trainMQRBF: Fit multiquadric RBF model to training data for d>1.
In SACOBRA: Self-Adjusting COBRA

Description Usage Arguments Details Value Author(s) See Also

The model for a point z=(z_1,...,z_d) is fitted using n sample points x_1, ..., x_n

s(z) = λ_1*Φ(||z-x_1||)+... +λ_n*Φ(||z-x_n||) + c_0 + c_1*z_1 + ... + c_d*z_d

where Φ(r)=√{(1+(r/σ)^2)} denotes the multiquadrics radial basis function with width σ. The coefficients λ_1, ..., λ_n, c_0, c_1, ..., c_d are determined by this training procedure.
This is for the default case squares==FALSE. In case squares==TRUE there are d additional pure square terms and the model is

s_{sq}(z) = s(z) + c_{d+1}*z_1^2 + ... + c_{d+d}*z_d^2

In case ptail==FALSE the polynomial tail (all coefficients c_i) is omitted completely.

trainMQRBF(
  xp,
  U,
  ptail = TRUE,
  squares = FALSE,
  width,
  RULE = "One",
  widthFactor = 1,
  rho = 0,
  DEBUG2 = F
)

`xp`	n points x_i of dimension d are arranged in (n x d) matrix `xp`
`U`	vector of length n, containing samples u(x_i) of the scalar function u to be fitted - or - (n x m) matrix, where each column 1,...,m contains one vector of samples u_j(x_i) for the m'th model, j=1,...,m
`ptail`	[TRUE] flag, see description
`squares`	[FALSE] flag, see 'Description'
`width`	[-1] either a positive real value which is the constant width σ for all Gaussians in all iterations, or -1. If -1, the appropriate width σ is calculated anew in each iteration with one of the rules `RULE`, based on the distribution of data points `xp`.
`RULE`	["One"] one out of ["One" \| "Two" \| "Three"], different rules for automatic estimation of width σ. Only relevant if `width = -1`,
`widthFactor`	[1.0] additional constant factor applied to each width σ
`rho`	[0.0] experimental: 0.0: interpolating, >0.0, approximating (spline-like) Gaussian RBFs
`DEBUG2`	[FALSE] if TRUE, save `M` and `rhs` on return value

The linear equation system is solved via SVD inversion. Near-zero elements in the diagonal matrix D are set to zero in D^{-1}. This makes rank-deficient systems numerically stable.

rbf.model, an object of class RBFinter, which is basically a list with elements:

`coef`	(n+d+1 x m) matrix holding in column m the coefficients for the m'th model: λ_1, ..., λ_n, c_0, c_1, ..., c_d. In case `squares==TRUE` it is an (n+2d+1 x m) matrix holding additionally the coefficients c_{d+1}, ..., c_{d+d}.
`xp`	matrix xp
`d`	size of the polynomial tail. If `length(d)==0` it means no polynomial tail will be used for the model. In case of ptail==T && squares==F d will be dimension+1 and in case of ptail==T && squares==T d will be 2*dimension+1
`npts`	number n of points x_i
`ptail`	TRUE or FALSE (see description)
`squares`	TRUE or FALSE (see description)
`width`	the calculated width σ
`type`	"MQ"