%# fields  is a package for analysis of spatial data written for
%# the R software environment .
%# Copyright (C) 2018
%# University Corporation for Atmospheric Research (UCAR)
%# Contact: Douglas Nychka, nychka@mines.edu,
%# National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307-3000
%#
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%# This program is distributed in the hope that it will be useful,
%# but WITHOUT ANY WARRANTY; without even the implied warranty of
%# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%# GNU General Public License for more details.
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\name{fields-stuff}
\alias{fields.diagonalize2}
\alias{fields.diagonalize}
\alias{fields.duplicated.matrix}
\alias{fields.mkpoly}
\alias{fields.derivative.poly}
\alias{fields.evlpoly}
\alias{fields.evlpoly2}

\title{Fields supporting functions}
\description{
Some supporting functions that are internal to fields top level
methods. Variants of these might be found in the R base but these
have been written for cleaner code or efficiency.
}
\usage{
fields.diagonalize2(A,B, verbose=FALSE)
fields.diagonalize(A,B)
fields.duplicated.matrix(mat, digits = 8) 

fields.mkpoly(x, m = 2, tag = "term")

fields.derivative.poly(x, m,dcoef)

fields.evlpoly( x, coef)

fields.evlpoly2( x, coef, ptab)

}
%- maybe also 'usage' for other objects documented here.
\arguments{
\item{A}{
A positive definite matrix}
\item{B}{
A positive definite matrix}
\item{mat}{
Arbitrary matrix for examining rows}
\item{digits}{Number of significant digits to use for comparing
elements to determine duplciate values. }

\item{x}{
Arbitrary matrix where rows are components of a multidimensional
vector}
\item{m}{ The null space degree  -- results in a polynomial of degree
(m-1) }
\item{dcoef}{ Coefficients of a multidimensional polynomial}

\item{coef}{Polynomial coefficients.}
\item{ptab}{Table of powers of different polnomial terms.}
\item{tag}{\code{mkpoly} fills in as columns names the higher order terms of the polynomial terms. The tag is the  text string prefix for thes column names and the powers of the individual variables arer appended. Default is just "terms".
}
\item{verbose}{If TRUE prints condition number of A+B}
}

\details{
Given two matrices A (positive definite) and B (nonnegative definite)
 \code{fields.diagonalize} and \code{fields.diagonalize2} finds the 
matrix transformation G that will convert A to a identity matrix and B 
to a diagonal matrix:

 G\^T A G= I       G\^T B G= D. 


\code{fields.diagonalize2} is not as easy to follow as \code{fields.diagonalize} but may be more stable
 and is the version used in 
the Krig engine. 

\code{fields.duplicated} finds duplicate rows in a matrix.  The digits
arguments is the number of digits that are considered in the
comparison. 
The returned value is an array of integers from 1:M where M is the
number of unique rows and duplicate rows are referenced in the same
order that they appear as the rows of \code{mat}. 

\code{fields.mkpoly}  computes the complete matrix of all monomial
terms up to degree (m-1). Each row of \code{x} is are the componets of
a vector.  (The fields function mkpoly returns the number of these
terms.) In 2 dimensions with m=3 there   6 polynomial terms  up to
quadratic ( 3-1 =2) order and will be returned as the matrix:

cbind(  1 , x[,1], x[,2], x[,1]**2, x[,1]*x[,2], x[,2]**2  )

This function is used for the fixed effects polynomial  or spatial
drift used in spatial estimating functions Krig, Tps and mKrig. 
The matrix ptab is a table of the powers in each term for each
variable and is included as an attribute to the matrix returned by
this function. 
See the \code{attr} function for extracting an attribute from an
object. 

\code{ptab} for the example above is 
\preformatted{
    [,1] [,2]
[1,]    0    0
[2,]    1    0
[3,]    0    1
[4,]    2    0
[5,]    1    1
[6,]    0    2
}

This information is used in finding derivatives of the polynomial is also used
to create column names  for the terms that are of higher order than linear. 

\code{fields.deriviative.poly} finds the partial derivative matrix of
a multidimensional polynomial of degree (m-1) at different vector
values and with coefficients \code{dcoef}. 
This function has been orgainzed to be a clean utility for the
predicting the derivative of the estimated function from Krig or
mKrig. 
Within the fields context
the polynomial itself would be evaluated as 
     fields.mkpoly( x,m)\%*\%dcoef.
If x has  d columns ( also the dimension of the polynomial) and n rows
the partial derivatives of this polynomial at the locations x can be
organized in a nXd matrix.  This is the object returned by ths
function.   

\code{evlpoly} and \code{evlpoly2} are FORTRAN based functions for 
evaluating univariate polynomials and multivariate polynomials.  The
table of powers (ptab) needed for evlpoly2 is the same format as that
returned my the fields.mkpoly function. 


}
\author{Doug Nychka}
\seealso{Krig, Tps, as.image, predict.Krig, predict.mKrig, 
Krig.engine.default, Wendland}
\keyword{spatial}
% at least one, from doc/KEYWORDS