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## 36.4 Multi-parameter fitting

The functions described in this section perform least-squares fits to a
general linear model, *y = X c* where *y* is a vector of
*n* observations, *X* is an *n* by *p* matrix of
predictor variables, and the elements of the vector *c* are the *p* unknown best-fit parameters which are to be estimated. The chi-squared value is given by *\chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2*.

This formulation can be used for fits to any number of functions and/or
variables by preparing the *n*-by-*p* matrix *X*
appropriately. For example, to fit to a *p*-th order polynomial in
`x`, use the following matrix,
where the index *i* runs over the observations and the index
*j* runs from 0 to *p-1*.

To fit to a set of *p* sinusoidal functions with fixed frequencies
*\omega_1*, *\omega_2*, …, *\omega_p*, use,
To fit to *p* independent variables *x_1*, *x_2*, …,
*x_p*, use,
where *x_j(i)* is the *i*-th value of the predictor variable
*x_j*.

The functions described in this section are declared in the header file
‘`gsl_multifit.h`’.

The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix *X*.

__Function:__gsl_multifit_linear_workspace ***gsl_multifit_linear_alloc***(size_t*`n`, size_t`p`)This function allocates a workspace for fitting a model to

`n`observations using`p`parameters.

__Function:__void**gsl_multifit_linear_free***(gsl_multifit_linear_workspace **`work`)This function frees the memory associated with the workspace

`w`.

__Function:__int**gsl_multifit_linear***(const gsl_matrix **`X`, const gsl_vector *`y`, gsl_vector *`c`, gsl_matrix *`cov`, double *`chisq`, gsl_multifit_linear_workspace *`work`)__Function:__int**gsl_multifit_linear_svd***(const gsl_matrix **`X`, const gsl_vector *`y`, double`tol`, size_t *`rank`, gsl_vector *`c`, gsl_matrix *`cov`, double *`chisq`, gsl_multifit_linear_workspace *`work`)These functions compute the best-fit parameters

`c`of the model*y = X c*for the observations`y`and the matrix of predictor variables`X`. The variance-covariance matrix of the model parameters`cov`is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit,*\chi^2*, is returned in`chisq`. If the coefficient of determination is desired, it can be computed from the expression*R^2 = 1 - \chi^2 / TSS*, where the total sum of squares (TSS) of the observations`y`may be computed from`gsl_stats_tss`

.The best-fit is found by singular value decomposition of the matrix

`X`using the preallocated workspace provided in`work`. The modified Golub-Reinsch SVD algorithm is used, with column scaling to improve the accuracy of the singular values. Any components which have zero singular value (to machine precision) are discarded from the fit. In the second form of the function the components are discarded if the ratio of singular values*s_i/s_0*falls below the user-specified tolerance`tol`, and the effective rank is returned in`rank`.

__Function:__int**gsl_multifit_wlinear***(const gsl_matrix **`X`, const gsl_vector *`w`, const gsl_vector *`y`, gsl_vector *`c`, gsl_matrix *`cov`, double *`chisq`, gsl_multifit_linear_workspace *`work`)__Function:__int**gsl_multifit_wlinear_svd***(const gsl_matrix **`X`, const gsl_vector *`w`, const gsl_vector *`y`, double`tol`, size_t *`rank`, gsl_vector *`c`, gsl_matrix *`cov`, double *`chisq`, gsl_multifit_linear_workspace *`work`)This function computes the best-fit parameters

`c`of the weighted model*y = X c*for the observations`y`with weights`w`and the matrix of predictor variables`X`. The covariance matrix of the model parameters`cov`is computed with the given weights. The weighted sum of squares of the residuals from the best-fit,*\chi^2*, is returned in`chisq`. If the coefficient of determination is desired, it can be computed from the expression*R^2 = 1 - \chi^2 / WTSS*, where the weighted total sum of squares (WTSS) of the observations`y`may be computed from`gsl_stats_wtss`

.The best-fit is found by singular value decomposition of the matrix

`X`using the preallocated workspace provided in`work`. Any components which have zero singular value (to machine precision) are discarded from the fit. In the second form of the function the components are discarded if the ratio of singular values*s_i/s_0*falls below the user-specified tolerance`tol`, and the effective rank is returned in`rank`.

__Function:__int**gsl_multifit_linear_est***(const gsl_vector **`x`, const gsl_vector *`c`, const gsl_matrix *`cov`, double *`y`, double *`y_err`)This function uses the best-fit multilinear regression coefficients

`c`and their covariance matrix`cov`to compute the fitted function value`y`and its standard deviation`y_err`for the model*y = x.c*at the point`x`.

__Function:__int**gsl_multifit_linear_residuals***(const gsl_matrix **`X`, const gsl_vector *`y`, const gsl_vector *`c`, gsl_vector *`r`)This function computes the vector of residuals

*r = y - X c*for the observations`y`, coefficients`c`and matrix of predictor variables`X`.

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