constructed by using the linearization F ( x + Δ x) ≈ F ( x) + J ( x) Δ x , which leads to the following linear least squares problem: (2) min Δ x 1 2 ‖ J ( x) Δ x + F ( x) ‖ 2. Unfortunately, naively solving a sequence of these problems and updating x ← x + Δ x leads to an ...

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Only the linear and polynomial fits are true linear least squares fits. The nonlinear fits (power, exponential, and logarithmic) are approximated through transforming the model to a linear form and then applying a least squares fit. Taking the logarithm of a negative number produces a complex number. When linearizing, for simplicity, this ...Iteratively Reweighted Least Squares. In weighted least squares, the fitting process includes the weight as an additional scale factor, which improves the fit. The weights determine how much each response value influences the final parameter estimates. A low-quality data point (for example, an outlier) should have less influence on the fit.Keyword arguments passed to leastsq for method='lm' or least_squares otherwise. If you have an unbound problem, by default method='lm' is used which uses leastsq which does not accept f_scale as a keyword. Therefore, we can use method='trf' which then uses least_squares which accepts f_scale.Nonlinear Least-Squares Fitting. This chapter describes functions for multidimensional nonlinear least-squares fitting. There are generally two classes of algorithm s for solving nonlinear least squares problems, which fall under line search methods and trust region methods. GSL currently implements only trust region methods and provides the ...Indices Commodities Currencies StocksIntroduction. Ceres can solve bounds constrained robustified non-linear least squares problems of the form. (1) min x 1 2 ∑ i ρ i ( ‖ f. i. ( x i 1,..., x i k) ‖ 2) s.t. l j ≤ x j ≤ u j. Problems of this form comes up in a broad range of areas across science and engineering - from fitting curves in statistics, to constructing 3D ...After years of hype, big investments, and a skyrocketing valuation, the mobile payments startup Square is coming to terms with the fact that even though its core business is wildly...I'm wondering if anyone has thought about using lsqnonlin to solve non-linear least squares problems with relative constraints on parameter estimates. Whereas it's straightforward to limit parameter estimates in an absolute sense by specifying lower and/or upper bounds, I'm wondering if it's possible to specify parameter values relative to one another.Before calling nlparci, get the estimated coefficients beta, residuals r, and Jacobian J by using the nlinfit function to fit a nonlinear regression model. example ci = nlparci( ___ ,"Alpha", alpha ) returns the 100(1 — alpha) % confidence intervals, using any of the input argument combinations in the previous syntaxes. ….

Introduction. In this Chapter, you will learn to fit non-linear mathematical models to data using Non-Linear Least Squares (NLLS). Specifically, you will learn to. Visualize the data and the mathematical model you want to fit to them. Fit a non-linear model. Assess the quality of the fit, and whether the model is appropriate for your data.To illustrate the differences between ML and GLS fitting, generate some example data. Assume that x i is one dimensional and suppose the true function f in the nonlinear logistic regression model is the Michaelis-Menten model parameterized by a 2 × 1 vector β: f ( x i, β) = β 1 x i β 2 + x i. myf = @(beta,x) beta(1)*x./(beta(2) + x);After you take the log, it's linear in all the coefficients so I don't see why any non-linear stuff is needed. Here's a snippet from a demo of mine that may help you: Theme. Copy. % Do a least squares fit of the histogram to a Gaussian. % Assume y = A*exp (- (x-mu)^2/sigma^2) % Take log of both sides.I am using non-linear least squares to estimate the parameters using Matlab through the function lsqnolin. The code is as below and I would like to know if the way I am estimating the initial condition is correct. The actual model is more complex and the data is different but I want to clarify of a way to estimate ODE initial conditions.Open in MATLAB Online. 1. Using "nlinfit". The "nlinfit" function expects a response vector "Y" and a function of unknown parameters. Simply encapsulate the implicit model in a function of the form: Theme. Copy. 0 = y - model (x,y,beta) The response vector to be passed to "nlinfit" becomes.For more information, see Large Scale Nonlinear Least Squares. PrecondBandWidth: Upper bandwidth of preconditioner for PCG, a nonnegative integer. ... You must have a MATLAB Coder license to generate code. The target hardware must support standard double-precision floating-point computations. You cannot generate code for single-precision or ...This fit gives greater weights to small values so, in order to weight the points equally, it is often better to minimize the function. Applying least squares fitting gives. Solving for and , In the plot above, the short-dashed curve is the fit computed from ( ) and ( ) and the long-dashed curve is the fit computed from ( 9 ) and ( 10 ).Iteratively Reweighted Least Squares. In weighted least squares, the fitting process includes the weight as an additional scale factor, which improves the fit. The weights determine how much each response value influences the final parameter estimates. A low-quality data point (for example, an outlier) should have less influence on the fit.The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are contaminated by some noise. For ill-posed TLS poblems, regularization techniques are necessary to stabilize the computed solution; otherwise, TLS produces a noise-dominant ... Matlab nonlinear least squares, Nonlinear least-squares solves min (∑|| F ( xi ) - yi || 2 ), where F ( xi ) is a nonlinear function and yi is data. The problem can have bounds, linear constraints, or nonlinear constraints. For the problem-based approach, create problem variables, and then represent the objective function and constraints in terms of these symbolic variables., If mu, Sigma, kappa, and y0 are your decision variables, then this is a nonlinear constraint, and the only solver that addresses problems with nonlinear constraints is fmincon. You would include the constraint as follows (I assume that the vector x is [mu, Sigma, kappa, y0]): Theme. Copy. function [c,ceq] = confun (x), Description. Solve nonnegative least-squares curve fitting problems of the form. min x ‖ C ⋅ x − d ‖ 2 2, where x ≥ 0. Note. lsqnonneg applies only to the solver-based approach. For a discussion of the two optimization approaches, see First Choose Problem-Based or Solver-Based Approach. example. x = lsqnonneg(C,d) returns the vector x ..., For non-linear least squares, an approximation can be constructed by using the linearization F ( x + Δ x) ≈ F ( x) + J ( x) Δ x , which leads to the following linear least squares problem: (2) min Δ x 1 2 ‖ J ( x) Δ x + F ( x) ‖ 2. Unfortunately, naively solving a sequence of these problems and updating x ← x + Δ x leads to an ..., lsqcurvefit enables you to fit parameterized nonlinear functions to data easily. You can also use lsqnonlin; lsqcurvefit is simply a convenient way to call lsqnonlin for curve fitting. In this example, the vector xdata represents 100 data points, and the vector ydata represents the associated measurements. Generate the data for the problem. , Generate Example Data. To illustrate the differences between ML and GLS fitting, generate some example data. Assume that x i is one dimensional and suppose the true function f in the nonlinear logistic regression model is the Michaelis-Menten model parameterized by a 2 × 1 vector β: f ( x i, β) = β 1 x i β 2 + x i., This example shows how to solve a nonlinear least-squares problem in two ways. The example first solves the problem without using a Jacobian function. Then it shows how to include a Jacobian, and illustrates the resulting improved efficiency. The problem has 10 terms with two unknowns: find x, a two-dimensional vector, that minimizes., 6 Least Squares Adjustment and find the partial derivatives of ϵ with respect to the intercept θ0 and the slope θ1 ∂ϵ ∂θ0 ∑ n i=1 (yi −(θ0 +θ1xi))(−1) = −∑n i=1 yi +nθ0 +θ1 ∑ i=1 xi (23) ∂ϵ ∂θ1 ∑n i=1 (yi −(θ0 +θ1xi))(−xi) = −∑ n i=1 xiyi +θ0 ∑n i=1 xi +θ1 ∑ i=1 x2 i. (24) Setting the partial derivatives equal to zero and denoting the solutions ..., 1 Answer. Sorted by: 0. Your least squares criteria, which is what you want to minimize, are different: in the first case, you have. ∑i=1n ( Ei−−√ − 3 4R∞− −−−−√ Zi + 3 …, A Levenberg-Marquardt least-squares algorithm was used in this procedure. I have used curve fitting option in Igor Pro software. I defined new fit function and tried to define independent and dependent variable. Nevertheless, I don't know what is the reason that I got the this error: "The fitting function returned INF for at least one X variable", Example of code generation for nonlinear least squares. Solve Generating Code for lsqnonlin Solver Approach. The goal is to find parameters for the model a ^ i, i = 1, 2, 3 that best fit the data.. To fit the parameters to the data using lsqnonlin, you need to define a fitting function.For lsqnonlin, the fitting function takes a parameter vector a, the data …, Linear least-squares solves min||C*x - d|| 2, possibly with bounds or linear constraints. For the problem-based approach, create problem variables, and then represent the objective function and constraints in terms of these symbolic variables., 1e-10<g<3e-10, g=2.5e-10. However, I both tried matlab and rigin to fit data with the model, but they all failed to find a good fit. I am appreciate if you can provide any suggestions. In fact, I understand there are too many parameters, and I also tried to fix parameter b, d, e and g while free others, but still no good results., , We review Square POS, including features such as integrations, multiple ways to pay, inventory management and more. By clicking "TRY IT", I agree to receive newsletters and promoti..., x = lsqr(A,b) attempts to solve the system of linear equations A*x = b for x using the Least Squares Method . lsqr finds a least squares solution for x that minimizes norm(b-A*x). When A is consistent, the least squares solution is also a solution of the linear system. When the attempt is successful, lsqr displays a message to confirm convergence. , As shown below, we had MATLAB generate a 3D plot to roughly show the positioning of the satellites. In order to determine the sensitivity of the measured coordinates to errors in the timing of the input data, the algorithm applied tiny changes (\(\pm d\) for some small \(d\), such as 10 nanoseconds) to the observed time delays and measured the ..., Keyword arguments passed to leastsq for method='lm' or least_squares otherwise. If you have an unbound problem, by default method='lm' is used which uses leastsq which does not accept f_scale as a keyword. Therefore, we can use method='trf' which then uses least_squares which accepts f_scale., The following file illustrates how to solve an NLLS problem in TOMLAB. Also view the m-files specified above for more information. File: tomlab/quickguide/nllsQG.m. Open the file for viewing, and execute nllsQG in Matlab. % nllsQG is a small example problem for defining and solving. % nonlinear least squares using the TOMLAB format., A perfect square is a number, but it can also be explained using an actual square. Advertisement You know what a square is: It's a shape with four equal sides. Seems hard to improv..., The simplified code used is reported below. The problem is divided in four functions: parameterEstimation - (a wrapper for the lsqnonlin function) objectiveFunction_lsq - (the objective function for the param estimation) yFun - (the function returing the value of the variable y) objectiveFunction_zero - (the objective function of the non-linear ..., Nonlinear Optimization. Solve constrained or unconstrained nonlinear problems with one or more objectives, in serial or parallel. To set up a nonlinear optimization problem for solution, first decide between a problem-based approach and solver-based approach. See First Choose Problem-Based or Solver-Based Approach., In MATLAB, you can find B using the mldivide operator as B = X\Y. From the dataset accidents, load accident data in y and state population data in x. Find the linear regression relation y = β 1 x between the accidents in a state and the population of a state using the \ operator. The \ operator performs a least-squares regression., The objective function of this fully linear least square problem is non-linear. I agree with your comment that lsqlin() is a possible solution, but fmincon is solving the same problem in a more compact and intuitive way to solve a non-linear objective function., The function is an explicit sum of squares. Therefore, the example also shows the efficiency of using a least-squares solver. For the least-squares solver lsqnonlin, the example uses the hlsqnonlin0obj helper function shown at the end of this example as a vector objective function that is equivalent to the hfminunc0obj function., nonlinear least-squares Gauss-Newton method 1. Nonlinear least-squares nonlinear least-squares (NLLS) problem: find that minimizes where is a vector of ‘residuals, This example shows how to perform nonlinear fitting of complex-valued data. While most Optimization Toolbox™ solvers and algorithms operate only on real-valued data, least-squares solvers and fsolve can work on both real-valued and complex-valued data for unconstrained problems. The objective function must be analytic in the complex function ... , The function LMFsolve.m serves for finding optimal solution of an overdetermined system of nonlinear equations in the least-squares sense. The standard Levenberg- Marquardt algorithm was modified by Fletcher and coded in …, The Least-Abs curve is much less affected by outliers than the Least Squares curve. It will also have the property that about 50% of the points will fall above the curve and 50% below. Alternatively, you can specify the percentage of points you want to fall below the curve here: (as a number greater than 0 and less than 100)., After you take the log, it's linear in all the coefficients so I don't see why any non-linear stuff is needed. Here's a snippet from a demo of mine that may help you: Theme. Copy. % Do a least squares fit of the histogram to a Gaussian. % Assume y = A*exp (- (x-mu)^2/sigma^2) % Take log of both sides., Description. beta = nlinfit(X,Y,modelfun,beta0) returns a vector of estimated coefficients for the nonlinear regression of the responses in Y on the predictors in X using the model specified by modelfun. The coefficients are estimated using iterative least squares estimation, with initial values specified by beta0., Next, I wanted to do the same thing but with non-linear least squares. However, the fit always looks wrong, here is the code for that attempt: However, the fit always looks wrong, here is the code for that attempt:, Least squares problems have two types. Linear least-squares solves min|| C * x - d || 2, possibly with bounds or linear constraints. See Linear Least Squares. Nonlinear least-squares solves min (∑|| F ( xi ) – yi || 2 ), where F ( xi ) is a nonlinear function and yi is data. See Nonlinear Least Squares (Curve Fitting).