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quartic_surfaces.f90
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quartic_surfaces.f90
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!
! Copyright (C) 2015 Andrea Dal Corso
! This file is distributed under the terms of the
! GNU General Public License. See the file `License'
! in the root directory of the present distribution,
! or http://www.gnu.org/copyleft/gpl.txt .
!
MODULE quartic_surfaces
!
! This module contains the support routines for dealing with quartic
! surfaces interpolation with 1 up to 6 variables.
!
! It provides the following routines:
!
! fit_multi_quartic receives as input the coordinates of some
! points and the value of a function in these points and
! finds the coefficients of a quartic interpolating polynomial.
!
! evaluate_fit_quartic, given the coordinates of a point, and the
! coefficients of the quartic polynomial, evaluates the quartic polynomial
! at that point.
!
! evaluate_quartic_grad, given the coordinates of a point, and the
! coefficients of the quartic polynomial, evaluates the gradient of the
! function at that point.
!
! evaluate_quartic_hessian, given the coordinates of a point, and the
! coefficients of the quartic polynomial, evaluates the hessian of the
! function at that point.
!
! quartic_ncoeff, given the number of variables gives the number of
! coefficients of the quartic polynomial
!
! find_quartic_extremum, find the extremum closest to the input point
! of the quartic polynomium
!
! print_quartic_polynomium, write on output the coefficients of the
! polynomium
!
! introduce_quartic_fit, gives a few information on output on the
! number of data used for the fit and the number of
! coefficients to be found
!
! print_chisq_quartic, writes on output the chi square of a given
! quartic polynomial interpolation
!
! evaluate_two_quartic, adds the coefficients of two quartic before
! evaluating them at a input point x
!
! find_two_quartic_extremum, find the extremum of two quartic starting
! the search from the input x point
!
! print_chisq_two_quartic, writes on output the chi square of the sum
! of two quartic polynomial interpolation
!
! set_quartic_linear_coefficients, adds the coefficients of a
! quartic and a linear polynomial
!
! find_quartic_linear_extremum, finds the extremum of the sum of
! a quartic and a linear polynomial starting from the
! point x given as input
!
! set_quartic_quadratic_coefficients, adds the coefficients of a
! quartic and a quadratic polynomial
!
! evaluate_quartic_quadratic, evaluate the sum of a quartic and a
! quadratic polynomial in a input x point
!
! find_quartic_quadratic_extremum, finds the extremum of the sum of
! a quartic and a quadratic polynomial starting from the
! point x given as input
!
! print_chisq_quartic_quadratic, writes on output the chi square of the sum
! of a quartic and quadratic polynomial interpolation
!
! set_quartic_cubic_coefficients, adds the coefficients of a
! quartic and a cubic polynomial
!
! evaluate_quartic_cubic, evaluate the sum of a quartic and a
! cubic polynomial in an input x point
!
! find_quartic_cubic_extremum, finds the extremum of the sum of
! a quartic and a cubic polynomial starting from the
! point x given as input
!
! print_chisq_quartic_cubic, writes on output the chi square of the sum
! of a quartic and cubic polynomial interpolation
!
!
! The following number of coefficients are necessary, depending on the
! dimension of the quadratic function
!
! dimension number of coefficients
! 1 5
! 2 15
! 3 35
! 4 70
! 5 126
! 6 210
! To interpolate the function it is better to give to
! multi_quartic a number of points equal or larger than
! the number of coefficients. The routines makes a least square fit
! of the data.
!
USE kinds, ONLY : DP
USE io_global, ONLY : stdout
IMPLICIT NONE
PRIVATE
SAVE
PUBLIC :: fit_multi_quartic, evaluate_fit_quartic, &
evaluate_quartic_grad, evaluate_quartic_hessian, &
quartic_ncoeff, find_quartic_extremum, &
print_quartic_polynomial, introduce_quartic_fit, &
print_chisq_quartic, &
evaluate_two_quartic, &
find_two_quartic_extremum, &
print_chisq_two_quartic, &
set_quartic_linear_coefficients, &
find_quartic_linear_extremum, &
set_quartic_quadratic_coefficients, &
evaluate_quartic_quadratic, &
find_quartic_quadratic_extremum, &
print_chisq_quartic_quadratic, &
set_quartic_cubic_coefficients, &
evaluate_quartic_cubic, &
find_quartic_cubic_extremum, &
print_chisq_quartic_cubic
CONTAINS
SUBROUTINE fit_multi_quartic(ndata,nvar,ncoeff,lsolve,x,f,coeff)
!
! This routine receives as input a set of vectors x(nvar,ndata) and
! function values f(ndata) and gives as output the coefficients of
! a quartic interpolating polynomial coeff(ncoeff). In input
! ndata is the number of data points. nvar is the number of variables
! (the maximum is 6), and ncoeff is the number of coefficients of the
! intepolating quartic polynomial.
! number of variables number of coeffients
! 1 5
! 2 15
! 3 35
! 4 70
! 5 126
! 6 210
!
! lsolve can be 1, 2 or 3. It chooses the method to compute the
! polynomial coefficients. Using 1 a matrix nvar x nvar is calculated
! Using 2 the overdetemined linear system is solved
! using QR or LQ factorization
! Using 3 the overdetermined linear system is solved
! using SVD decomposition
! If lsolve is not one of these values method 2
! is used
!
! The coefficients are organized as follows:
! a_1 + a_2 x(1,i) + a_3 x(1,i)**2 + a_4 x(1,i)**3 + a_5 x(1,i)**4
! + a_6 x(2,i) + a_7 x(2,i)**2 + a_8 x(2,i)**3 + a_9 x(2,i)**4 2
! + a_10 x(1,i)*x(2,i) + a_11 x(1,i)*x(2,i)**2 + a_12 x(1,i)*x(2,i)**3
! + a_13 x(1,i)**2*x(2,i) + a_14 x(1,i)**2*x(2,i)**2
! + a_15 x(1,i)**3*x(2,i)
! + a_16 x(3,i) + a_17 x(3,i)**2 + a_18 x(3,i)**3 + a_19 x(3,i)**4 3
! + a_20 x(1,i)*x(3,i) + a_21 x(1,i)*x(3,i)**2 + a_22 x(1,i)*x(3,i)**3
! + a_23 x(1,i)**2*x(3,i) + a_24 x(1,i)**2*x(3,i)**2
! + a_25 x(1,i)**3*x(3,i)
! + a_26 x(2,i)*x(3,i) + a_27 x(2,i)*x(3,i)**2 + a_28 x(2,i)*x(3,i)**3
! + a_29 x(2,i)**2*x(3,i) + a_30 x(2,i)**2*x(3,i)**2
! + a_31 x(2,i)**3*x(3,i)
! + a_32 x(1,i) * x(2,i) * x(3,i) + a_33 x(1,i) * x(2,i)**2 * x(3,i)
! + a_34 x(1,i) * x(2,i) * x(3,i)**2 + a_35 x(1,i)**2 * x(2,i) * x(3,i)
! + a_36 x(4,i) + a_37 x(4,i)**2 + a_38 x(4,i)**3 + a_39 x(4,i)**4 4
! + a_40 x(1,i)*x(4,i) + a_41 x(1,i)*x(4,i)**2 + a_42 x(1,i)*x(4,i)**3
! + a_43 x(1,i)**2*x(4,i) + a_44 x(1,i)**2*x(4,i)**2
! + a_45 x(1,i)**3*x(4,i)
! + a_46 x(2,i)*x(4,i) + a_47 x(2,i)*x(4,i)**2 + a_48 x(2,i)*x(4,i)**3
! + a_49 x(2,i)**2*x(4,i) + a_50 x(2,i)**2*x(4,i)**2
! + a_51 x(2,i)**3*x(4,i)
! + a_52 x(3,i)*x(4,i) + a_53 x(3,i)*x(4,i)**2 + a_54 x(3,i)*x(4,i)**3
! + a_55 x(3,i)**2*x(4,i) + a_56 x(3,i)**2*x(4,i)**2
! + a_57 x(3,i)**3*x(4,i)
! + a_58 x(1,i) * x(2,i) * x(4,i) + a_59 x(1,i) * x(2,i)**2 * x(4,i)
! + a_60 x(1,i) * x(2,i) * x(4,i)**2 + a_61 x(1,i)**2 * x(2,i) * x(4,i)
! + a_62 x(1,i) * x(3,i) * x(4,i) + a_63 x(1,i) * x(3,i)**2 * x(4,i)
! + a_64 x(1,i) * x(3,i) * x(4,i)**2 + a_65 x(1,i)**2 * x(3,i) * x(4,i)
! + a_66 x(2,i) * x(3,i) * x(4,i) + a_67 x(2,i) * x(3,i)**2 * x(4,i)
! + a_68 x(2,i) * x(3,i) * x(4,i)**2 + a_69 x(2,i)**2 * x(3,i) * x(4,i)
! + a_70 x(1,i) * x(2,i) * x(3,i) * x(4,i)
! + a_71 x(5,i) + a_72 x(5,i)**2 + a_73 x(5,i)**3 + a_74 x(5,i)**4 5
! + a_75 x(1,i)*x(5,i) + a_76 x(1,i)*x(5,i)**2 + a_77 x(1,i)*x(5,i)**3
! + a_78 x(1,i)**2*x(5,i) + a_79 x(1,i)**2*x(5,i)**2
! + a_80 x(1,i)**3*x(5,i)
! + a_81 x(2,i)*x(5,i) + a_82 x(2,i)*x(5,i)**2 + a_83 x(2,i)*x(5,i)**3
! + a_84 x(2,i)**2*x(5,i) + a_85 x(2,i)**2*x(5,i)**2
! + a_86 x(2,i)**3*x(5,i)
! + a_87 x(3,i)*x(5,i) + a_88 x(3,i)*x(5,i)**2 + a_89 x(3,i)*x(5,i)**3
! + a_90 x(3,i)**2*x(5,i) + a_91 x(3,i)**2*x(5,i)**2
! + a_92 x(3,i)**3*x(5,i)
! + a_93 x(4,i)*x(5,i) + a_94 x(4,i)*x(5,i)**2 + a_95 x(4,i)*x(5,i)**3
! + a_96 x(4,i)**2*x(5,i) + a_97 x(4,i)**2*x(5,i)**2
! + a_98 x(4,i)**3*x(5,i)
! + a_99 x(1,i) * x(2,i) * x(5,i) + a_100 x(1,i) * x(2,i)**2 * x(5,i)
! + a_101 x(1,i) * x(2,i) * x(5,i)**2 + a_102 x(1,i)**2 * x(2,i) * x(5,i)
! + a_103 x(1,i) * x(3,i) * x(5,i) + a_104 x(1,i) * x(3,i)**2 * x(5,i)
! + a_105 x(1,i) * x(3,i) * x(5,i)**2 + a_106 x(1,i)**2 * x(3,i) * x(5,i)
! + a_107 x(1,i) * x(4,i) * x(5,i) + a_108 x(1,i) * x(4,i)**2 * x(5,i)
! + a_109 x(1,i) * x(4,i) * x(5,i)**2 + a_110 x(1,i)**2 * x(4,i) * x(5,i)
! + a_111 x(2,i) * x(3,i) * x(5,i) + a_112 x(2,i) * x(3,i)**2 * x(5,i)
! + a_113 x(2,i) * x(3,i) * x(5,i)**2 + a_114 x(2,i)**2 * x(3,i) * x(5,i)
! + a_115 x(2,i) * x(4,i) * x(5,i) + a_116 x(2,i) * x(4,i)**2 * x(5,i)
! + a_117 x(2,i) * x(4,i) * x(5,i)**2 + a_118 x(2,i)**2 * x(4,i) * x(5,i)
! + a_119 x(3,i) * x(4,i) * x(5,i) + a_120 x(3,i) * x(4,i)**2 * x(5,i)
! + a_121 x(3,i) * x(4,i) * x(5,i)**2 + a_122 x(3,i)**2 * x(4,i) * x(5,i)
! + a_123 x(1,i) * x(2,i) * x(3,i) * x(5,i)
! + a_124 x(1,i) * x(2,i) * x(4,i) * x(5,i)
! + a_125 x(1,i) * x(3,i) * x(4,i) * x(5,i)
! + a_126 x(2,i) * x(3,i) * x(4,i) * x(5,i)
! + a_127 x(6,i) + a_128 x(6,i)**2 + a_129 x(6,i)**3 + a_130 x(6,i)**4 6
! + a_131 x(1,i)*x(6,i) + a_132 x(1,i)*x(6,i)**2 + a_133 x(1,i)*x(6,i)**3
! + a_134 x(1,i)**2*x(6,i) + a_135 x(1,i)**2*x(6,i)**2
! + a_136 x(1,i)**3*x(6,i)
! + a_137 x(2,i)*x(6,i) + a_138 x(2,i)*x(6,i)**2 + a_139 x(2,i)*x(6,i)**3
! + a_140 x(2,i)**2*x(6,i) + a_141 x(2,i)**2*x(6,i)**2
! + a_142 x(2,i)**3*x(6,i)
! + a_143 x(3,i)*x(6,i) + a_144 x(3,i)*x(6,i)**2 + a_145 x(3,i)*x(6,i)**3
! + a_146 x(3,i)**2*x(6,i) + a_147 x(3,i)**2*x(6,i)**2
! + a_148 x(3,i)**3*x(6,i)
! + a_149 x(4,i)*x(6,i) + a_150 x(4,i)*x(6,i)**2 + a_151 x(4,i)*x(6,i)**3
! + a_152 x(4,i)**2*x(6,i) + a_153 x(4,i)**2*x(6,i)**2
! + a_154 x(4,i)**3*x(6,i)
! + a_155 x(5,i)*x(6,i) + a_156 x(5,i)*x(6,i)**2 + a_157 x(5,i)*x(6,i)**3
! + a_158 x(5,i)**2*x(6,i) + a_159 x(5,i)**2*x(6,i)**2
! + a_160 x(5,i)**3*x(6,i)
! + a_161 x(1,i) * x(2,i) * x(6,i) + a_162 x(1,i) * x(2,i)**2 * x(6,i)
! + a_163 x(1,i) * x(2,i) * x(6,i)**2 + a_164 x(1,i)**2 * x(2,i) * x(6,i)
! + a_165 x(1,i) * x(3,i) * x(6,i) + a_166 x(1,i) * x(3,i)**2 * x(6,i)
! + a_167 x(1,i) * x(3,i) * x(6,i)**2 + a_168 x(1,i)**2 * x(3,i) * x(6,i)
! + a_169 x(1,i) * x(4,i) * x(6,i) + a_170 x(1,i) * x(4,i)**2 * x(6,i)
! + a_171 x(1,i) * x(4,i) * x(6,i)**2 + a_172 x(1,i)**2 * x(4,i) * x(6,i)
! + a_173 x(1,i) * x(5,i) * x(6,i) + a_174 x(1,i) * x(5,i)**2 * x(6,i)
! + a_175 x(1,i) * x(5,i) * x(6,i)**2 + a_176 x(1,i)**2 * x(5,i) * x(6,i)
! + a_177 x(2,i) * x(3,i) * x(6,i) + a_178 x(2,i) * x(3,i)**2 * x(6,i)
! + a_179 x(2,i) * x(3,i) * x(6,i)**2 + a_180 x(2,i)**2 * x(3,i) * x(6,i)
! + a_181 x(2,i) * x(4,i) * x(6,i) + a_182 x(2,i) * x(4,i)**2 * x(6,i)
! + a_183 x(2,i) * x(4,i) * x(6,i)**2 + a_184 x(2,i)**2 * x(4,i) * x(6,i)
! + a_185 x(2,i) * x(5,i) * x(6,i) + a_186 x(2,i) * x(5,i)**2 * x(6,i)
! + a_187 x(2,i) * x(5,i) * x(6,i)**2 + a_188 x(2,i)**2 * x(5,i) * x(6,i)
! + a_189 x(3,i) * x(4,i) * x(6,i) + a_190 x(3,i) * x(4,i)**2 * x(6,i)
! + a_191 x(3,i) * x(4,i) * x(6,i)**2 + a_192 x(3,i)**2 * x(4,i) * x(6,i)
! + a_193 x(3,i) * x(5,i) * x(6,i) + a_194 x(3,i) * x(5,i)**2 * x(6,i)
! + a_195 x(3,i) * x(5,i) * x(6,i)**2 + a_196 x(3,i)**2 * x(5,i) * x(6,i)
! + a_197 x(4,i) * x(5,i) * x(6,i) + a_198 x(4,i) * x(5,i)**2 * x(6,i)
! + a_199 x(4,i) * x(5,i) * x(6,i)**2 + a_200 x(4,i)**2 * x(5,i) * x(6,i)
! + a_201 x(1,i) * x(2,i) * x(3,i) * x(6,i)
! + a_202 x(1,i) * x(2,i) * x(4,i) * x(6,i)
! + a_203 x(1,i) * x(2,i) * x(5,i) * x(6,i)
! + a_204 x(1,i) * x(3,i) * x(4,i) * x(6,i)
! + a_205 x(1,i) * x(3,i) * x(5,i) * x(6,i)
! + a_206 x(1,i) * x(4,i) * x(5,i) * x(6,i)
! + a_207 x(2,i) * x(3,i) * x(4,i) * x(6,i)
! + a_208 x(2,i) * x(3,i) * x(5,i) * x(6,i)
! + a_209 x(2,i) * x(4,i) * x(5,i) * x(6,i)
! + a_210 x(3,i) * x(4,i) * x(5,i) * x(6,i)
!
USE linear_solvers, ONLY : linsolvx, linsolvms, linsolvsvd
IMPLICIT NONE
INTEGER, INTENT(IN) :: nvar, ncoeff, ndata
INTEGER, INTENT(INOUT) :: lsolve
REAL(DP), INTENT(IN) :: x(nvar,ndata), f(ndata)
REAL(DP), INTENT(INOUT) :: coeff(ncoeff)
REAL(DP) :: amat(ndata,ncoeff), aa(ncoeff,ncoeff), b(ncoeff)
INTEGER :: ivar, jvar, idata
IF (nvar>6.OR.nvar<1) &
CALL errore('multi_quartic','nvar must be from 1 to 6',1)
IF (ndata < 3) &
CALL errore('multi_quartic','Too few sampling data',1)
IF (ndata < ncoeff) &
WRITE(stdout,'(/,5x,"Be careful: there are too few sampling data")')
!
! prepare the auxiliary matrix
!
amat=0.0_DP
DO idata=1,ndata
amat(idata,1) = 1.0_DP
amat(idata,2) = x(1,idata)
amat(idata,3) = x(1,idata)*x(1,idata)
amat(idata,4) = x(1,idata)*x(1,idata)*x(1,idata)
amat(idata,5) = x(1,idata)*x(1,idata)*x(1,idata)*x(1,idata)
IF (nvar>1) THEN
amat(idata,6) = x(2,idata)
amat(idata,7) = x(2,idata) * x(2,idata)
amat(idata,8) = x(2,idata) * x(2,idata) * x(2,idata)
amat(idata,9) = x(2,idata) * x(2,idata) * x(2,idata) * x(2,idata)
amat(idata,10) = x(1,idata) * x(2,idata)
amat(idata,11) = x(1,idata) * x(2,idata) * x(2,idata)
amat(idata,12) = x(1,idata) * x(2,idata) * x(2,idata) * x(2,idata)
amat(idata,13) = x(1,idata) * x(1,idata) * x(2,idata)
amat(idata,14) = x(1,idata) * x(1,idata) * x(2,idata) * x(2,idata)
amat(idata,15) = x(1,idata) * x(1,idata) * x(1,idata) * x(2,idata)
ENDIF
IF (nvar>2) THEN
amat(idata,16) = x(3,idata)
amat(idata,17) = x(3,idata)**2
amat(idata,18) = x(3,idata)**3
amat(idata,19) = x(3,idata)**4
amat(idata,20) = x(1,idata) * x(3,idata)
amat(idata,21) = x(1,idata) * x(3,idata)**2
amat(idata,22) = x(1,idata) * x(3,idata)**3
amat(idata,23) = x(1,idata) **2 * x(3,idata)
amat(idata,24) = x(1,idata) **2 * x(3,idata)**2
amat(idata,25) = x(1,idata) **3 * x(3,idata)
amat(idata,26) = x(2,idata) * x(3,idata)
amat(idata,27) = x(2,idata) * x(3,idata)**2
amat(idata,28) = x(2,idata) * x(3,idata)**3
amat(idata,29) = x(2,idata) **2 * x(3,idata)
amat(idata,30) = x(2,idata) **2 * x(3,idata)**2
amat(idata,31) = x(2,idata) **3 * x(3,idata)
amat(idata,32) = x(1,idata) * x(2,idata) * x(3,idata)
amat(idata,33) = x(1,idata) * x(2,idata)**2 * x(3,idata)
amat(idata,34) = x(1,idata) * x(2,idata) * x(3,idata)**2
amat(idata,35) = x(1,idata)**2 * x(2,idata) * x(3,idata)
ENDIF
IF (nvar>3) THEN
amat(idata,36) = x(4,idata)
amat(idata,37) = x(4,idata)**2
amat(idata,38) = x(4,idata)**3
amat(idata,39) = x(4,idata)**4
amat(idata,40) = x(1,idata) * x(4,idata)
amat(idata,41) = x(1,idata) * x(4,idata)**2
amat(idata,42) = x(1,idata) * x(4,idata)**3
amat(idata,43) = x(1,idata) **2 * x(4,idata)
amat(idata,44) = x(1,idata) **2 * x(4,idata)**2
amat(idata,45) = x(1,idata) **3 * x(4,idata)
amat(idata,46) = x(2,idata) * x(4,idata)
amat(idata,47) = x(2,idata) * x(4,idata)**2
amat(idata,48) = x(2,idata) * x(4,idata)**3
amat(idata,49) = x(2,idata) **2 * x(4,idata)
amat(idata,50) = x(2,idata) **2 * x(4,idata)**2
amat(idata,51) = x(2,idata) **3 * x(4,idata)
amat(idata,52) = x(3,idata) * x(4,idata)
amat(idata,53) = x(3,idata) * x(4,idata)**2
amat(idata,54) = x(3,idata) * x(4,idata)**3
amat(idata,55) = x(3,idata) **2 * x(4,idata)
amat(idata,56) = x(3,idata) **2 * x(4,idata)**2
amat(idata,57) = x(3,idata) **3 * x(4,idata)
amat(idata,58) = x(1,idata) * x(2,idata) * x(4,idata)
amat(idata,59) = x(1,idata) * x(2,idata)**2 * x(4,idata)
amat(idata,60) = x(1,idata) * x(2,idata) * x(4,idata)**2
amat(idata,61) = x(1,idata)**2 * x(2,idata) * x(4,idata)
amat(idata,62) = x(1,idata) * x(3,idata) * x(4,idata)
amat(idata,63) = x(1,idata) * x(3,idata)**2 * x(4,idata)
amat(idata,64) = x(1,idata) * x(3,idata) * x(4,idata)**2
amat(idata,65) = x(1,idata)**2 * x(3,idata) * x(4,idata)
amat(idata,66) = x(2,idata) * x(3,idata) * x(4,idata)
amat(idata,67) = x(2,idata) * x(3,idata)**2 * x(4,idata)
amat(idata,68) = x(2,idata) * x(3,idata) * x(4,idata)**2
amat(idata,69) = x(2,idata)**2 * x(3,idata) * x(4,idata)
amat(idata,70) = x(1,idata) * x(2,idata) * x(3,idata) * x(4,idata)
ENDIF
IF (nvar>4) THEN
amat(idata,71) = x(5,idata)
amat(idata,72) = x(5,idata)**2
amat(idata,73) = x(5,idata)**3
amat(idata,74) = x(5,idata)**4
amat(idata,75) = x(1,idata) * x(5,idata)
amat(idata,76) = x(1,idata) * x(5,idata)**2
amat(idata,77) = x(1,idata) * x(5,idata)**3
amat(idata,78) = x(1,idata) **2 * x(5,idata)
amat(idata,79) = x(1,idata) **2 * x(5,idata)**2
amat(idata,80) = x(1,idata) **3 * x(5,idata)
amat(idata,81) = x(2,idata) * x(5,idata)
amat(idata,82) = x(2,idata) * x(5,idata)**2
amat(idata,83) = x(2,idata) * x(5,idata)**3
amat(idata,84) = x(2,idata) **2 * x(5,idata)
amat(idata,85) = x(2,idata) **2 * x(5,idata)**2
amat(idata,86) = x(2,idata) **3 * x(5,idata)
amat(idata,87) = x(3,idata) * x(5,idata)
amat(idata,88) = x(3,idata) * x(5,idata)**2
amat(idata,89) = x(3,idata) * x(5,idata)**3
amat(idata,90) = x(3,idata) **2 * x(5,idata)
amat(idata,91) = x(3,idata) **2 * x(5,idata)**2
amat(idata,92) = x(3,idata) **3 * x(5,idata)
amat(idata,93) = x(4,idata) * x(5,idata)
amat(idata,94) = x(4,idata) * x(5,idata)**2
amat(idata,95) = x(4,idata) * x(5,idata)**3
amat(idata,96) = x(4,idata) **2 * x(5,idata)
amat(idata,97) = x(4,idata) **2 * x(5,idata)**2
amat(idata,98) = x(4,idata) **3 * x(5,idata)
amat(idata,99) = x(1,idata) * x(2,idata) * x(5,idata)
amat(idata,100) = x(1,idata) * x(2,idata)**2 * x(5,idata)
amat(idata,101) = x(1,idata) * x(2,idata) * x(5,idata)**2
amat(idata,102) = x(1,idata)**2 * x(2,idata) * x(5,idata)
amat(idata,103) = x(1,idata) * x(3,idata) * x(5,idata)
amat(idata,104) = x(1,idata) * x(3,idata)**2 * x(5,idata)
amat(idata,105) = x(1,idata) * x(3,idata) * x(5,idata)**2
amat(idata,106) = x(1,idata)**2 * x(3,idata) * x(5,idata)
amat(idata,107) = x(1,idata) * x(4,idata) * x(5,idata)
amat(idata,108) = x(1,idata) * x(4,idata)**2 * x(5,idata)
amat(idata,109) = x(1,idata) * x(4,idata) * x(5,idata)**2
amat(idata,110) = x(1,idata)**2 * x(4,idata) * x(5,idata)
amat(idata,111) = x(2,idata) * x(3,idata) * x(5,idata)
amat(idata,112) = x(2,idata) * x(3,idata)**2 * x(5,idata)
amat(idata,113) = x(2,idata) * x(3,idata) * x(5,idata)**2
amat(idata,114) = x(2,idata)**2 * x(3,idata) * x(5,idata)
amat(idata,115) = x(2,idata) * x(4,idata) * x(5,idata)
amat(idata,116) = x(2,idata) * x(4,idata)**2 * x(5,idata)
amat(idata,117) = x(2,idata) * x(4,idata) * x(5,idata)**2
amat(idata,118) = x(2,idata)**2 * x(4,idata) * x(5,idata)
amat(idata,119) = x(3,idata) * x(4,idata) * x(5,idata)
amat(idata,120) = x(3,idata) * x(4,idata)**2 * x(5,idata)
amat(idata,121) = x(3,idata) * x(4,idata) * x(5,idata)**2
amat(idata,122) = x(3,idata)**2 * x(4,idata) * x(5,idata)
amat(idata,123) = x(1,idata) * x(2,idata) * x(3,idata) * x(5,idata)
amat(idata,124) = x(1,idata) * x(2,idata) * x(4,idata) * x(5,idata)
amat(idata,125) = x(1,idata) * x(3,idata) * x(4,idata) * x(5,idata)
amat(idata,126) = x(2,idata) * x(3,idata) * x(4,idata) * x(5,idata)
ENDIF
IF (nvar>5) THEN
amat(idata,127) = x(6,idata)
amat(idata,128) = x(6,idata)**2
amat(idata,129) = x(6,idata)**3
amat(idata,130) = x(6,idata)**4
amat(idata,131) = x(1,idata) * x(6,idata)
amat(idata,132) = x(1,idata) * x(6,idata)**2
amat(idata,133) = x(1,idata) * x(6,idata)**3
amat(idata,134) = x(1,idata) **2 * x(6,idata)
amat(idata,135) = x(1,idata) **2 * x(6,idata)**2
amat(idata,136) = x(1,idata) **3 * x(6,idata)
amat(idata,137) = x(2,idata) * x(6,idata)
amat(idata,138) = x(2,idata) * x(6,idata)**2
amat(idata,139) = x(2,idata) * x(6,idata)**3
amat(idata,140) = x(2,idata) **2 * x(6,idata)
amat(idata,141) = x(2,idata) **2 * x(6,idata)**2
amat(idata,142) = x(2,idata) **3 * x(6,idata)
amat(idata,143) = x(3,idata) * x(6,idata)
amat(idata,144) = x(3,idata) * x(6,idata)**2
amat(idata,145) = x(3,idata) * x(6,idata)**3
amat(idata,146) = x(3,idata) **2 * x(6,idata)
amat(idata,147) = x(3,idata) **2 * x(6,idata)**2
amat(idata,148) = x(3,idata) **3 * x(6,idata)
amat(idata,149) = x(4,idata) * x(6,idata)
amat(idata,150) = x(4,idata) * x(6,idata)**2
amat(idata,151) = x(4,idata) * x(6,idata)**3
amat(idata,152) = x(4,idata) **2 * x(6,idata)
amat(idata,153) = x(4,idata) **2 * x(6,idata)**2
amat(idata,154) = x(4,idata) **3 * x(6,idata)
amat(idata,155) = x(5,idata) * x(6,idata)
amat(idata,156) = x(5,idata) * x(6,idata)**2
amat(idata,157) = x(5,idata) * x(6,idata)**3
amat(idata,158) = x(5,idata) **2 * x(6,idata)
amat(idata,159) = x(5,idata) **2 * x(6,idata)**2
amat(idata,160) = x(5,idata) **3 * x(6,idata)
amat(idata,161) = x(1,idata) * x(2,idata) * x(6,idata)
amat(idata,162) = x(1,idata) * x(2,idata)**2 * x(6,idata)
amat(idata,163) = x(1,idata) * x(2,idata) * x(6,idata)**2
amat(idata,164) = x(1,idata)**2 * x(2,idata) * x(6,idata)
amat(idata,165) = x(1,idata) * x(3,idata) * x(6,idata)
amat(idata,166) = x(1,idata) * x(3,idata)**2 * x(6,idata)
amat(idata,167) = x(1,idata) * x(3,idata) * x(6,idata)**2
amat(idata,168) = x(1,idata)**2 * x(3,idata) * x(6,idata)
amat(idata,169) = x(1,idata) * x(4,idata) * x(6,idata)
amat(idata,170) = x(1,idata) * x(4,idata)**2 * x(6,idata)
amat(idata,171) = x(1,idata) * x(4,idata) * x(6,idata)**2
amat(idata,172) = x(1,idata)**2 * x(4,idata) * x(6,idata)
amat(idata,173) = x(1,idata) * x(5,idata) * x(6,idata)
amat(idata,174) = x(1,idata) * x(5,idata)**2 * x(6,idata)
amat(idata,175) = x(1,idata) * x(5,idata) * x(6,idata)**2
amat(idata,176) = x(1,idata)**2 * x(5,idata) * x(6,idata)
amat(idata,177) = x(2,idata) * x(3,idata) * x(6,idata)
amat(idata,178) = x(2,idata) * x(3,idata)**2 * x(6,idata)
amat(idata,179) = x(2,idata) * x(3,idata) * x(6,idata)**2
amat(idata,180) = x(2,idata)**2 * x(3,idata) * x(6,idata)
amat(idata,181) = x(2,idata) * x(4,idata) * x(6,idata)
amat(idata,182) = x(2,idata) * x(4,idata)**2 * x(6,idata)
amat(idata,183) = x(2,idata) * x(4,idata) * x(6,idata)**2
amat(idata,184) = x(2,idata)**2 * x(4,idata) * x(6,idata)
amat(idata,185) = x(2,idata) * x(5,idata) * x(6,idata)
amat(idata,186) = x(2,idata) * x(5,idata)**2 * x(6,idata)
amat(idata,187) = x(2,idata) * x(5,idata) * x(6,idata)**2
amat(idata,188) = x(2,idata)**2 * x(5,idata) * x(6,idata)
amat(idata,189) = x(3,idata) * x(4,idata) * x(6,idata)
amat(idata,190) = x(3,idata) * x(4,idata)**2 * x(6,idata)
amat(idata,191) = x(3,idata) * x(4,idata) * x(6,idata)**2
amat(idata,192) = x(3,idata)**2 * x(4,idata) * x(6,idata)
amat(idata,193) = x(3,idata) * x(5,idata) * x(6,idata)
amat(idata,194) = x(3,idata) * x(5,idata)**2 * x(6,idata)
amat(idata,195) = x(3,idata) * x(5,idata) * x(6,idata)**2
amat(idata,196) = x(3,idata)**2 * x(5,idata) * x(6,idata)
amat(idata,197) = x(4,idata) * x(5,idata) * x(6,idata)
amat(idata,198) = x(4,idata) * x(5,idata)**2 * x(6,idata)
amat(idata,199) = x(4,idata) * x(5,idata) * x(6,idata)**2
amat(idata,200) = x(4,idata)**2 * x(5,idata) * x(6,idata)
amat(idata,201) = x(1,idata) * x(2,idata) * x(3,idata) * x(6,idata)
amat(idata,202) = x(1,idata) * x(2,idata) * x(4,idata) * x(6,idata)
amat(idata,203) = x(1,idata) * x(2,idata) * x(5,idata) * x(6,idata)
amat(idata,204) = x(1,idata) * x(3,idata) * x(4,idata) * x(6,idata)
amat(idata,205) = x(1,idata) * x(3,idata) * x(5,idata) * x(6,idata)
amat(idata,206) = x(1,idata) * x(4,idata) * x(5,idata) * x(6,idata)
amat(idata,207) = x(2,idata) * x(3,idata) * x(4,idata) * x(6,idata)
amat(idata,208) = x(2,idata) * x(3,idata) * x(5,idata) * x(6,idata)
amat(idata,209) = x(2,idata) * x(4,idata) * x(5,idata) * x(6,idata)
amat(idata,210) = x(3,idata) * x(4,idata) * x(5,idata) * x(6,idata)
ENDIF
ENDDO
aa=0.0_DP
b =0.0_DP
DO ivar=1,ncoeff
DO jvar=1,ncoeff
DO idata=1,ndata
aa(ivar,jvar)= aa(ivar,jvar) + amat(idata,ivar) * amat(idata,jvar)
END DO
END DO
DO idata=1,ndata
b(ivar) = b(ivar) + amat(idata,ivar) * f(idata)
END DO
END DO
!
! solve the linear system and find the coefficients
!
coeff=0.0_DP
IF (lsolve<1.OR.lsolve>3) lsolve=2
IF (lsolve==1) THEN
WRITE(stdout,'(5x,"Finding the quartic polynomial using &
&ncoeff x ncoeff matrix")')
CALL linsolvx(aa,ncoeff,b,coeff)
ELSEIF(lsolve==2) THEN
WRITE(stdout,'(5x,"Finding the quartic polynomial using &
&QR factorization")')
CALL linsolvms(amat,ndata,ncoeff,f,coeff)
ELSEIF(lsolve==3) THEN
WRITE(stdout,'(5x,"Finding the quartic polynomial using SVD")')
CALL linsolvsvd(amat,ndata,ncoeff,f,coeff)
ENDIF
RETURN
END SUBROUTINE fit_multi_quartic
SUBROUTINE evaluate_fit_quartic(nvar,ncoeff,x,f,coeff)
!
! This routine evaluates the quartic polynomial at the point x
!
IMPLICIT NONE
INTEGER, INTENT(IN) :: nvar, ncoeff
REAL(DP), INTENT(IN) :: x(nvar)
REAL(DP), INTENT(IN) :: coeff(ncoeff)
REAL(DP), INTENT(INOUT) :: f
REAL(DP) :: aux
!
! one variable
!
aux = coeff(1) + x(1)*(coeff(2)+x(1)*(coeff(3)+x(1)*(coeff(4)+coeff(5)*x(1))))
!
! two variables
!
IF (nvar>1) THEN
aux=aux+x(2)*(coeff(6) + x(2)*( coeff(7) + coeff(11) * x(1) + &
coeff(14) * x(1)**2 + &
x(2)*(coeff(8) + coeff(9) * x(2) + coeff(12) * x(1))) &
+ x(1) *( coeff(10) &
+ x(1) * (coeff(13) + coeff(15) * x(1) ) ) )
ENDIF
!
! three variables
!
IF (nvar>2) THEN
aux = aux + coeff(16) * x(3) + coeff(17) * x(3)**2 + coeff(18) * x(3)**3 + &
coeff(19) * x(3)**4 &
+ coeff(20) * x(1) * x(3) + coeff(21) * x(1) * x(3)**2 &
+ coeff(22) * x(1) * x(3)**3 + coeff(23) * x(1)**2 * x(3) &
+ coeff(24) * x(1)**2 * x(3)**2 + coeff(25) * x(1)**3 * x(3) &
+ coeff(26) * x(2) * x(3) + coeff(27) * x(2) * x(3)**2 &
+ coeff(28) * x(2) * x(3)**3 + coeff(29) * x(2)**2 * x(3) &
+ coeff(30) * x(2)**2 * x(3)**2 + coeff(31) * x(2)**3 * x(3) &
+ coeff(32) * x(1) * x(2) * x(3) &
+ coeff(33) * x(1) * x(2)**2 * x(3) &
+ coeff(34) * x(1) * x(2) * x(3)**2 &
+ coeff(35) * x(1)**2 * x(2) * x(3)
ENDIF
!
! four variables
!
IF (nvar>3) THEN
aux = aux + coeff(36) * x(4) + coeff(37) * x(4)**2 + coeff(38) * x(4)**3 + &
coeff(39) * x(4)**4 &
+ coeff(40) * x(1) * x(4) + coeff(41) * x(1) * x(4)**2 &
+ coeff(42) * x(1) * x(4)**3 + coeff(43) * x(1)**2 * x(4) &
+ coeff(44) * x(1)**2 * x(4)**2 + coeff(45) * x(1)**3 * x(4) &
+ coeff(46) * x(2) * x(4) + coeff(47) * x(2) * x(4)**2 &
+ coeff(48) * x(2) * x(4)**3 + coeff(49) * x(2)**2 * x(4) &
+ coeff(50) * x(2)**2 * x(4)**2 + coeff(51) * x(2)**3 * x(4) &
+ coeff(52) * x(3) * x(4) + coeff(53) * x(3) * x(4)**2 &
+ coeff(54) * x(3) * x(4)**3 + coeff(55) * x(3)**2 * x(4) &
+ coeff(56) * x(3)**2 * x(4)**2 + coeff(57) * x(3)**3 * x(4) &
+ coeff(58) * x(1) * x(2) * x(4) &
+ coeff(59) * x(1) * x(2)**2 * x(4) &
+ coeff(60) * x(1) * x(2) * x(4)**2 &
+ coeff(61) * x(1)**2 * x(2) * x(4) &
+ coeff(62) * x(1) * x(3) * x(4) &
+ coeff(63) * x(1) * x(3)**2 * x(4) &
+ coeff(64) * x(1) * x(3) * x(4)**2 &
+ coeff(65) * x(1)**2 * x(3) * x(4) &
+ coeff(66) * x(2) * x(3) * x(4) &
+ coeff(67) * x(2) * x(3)**2 * x(4) &
+ coeff(68) * x(2) * x(3) * x(4)**2 &
+ coeff(69) * x(2)**2 * x(3) * x(4) &
+ coeff(70) * x(1) * x(2) * x(3) * x(4)
ENDIF
!
! five variables
!
IF (nvar>4) THEN
aux = aux + coeff(71) * x(5) + coeff(72) * x(5)**2 + coeff(73) * x(5)**3 + &
coeff(74) * x(5)**4 &
+ coeff(75) * x(1) * x(5) + coeff(76) * x(1) * x(5)**2 &
+ coeff(77) * x(1) * x(5)**3 + coeff(78) * x(1)**2 * x(5) &
+ coeff(79) * x(1)**2 * x(5)**2 + coeff(80) * x(1)**3 * x(5) &
+ coeff(81) * x(2) * x(5) + coeff(82) * x(2) * x(5)**2 &
+ coeff(83) * x(2) * x(5)**3 + coeff(84) * x(2)**2 * x(5) &
+ coeff(85) * x(2)**2 * x(5)**2 + coeff(86) * x(2)**3 * x(5) &
+ coeff(87) * x(3) * x(5) + coeff(88) * x(3) * x(5)**2 &
+ coeff(89) * x(3) * x(5)**3 + coeff(90) * x(3)**2 * x(5) &
+ coeff(91) * x(3)**2 * x(5)**2 + coeff(92) * x(3)**3 * x(5) &
+ coeff(93) * x(4) * x(5) + coeff(94) * x(4) * x(5)**2 &
+ coeff(95) * x(4) * x(5)**3 + coeff(96) * x(4)**2 * x(5) &
+ coeff(97) * x(4)**2 * x(5)**2 + coeff(98) * x(4)**3 * x(5) &
+ coeff(99) * x(1) * x(2) * x(5) &
+ coeff(100) * x(1) * x(2)**2 * x(5) &
+ coeff(101) * x(1) * x(2) * x(5)**2 &
+ coeff(102) * x(1)**2 * x(2) * x(5) &
+ coeff(103) * x(1) * x(3) * x(5) &
+ coeff(104) * x(1) * x(3)**2 * x(5) &
+ coeff(105) * x(1) * x(3) * x(5)**2 &
+ coeff(106) * x(1)**2 * x(3) * x(5) &
+ coeff(107) * x(1) * x(4) * x(5) &
+ coeff(108) * x(1) * x(4)**2 * x(5) &
+ coeff(109) * x(1) * x(4) * x(5)**2 &
+ coeff(110) * x(1)**2 * x(4) * x(5) &
+ coeff(111) * x(2) * x(3) * x(5) &
+ coeff(112) * x(2) * x(3)**2 * x(5) &
+ coeff(113) * x(2) * x(3) * x(5)**2 &
+ coeff(114) * x(2)**2 * x(3) * x(5) &
+ coeff(115) * x(2) * x(4) * x(5) &
+ coeff(116) * x(2) * x(4)**2 * x(5) &
+ coeff(117) * x(2) * x(4) * x(5)**2 &
+ coeff(118) * x(2)**2 * x(4) * x(5) &
+ coeff(119) * x(3) * x(4) * x(5) &
+ coeff(120) * x(3) * x(4)**2 * x(5) &
+ coeff(121) * x(3) * x(4) * x(5)**2 &
+ coeff(122) * x(3)**2 * x(4) * x(5) &
+ coeff(123) * x(1) * x(2) * x(3) * x(5) &
+ coeff(124) * x(1) * x(2) * x(4) * x(5) &
+ coeff(125) * x(1) * x(3) * x(4) * x(5) &
+ coeff(126) * x(2) * x(3) * x(4) * x(5)
ENDIF
!
! six variables
!
IF (nvar>5) THEN
aux = aux + coeff(127) * x(6) + coeff(128) * x(6)**2 + &
coeff(129) * x(6)**3 + coeff(130) * x(6)**4 &
+ coeff(131) * x(1) * x(6) + coeff(132) * x(1) * x(6)**2 &
+ coeff(133) * x(1) * x(6)**3 + coeff(134) * x(1)**2 * x(6) &
+ coeff(135) * x(1)**2 * x(6)**2 + coeff(136) * x(1)**3 * x(6) &
+ coeff(137) * x(2) * x(6) + coeff(138) * x(2) * x(6)**2 &
+ coeff(139) * x(2) * x(6)**3 + coeff(140) * x(2)**2 * x(6) &
+ coeff(141) * x(2)**2 * x(6)**2 + coeff(142) * x(2)**3 * x(6) &
+ coeff(143) * x(3) * x(6) + coeff(144) * x(3) * x(6)**2 &
+ coeff(145) * x(3) * x(6)**3 + coeff(146) * x(3)**2 * x(6) &
+ coeff(147) * x(3)**2 * x(6)**2 + coeff(148) * x(3)**3 * x(6) &
+ coeff(149) * x(4) * x(6) + coeff(150) * x(4) * x(6)**2 &
+ coeff(151) * x(4) * x(6)**3 + coeff(152) * x(4)**2 * x(6) &
+ coeff(153) * x(4)**2 * x(6)**2 + coeff(154) * x(4)**3 * x(6) &
+ coeff(155) * x(5) * x(6) + coeff(156) * x(5) * x(6)**2 &
+ coeff(157) * x(5) * x(6)**3 + coeff(158) * x(5)**2 * x(6) &
+ coeff(159) * x(5)**2 * x(6)**2 + coeff(160) * x(5)**3 * x(6) &
+ coeff(161) * x(1) * x(2) * x(6) &
+ coeff(162) * x(1) * x(2)**2 * x(6) &
+ coeff(163) * x(1) * x(2) * x(6)**2 &
+ coeff(164) * x(1)**2 * x(2) * x(6) &
+ coeff(165) * x(1) * x(3) * x(6) &
+ coeff(166) * x(1) * x(3)**2 * x(6) &
+ coeff(167) * x(1) * x(3) * x(6)**2 &
+ coeff(168) * x(1)**2 * x(3) * x(6) &
+ coeff(169) * x(1) * x(4) * x(6) &
+ coeff(170) * x(1) * x(4)**2 * x(6) &
+ coeff(171) * x(1) * x(4) * x(6)**2 &
+ coeff(172) * x(1)**2 * x(4) * x(6) &
+ coeff(173) * x(1) * x(5) * x(6) &
+ coeff(174) * x(1) * x(5)**2 * x(6) &
+ coeff(175) * x(1) * x(5) * x(6)**2 &
+ coeff(176) * x(1)**2 * x(5) * x(6) &
+ coeff(177) * x(2) * x(3) * x(6) &
+ coeff(178) * x(2) * x(3)**2 * x(6) &
+ coeff(179) * x(2) * x(3) * x(6)**2 &
+ coeff(180) * x(2)**2 * x(3) * x(6) &
+ coeff(181) * x(2) * x(4) * x(6) &
+ coeff(182) * x(2) * x(4)**2 * x(6) &
+ coeff(183) * x(2) * x(4) * x(6)**2 &
+ coeff(184) * x(2)**2 * x(4) * x(6) &
+ coeff(185) * x(2) * x(5) * x(6) &
+ coeff(186) * x(2) * x(5)**2 * x(6) &
+ coeff(187) * x(2) * x(5) * x(6)**2 &
+ coeff(188) * x(2)**2 * x(5) * x(6) &
+ coeff(189) * x(3) * x(4) * x(6) &
+ coeff(190) * x(3) * x(4)**2 * x(6) &
+ coeff(191) * x(3) * x(4) * x(6)**2 &
+ coeff(192) * x(3)**2 * x(4) * x(6) &
+ coeff(193) * x(3) * x(5) * x(6) &
+ coeff(194) * x(3) * x(5)**2 * x(6) &
+ coeff(195) * x(3) * x(5) * x(6)**2 &
+ coeff(196) * x(3)**2 * x(5) * x(6) &
+ coeff(197) * x(4) * x(5) * x(6) &
+ coeff(198) * x(4) * x(5)**2 * x(6) &
+ coeff(199) * x(4) * x(5) * x(6)**2 &
+ coeff(200) * x(4)**2 * x(5) * x(6) &
+ coeff(201) * x(1) * x(2) * x(3) * x(6) &
+ coeff(202) * x(1) * x(2) * x(4) * x(6) &
+ coeff(203) * x(1) * x(2) * x(5) * x(6) &
+ coeff(204) * x(1) * x(3) * x(4) * x(6) &
+ coeff(205) * x(1) * x(3) * x(5) * x(6) &
+ coeff(206) * x(1) * x(4) * x(5) * x(6) &
+ coeff(207) * x(2) * x(3) * x(4) * x(6) &
+ coeff(208) * x(2) * x(3) * x(5) * x(6) &
+ coeff(209) * x(2) * x(4) * x(5) * x(6) &
+ coeff(210) * x(3) * x(4) * x(5) * x(6)
ENDIF
f=aux
RETURN
END SUBROUTINE evaluate_fit_quartic
SUBROUTINE evaluate_quartic_grad(nvar,ncoeff,x,f,coeff)
!
! This routine evaluates the gradient of the quartic polynomial
! at the point x.
!
IMPLICIT NONE
INTEGER, INTENT(IN) :: nvar, ncoeff
REAL(DP), INTENT(IN) :: x(nvar)
REAL(DP), INTENT(IN) :: coeff(ncoeff)
REAL(DP), INTENT(INOUT) :: f(nvar)
REAL(DP) :: aux(nvar)
IF (nvar>6) CALL errore('evaluate_quartic_grad','gradient not availble',1)
aux(1) = coeff(2) + 2.0_DP * coeff(3) * x(1) + 3.0_DP * coeff(4)*x(1)**2 + &
4.0_DP * coeff(5) * x(1)**3
IF (nvar>1) THEN
aux(1) = aux(1) + coeff(10) * x(2) + coeff(11) * x(2)**2 + &
coeff(12) * x(2)**3 + 2.0_DP * coeff(13) * x(1) * x(2) + &
2.0_DP * coeff(14) * x(1) * x(2)**2 + &
3.0_DP * coeff(15) * x(1)**2 * x(2)
aux(2) = coeff(6) + 2.0_DP * coeff(7) * x(2) + 3.0_DP * coeff(8)*x(2)**2 + &
4.0_DP*coeff(9)*x(2)**3 + coeff(10) * x(1) &
+ 2.0_DP*coeff(11)*x(1)*x(2) &
+ 3.0_DP*coeff(12)*x(1)*x(2)**2 + coeff(13)*x(1)**2 + &
2.0_DP*coeff(14)*x(1)**2*x(2) + coeff(15)*x(1)**3
ENDIF
IF (nvar>2) THEN
aux(1) = aux(1) + coeff(20) * x(3) + coeff(21) * x(3)**2 + &
coeff(22) * x(3)**3 + 2.0_DP * coeff(23) * x(1)*x(3) + &
2.0_DP * coeff(24) * x(1)*x(3)**2 + &
3.0_DP * coeff(25) * x(1)**2 * x(3) + &
coeff(32) * x(2) * x(3) + coeff(33) * x(2)**2 * x(3) + &
coeff(34) * x(2) * x(3)**2 + &
2.0_DP * coeff(35) * x(1) * x(2) * x(3)
aux(2) = aux(2) + coeff(26) * x(3) + coeff(27) * x(3)**2 &
+ coeff(28) * x(3)**3 + 2.0_DP*coeff(29)*x(2)*x(3) &
+ 2.0_DP*coeff(30)*x(2)*x(3)**2 + 3.0_DP*coeff(31)*x(2)**2*x(3) &
+ coeff(32) * x(1) * x(3) + 2.0_DP * coeff(33) * x(1) * x(2) * x(3) &
+ coeff(34) * x(1) * x(3)**2 + coeff(35) * x(1)**2 * x(3)
aux(3) = coeff(16) + 2.0_DP*coeff(17)*x(3) + 3.0_DP*coeff(18)*x(3)**2 &
+ 4.0_DP*coeff(19)*x(3)**3 + coeff(20)*x(1) &
+ 2.0_DP*coeff(21)*x(1)*x(3) &
+ 3.0_DP*coeff(22)*x(1)*x(3)**2 + coeff(23)*x(1)**2 &
+ 2.0_DP*coeff(24)*x(1)**2*x(3) + coeff(25)*x(1)**3 &
+ coeff(26)*x(2) &
+ 2.0_DP*coeff(27)*x(2)*x(3) &
+ 3.0_DP*coeff(28)*x(2)*x(3)**2 + coeff(29)*x(2)**2 &
+ 2.0_DP*coeff(30)*x(2)**2*x(3) + coeff(31)*x(2)**3 &
+ coeff(32)*x(1)*x(2) + coeff(33)*x(1)*x(2)**2 &
+ 2.0_DP*coeff(34)*x(1)*x(2)*x(3) + coeff(35)*x(1)**2*x(2)
ENDIF
IF (nvar>3) THEN
aux(1) = aux(1) + coeff(40) * x(4) + coeff(41) * x(4)**2 + &
coeff(42) * x(4)**3 + 2.0_DP * coeff(43) * x(1)*x(4) + &
2.0_DP * coeff(44) * x(1)*x(4)**2 + &
3.0_DP * coeff(45) * x(1)**2 * x(4) + &
coeff(58) * x(2) * x(4) + coeff(59) * x(2)**2 * x(4) + &
coeff(60) * x(2) * x(4)**2 + &
2.0_DP * coeff(61) * x(1) * x(2) * x(4) + &
coeff(62) * x(3) * x(4) + coeff(63) * x(3)**2 * x(4) + &
coeff(64) * x(3) * x(4)**2 + &
2.0_DP * coeff(65) * x(1) * x(3) * x(4) + &
coeff(70) * x(2) * x(3) * x(4)
aux(2) = aux(2) + coeff(46) * x(4) + coeff(47) * x(4)**2 + &
coeff(48) * x(4)**3 + 2.0_DP * coeff(49) * x(2)*x(4) + &
2.0_DP * coeff(50) * x(2)*x(4)**2 + &
3.0_DP * coeff(51) * x(2)**2 * x(4) + &
coeff(58) * x(1) * x(4) + &
2.0_DP *coeff(59) * x(1)*x(2)*x(4) + &
coeff(60) * x(1) * x(4)**2 + &
coeff(61) * x(1) **2 * x(4) + &
coeff(66) * x(3) * x(4) + coeff(67) * x(3)**2 * x(4) + &
coeff(68) * x(3) * x(4)**2 + &
2.0_DP * coeff(69) * x(2) * x(3) * x(4) + &
coeff(70) * x(1) * x(3) * x(4)
aux(3) = aux(3) + coeff(52) * x(4) + coeff(53) * x(4)**2 + &
coeff(54) * x(4)**3 + 2.0_DP * coeff(55) * x(3)*x(4) + &
2.0_DP * coeff(56) * x(3)*x(4)**2 + &
3.0_DP * coeff(57) * x(3)**2 * x(4) + &
coeff(62) * x(1) * x(4) + &
2.0_DP *coeff(63) * x(1)*x(3)*x(4) + &
coeff(64) * x(1) * x(4)**2 + &
coeff(65) * x(1) **2 * x(4) + &
coeff(66) * x(2) * x(4) + &
2.0_DP*coeff(67)*x(2)*x(3)*x(4) + &
coeff(68) * x(2) * x(4)**2 + &
coeff(69) * x(2) **2 * x(4) + &
coeff(70) * x(1) * x(2) * x(4)
aux(4) = coeff(36) + 2.0_DP * coeff(37) * x(4) + &
3.0_DP * coeff(38) * x(4)**2 + &
4.0_DP * coeff(39) * x(4)**3 + &
coeff(40) * x(1) + &
2.0_DP * coeff(41) * x(1) * x(4) + &
3.0_DP * coeff(42) * x(1) * x(4) **2 + &
coeff(43) * x(1) ** 2 + &
2.0_DP * coeff(44) * x(1) **2 * x(4) + &
coeff(45) * x(1) ** 3 + &
coeff(46) * x(2) + &
2.0_DP * coeff(47) * x(2) * x(4) + &
3.0_DP * coeff(48) * x(2) * x(4) **2 + &
coeff(49) * x(2) ** 2 + &
2.0_DP * coeff(50) * x(2) **2 * x(4) + &
coeff(51) * x(2) ** 3 + &
coeff(52) * x(3) + &
2.0_DP * coeff(53) * x(3) * x(4) + &
3.0_DP * coeff(54) * x(3) * x(4) **2 + &
coeff(55) * x(3) ** 2 + &
2.0_DP * coeff(56) * x(3) **2 * x(4) + &
coeff(57) * x(3) ** 3 + &
coeff(58) * x(1) * x(2) + &
coeff(59) * x(1) * x(2) **2 + &
2.0_DP * coeff(60) * x(1) * x(2) * x(4) + &
coeff(61) * x(1) **2 * x(2) + &
coeff(62) * x(1) * x(3) + &
coeff(63) * x(1) * x(3) **2 + &
2.0_DP * coeff(64) * x(1) * x(3) * x(4) + &
coeff(65) * x(1) **2 * x(3) + &
coeff(66) * x(2) * x(3) + &
coeff(67) * x(2) * x(3) **2 + &
2.0_DP * coeff(68) * x(2) * x(3) * x(4) + &
coeff(69) * x(2) **2 * x(3) + &
coeff(70) * x(1) * x(2) * x(3)
ENDIF
IF (nvar>4) THEN
aux(1) = aux(1) + coeff(75) * x(5) + coeff(76) * x(5)**2 + &
coeff(77) * x(5)**3 + 2.0_DP * coeff(78) * x(1)*x(5) + &
2.0_DP * coeff(79) * x(1)*x(5)**2 + &
3.0_DP * coeff(80) * x(1)**2 * x(5) + &
coeff(99) * x(2) * x(5) + coeff(100) * x(2)**2 * x(5) + &
coeff(101) * x(2) * x(5)**2 + &
2.0_DP * coeff(102) * x(1) * x(2) * x(5) + &
coeff(103) * x(3) * x(5) + coeff(104) * x(3)**2 * x(5) + &
coeff(105) * x(3) * x(5)**2 + &
2.0_DP * coeff(106) * x(1) * x(3) * x(5) + &
coeff(107) * x(4) * x(5) + coeff(108) * x(4)**2 * x(5) + &
coeff(109) * x(4) * x(5)**2 + &
2.0_DP * coeff(110) * x(1) * x(4) * x(5) + &
coeff(123) * x(2) * x(3) * x(5) + &
coeff(124) * x(2) * x(4) * x(5) + &
coeff(125) * x(3) * x(4) * x(5)
aux(2) = aux(2) + coeff(81) * x(5) + coeff(82) * x(5)**2 + &
coeff(83) * x(5)**3 + 2.0_DP * coeff(84) * x(2)*x(5) + &
2.0_DP * coeff(85) * x(2)*x(5)**2 + &
3.0_DP * coeff(86) * x(2)**2 * x(5) + &
coeff(99) * x(1) * x(5) + &
2.0_DP *coeff(100) * x(1)*x(2)*x(5) + &
coeff(101) * x(1) * x(5)**2 + &
coeff(102) * x(1) **2 * x(5) + &
coeff(111) * x(3) * x(5) + coeff(112) * x(3)**2 * x(5) + &
coeff(113) * x(3) * x(5)**2 + &
2.0_DP * coeff(114) * x(2) * x(3) * x(5) + &
coeff(115) * x(4) * x(5) + coeff(116) * x(4)**2 * x(5) + &
coeff(117) * x(4) * x(5)**2 + &
2.0_DP * coeff(118) * x(2) * x(4) * x(5) + &
coeff(123) * x(1) * x(3) * x(5) + &
coeff(124) * x(1) * x(4) * x(5) + &
coeff(126) * x(3) * x(4) * x(5)
aux(3) = aux(3) + coeff(87) * x(5) + coeff(88) * x(5)**2 + &
coeff(89) * x(5)**3 + 2.0_DP * coeff(90) * x(3)*x(5) + &
2.0_DP * coeff(91) * x(3)*x(5)**2 + &
3.0_DP * coeff(92) * x(3)**2 * x(5) + &
coeff(103) * x(1) * x(5) + &
2.0_DP *coeff(104) * x(1)*x(3)*x(5) + &
coeff(105) * x(1) * x(5)**2 + &
coeff(106) * x(1) **2 * x(5) + &
coeff(111) * x(2) * x(5) + &
2.0_DP*coeff(112)*x(2)*x(3)*x(5) + &
coeff(113) * x(2) * x(5)**2 + &
coeff(114) * x(2) **2 * x(5) + &
coeff(119) * x(4) * x(5) + &
coeff(120) * x(4) **2 * x(5) + &
coeff(121) * x(4) * x(5)**2 + &
2.0_DP*coeff(122)*x(3)*x(4)*x(5) + &
coeff(123) * x(1) * x(2) * x(5) + &
coeff(125) * x(1) * x(4) * x(5) + &
coeff(126) * x(2) * x(4) * x(5)
aux(4) = aux(4) + coeff(93) * x(5) + coeff(94) * x(5)**2 + &
coeff(95) * x(5)**3 + 2.0_DP * coeff(96) * x(4)*x(5) + &
2.0_DP * coeff(97) * x(4)*x(5)**2 + &
3.0_DP * coeff(98) * x(4)**2 * x(5) + &
coeff(107) * x(1) * x(5) + &
2.0_DP *coeff(108) * x(1)*x(4)*x(5) + &
coeff(109) * x(1) * x(5)**2 + &
coeff(110) * x(1) **2 * x(5) + &
coeff(115) * x(2) * x(5) + &
2.0_DP*coeff(116)*x(2)*x(4)*x(5) + &
coeff(117) * x(2) * x(5)**2 + &
coeff(118) * x(2) **2 * x(5) + &
coeff(119) * x(3) * x(5) + &
2.0_DP*coeff(120)*x(3)*x(4)*x(5) + &
coeff(121) * x(3) * x(5)**2 + &
coeff(122) * x(3) **2 * x(5) + &
coeff(124) * x(1) * x(2) * x(5) + &
coeff(125) * x(1) * x(3) * x(5) + &
coeff(126) * x(2) * x(3) * x(5)
aux(5) = coeff(71) + 2.0_DP * coeff(72) * x(5) + &
3.0_DP * coeff(73) * x(5)**2 + &
4.0_DP * coeff(74) * x(5)**3 + &
coeff(75) * x(1) + &
2.0_DP * coeff(76) * x(1) * x(5) + &
3.0_DP * coeff(77) * x(1) * x(5) **2 + &
coeff(78) * x(1) ** 2 + &
2.0_DP * coeff(79) * x(1) **2 * x(5) + &
coeff(80) * x(1) ** 3 + &
coeff(81) * x(2) + &
2.0_DP * coeff(82) * x(2) * x(5) + &
3.0_DP * coeff(83) * x(2) * x(5) **2 + &
coeff(84) * x(2) ** 2 + &
2.0_DP * coeff(85) * x(2) **2 * x(5) + &
coeff(86) * x(2) ** 3 + &
coeff(87) * x(3) + &
2.0_DP * coeff(88) * x(3) * x(5) + &
3.0_DP * coeff(89) * x(3) * x(5) **2 + &
coeff(90) * x(3) ** 2 + &
2.0_DP * coeff(91) * x(3) **2 * x(5) + &
coeff(92) * x(3) ** 3 + &
coeff(93) * x(4) + &
2.0_DP * coeff(94) * x(4) * x(5) + &
3.0_DP * coeff(95) * x(4) * x(5) **2 + &
coeff(96) * x(4) ** 2 + &
2.0_DP * coeff(97) * x(4) **2 * x(5) + &
coeff(98) * x(4) ** 3 + &
coeff(99) * x(1) * x(2) + &
coeff(100) * x(1) * x(2) **2 + &
2.0_DP * coeff(101) * x(1) * x(2) * x(5) + &
coeff(102) * x(1) **2 * x(2) + &
coeff(103) * x(1) * x(3) + &
coeff(104) * x(1) * x(3) **2 + &
2.0_DP * coeff(105) * x(1) * x(3) * x(5) + &
coeff(106) * x(1) **2 * x(3) + &
coeff(107) * x(1) * x(4) + &
coeff(108) * x(1) * x(4) **2 + &
2.0_DP * coeff(109) * x(1) * x(4) * x(5) + &
coeff(110) * x(1) **2 * x(4) + &
coeff(111) * x(2) * x(3) + &
coeff(112) * x(2) * x(3) **2 + &
2.0_DP * coeff(113) * x(2) * x(3) * x(5) + &
coeff(114) * x(2) **2 * x(3) + &
coeff(115) * x(2) * x(4) + &
coeff(116) * x(2) * x(4) **2 + &
2.0_DP * coeff(117) * x(2) * x(4) * x(5) + &
coeff(118) * x(2) **2 * x(4) + &
coeff(119) * x(3) * x(4) + &
coeff(120) * x(3) * x(4) **2 + &
2.0_DP * coeff(121) * x(3) * x(4) * x(5) + &