update readme, add source files

43f1d3be · Lilli Frison · fcd72d31 · fcd72d31 · 43f1d3be · 43f1d3be
Commit 43f1d3be authored Nov 28, 2019 by Lilli Frison
29 changed files
--- a/README.md
+++ b/README.md
--- a/README.txt
+++ b/README.txt
+CONTENTS
+   Implementation of gPC (uniform, normal and truncated normal distribution) for 
+   solving optimal control problems with uncertain parameters and initial conditions.
+   This repository only contains the gPC extension files that must be used on top
+   of a numerical optimal control solver (at this time only MUSCOD-II possible).
+   In the future, we hope make an open source substitute publicly available.
+   
+src-pce
+	Contains necessary extension files for gPC projection. 
+
+REQUIREMENTS
+
+	A copy of the source tree of the MUSCOD-II optimal control problem solver is 
+	required to use the uocp extension. 
+	
+	As MUSCOD-II is not in the public domain, we are unfortunately bound by a 
+   closed-source license at this time and its source code cannot be provided. 
+   In the future, we hope make an open source substitute publicly available that 
+   is currently under active development, and we hope to use ICMS (International 
+   Conference on Mathematical Software) as one possible outlet.
+
+examples
+	A collection of OCPs with uncertain parameters and initial conditions. NISP (non-intrusive spectral projection) is implemented on top of problem equations.
+   - van der Pol problem (Bock/Plitt, 1984)
+   - container crane (Goh/Teo, 1988)
+   - fermentation process
+	
--- a/examples/ccrane0.cpp
+++ b/examples/ccrane0.cpp
+/*
+ *
+ *  MUSCOD-II/MC2_TEST/SRC/ccrane.c
+ *  (c) Daniel B. Leineweber, 1995
+ *
+ *  container crane (Goh/Teo, 1988)
+ *
+ *  $Id: ccrane.cpp 725 2011-08-10 23:04:16Z ckirches $
+ *
+ */
+
+#include <cmath>
+
+#include "def_usrmod.hpp"
+
+#define  NMOS   1
+#define  NP     0
+#define  NRC    0
+#define  NRCE   0
+
+#define  NXD    7
+#define  NXA    0
+#define  NU     2
+#define  NPR    0
+
+
+static void lfcn(double *t, double *xd, double *xa, double *u,
+  double *p, double *lval, double *rwh, long *iwh, InfoPtr *info)
+{
+ *lval = 0.5*(xd[2]*xd[2] + xd[5]*xd[5]);
+ // *lval = 0.5*(xd[2]*xd[2] + xd[5]*xd[5]) + 0.01*u[0]*u[0] + 0.1*u[1]*u[1];
+}
+
+static void mfcn(double *ts, double *sd, double *sa, double *p, double *pr, double *mval, long *dpnd, InfoPtr *info)
+{
+  if (*dpnd) { *dpnd = MFCN_DPND(0, *sd, 0, 0, 0); return; }
+  *mval = sd[NXD-1];
+}
+
+static void ffcn(double *t, double *xd, double *xa, double *u,double *p, double *rhs, double *rwh, long *iwh, InfoPtr *info)
+{
+  rhs[0] = xd[3];
+  rhs[1] = xd[4];
+  rhs[2] = xd[5];
+  rhs[3] = u[0] + 1.76*9.81*xd[2];
+  rhs[4] = u[1];
+  rhs[5] = -(u[0] + (1.0+1.76)*9.81*xd[2] + 2*xd[4]*xd[5])/xd[1];
+  rhs[NXD-1] = 0.5*(xd[2]*xd[2] + xd[5]*xd[5] + 1e-4*u[1]*u[1]);
+}
+
+
+static void rdfcnRe(double *ts, double *sd, double *sa, double *u, double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+    if (*dpnd) { *dpnd = RFCN_DPND(0, *sd, 0, 0, 0, *pr); return; }
+    res[0] = sd[0] - 9.9;
+    res[1] = sd[1] - 13.9;
+    res[2] = sd[2] + 0.001;
+    res[3] = sd[3] - 2.4;
+    res[4] = sd[4] + 0.001;
+    res[5] = sd[5] + 0.001;
+    res[6] = -sd[0] + 11.0;
+    res[7] = -sd[1] + 14.1;
+    res[8] = -sd[2] + 0.001;
+    res[9] = -sd[3] + 2.6;
+    res[10] = -sd[4] + 0.001;
+    res[11] = -sd[5] + 0.001;
+}
+
+
+
+/* -------------------------------------------- */
+extern "C" void def_model(void);
+void def_model(void)
+{
+  def_mdims(NMOS, NP, NRC, NRCE);
+  def_mstage(0,NXD, NXA, NU, mfcn, NULL, 0, 0, 0, NULL, ffcn, NULL,NULL, NULL);
+  def_mpc(0, "End Point", NPR, 12,0, rdfcnRe, NULL);
+  //def_mpc(0, "Inner Point", NPR, 2,0, rdfcnR, NULL);
+}
+
+
+/* old */
+static void rdfcn_s(double *ts, double *sd, double *sa, double *u,
+  double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+  if (*dpnd) {
+    *dpnd = RFCN_DPND(0, *sd, 0, 0, 0, 0);
+    return;
+  }
+
+  res[0] = sd[0];
+  res[1] = sd[1] - 22.0;
+  res[2] = sd[2];
+  res[3] = sd[3];
+  res[4] = sd[4] + 1.0;
+  res[5] = sd[5];
+}
+
+static void rdfcn_e(double *ts, double *sd, double *sa, double *u,double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+  if (*dpnd) {
+    *dpnd = RFCN_DPND(0, *sd, 0, 0, 0, 0);
+    return;
+  }
+  res[0] = sd[0] - 10.0;
+  res[1] = sd[1] - 14.0;
+  res[2] = sd[2];
+  res[3] = sd[3] - 2.5;
+  res[4] = sd[4];
+  res[5] = sd[5];
+}
+
--- a/examples/ccrane0_lin.dat
+++ b/examples/ccrane0_lin.dat
--- a/examples/ccraneAD.cpp
+++ b/examples/ccraneAD.cpp
--- a/examples/ccraneNISP.cpp
+++ b/examples/ccraneNISP.cpp
--- a/examples/ccraneNISP_lin.dat
+++ b/examples/ccraneNISP_lin.dat
--- a/examples/fermenter0.cpp
+++ b/examples/fermenter0.cpp
+/* model:  Model is adapted for simulation study by IWR
+ * author: Dr. Hergen Schultze, GVC/S - B009, BASF AG
+ * date:   2006-02-17
+ */
+
+#include <math.h>
+#include <stdio.h>
+#include "def_usrmod.hpp"
+
+#define  NMOS    1
+#define  NXD     9
+#define  NXA     0
+#define  NU      3
+#define  NP      7
+#define  NPR     0
+
+#define  NRC     0
+#define  NRCE    0
+
+#define  NRD_E   1
+#define  NRDE_E  1
+
+
+static void ffcn(double *t, double *xd, double *xa, double *u,
+		 double *p, double *rhs, double *rwh, long *iwh, InfoPtr *info)
+{
+/*
+c     * x(1) <--> Pr ... product concentration
+c     * x(2) <--> S1 ... (fermentable) substrate1 concentration
+c     * x(3) <--> S2 ... substrate2 concentration
+c     * x(4) <--> En ... (living) biomass concentration
+c     * x(5) <--> Vo ... (absolute) volume
+c     * x(6) <--> Go ... goodies (out of batch) concentration
+c                 #P
+c                 #S1
+c                 #S2
+c
+c     * u(1) <--> FS1 ... feed rate substrate1
+c     * u(2) <--> FS2 ... feed rate substrate2
+c     * u(3) <--> Zw  ... early harvest rate
+c
+c     * p(1) <--> mu_X10    ... growth rate biomass
+c     * p(2) <--> mu_P10    ... production rate
+c     * p(3) <--> Gamma_XS1 ... consumption of substrate1 by growth
+c     * p(4) <--> Gamma_PS1 ... consumption of substrate1 by production
+c     * p(5) <--> Gamma_PS2 ... consumption of substrate2 by production
+c     * p(6) <--> Gamma_XS2 ... consumption of substrate2 by growth
+c     * p(7) <--> Gamma_XG  ... consumption of goodies by growth
+*/
+
+    double pr, s1, s2, en, vo, go;
+    double mu_x10, mu_p10, gamma_xs1, gamma_ps1, gamma_ps2, gamma_xs2, gamma_xg;
+    double fs1, fs2, zw, s11, s22;
+
+    /* differential states in right hand side */
+    pr = xd[0];
+    s1 = xd[1];
+    s2 = xd[2];
+    en = xd[3];
+    vo = xd[4];
+    go = xd[5];
+
+    /* parameters */
+    mu_x10   = p[0] * 1e4;
+    mu_p10   = p[1] * 1e4; // 1e3;
+    gamma_xs1= p[2] * 1e4;
+    gamma_ps1= p[3] * 1e4; // 1e3;
+    gamma_ps2= p[4] * 1e4; // 1e3;
+    gamma_xs2= p[5] * 1e4;
+    gamma_xg = p[6] * 1e4;
+
+    /* process controls */
+    fs1= u[0] / 25.0;
+    fs2= u[1] / 25.0;
+    zw = u[2] / 25.0;
+
+    /* intern parameters */
+    s11 = 42.0 / 1e+2;
+    s22 = 33.3 / 1e+2;
+    
+    rhs[0] = mu_p10 * en * s1 * s2 - pr * (fs1 + fs2) / vo;  // product concentration
+    rhs[1] = - gamma_xs1 * en * s1 * s2 * go  - gamma_ps1 * en * s1 * s2    + s11 * fs1 / vo - s1 * (fs1 + fs2) / vo; // substrate1 fermentable
+    rhs[2] = - gamma_ps2 * en * s1 * s2 - gamma_xs2 * en * s1 * s2 * go  + s22 * fs2 / vo - s2 * (fs1 + fs2) / vo; // substrate2
+    rhs[3] = mu_x10 * en * s1 * s2 * go  - en * (fs1 + fs2) / vo; // cell volume
+    rhs[4] = fs1 + fs2 - zw; // fermentation volume
+    rhs[5] = - gamma_xg * en * s1 * s2 * go - go * (fs1 + fs2) / vo; // goodies
+    rhs[6] = zw * pr + (fs1 + fs2 - zw) * pr + vo * (mu_p10 * en * s1 * s2 - pr * (fs1 + fs2)/vo); // product accumulated
+    rhs[7] = fs1 * s11; // substrate1 accumulated
+    rhs[8] = fs2 * s22; // substrate2 accumulated
+    
+ // product accumulated:
+    // #P = int (z * p) + v * p
+    // dP = z * p + dv * p + v * dp
+   
+    return;
+}
+
+static void iffcn(double *t, double *xd, double *xdd, double *xa, double *u,double *p, double *rhs, double *rwh, long *iwh, InfoPtr *info)
+{
+
+    double pr, s1, s2, en, vo, go;
+    double mu_x10, mu_p10, gamma_xs1, gamma_ps1, gamma_ps2, gamma_xs2, gamma_xg;
+    double fs1, fs2, zw, s11, s22;
+
+    /* differential states in right hand side */
+    pr = xd[0];
+    s1 = xd[1];
+    s2 = xd[2];
+    en = xd[3];
+    vo = xd[4];
+    go = xd[5];
+
+    /* parameters */
+    mu_x10   = p[0] * 1e4;
+    mu_p10   = p[1] * 1e4; // 1e3;
+    gamma_xs1= p[2] * 1e4;
+    gamma_ps1= p[3] * 1e4; // 1e3;
+    gamma_ps2= p[4] * 1e4; // 1e3;
+    gamma_xs2= p[5] * 1e4;
+    gamma_xg = p[6] * 1e4;
+
+    /* process controls */
+    fs1= u[0] / 25.0;
+    fs2= u[1] / 25.0;
+    zw = u[2] / 25.0;
+
+    /* intern parameters */
+    s11 = 42.0 / 1e+2;
+    s22 = 33.3 / 1e+2;
+
+    
+    rhs[0] = mu_p10 * en * s1 * s2 - pr * (fs1 + fs2) / vo - xdd[0];  // product concentration
+    rhs[1] = - gamma_xs1 * en * s1 * s2 * go  - gamma_ps1 * en * s1 * s2    + s11 * fs1 / vo - s1 * (fs1 + fs2) / vo - xdd[1]; // substrate1 fermentable
+    rhs[2] = - gamma_ps2 * en * s1 * s2 - gamma_xs2 * en * s1 * s2 * go  + s22 * fs2 / vo - s2 * (fs1 + fs2) / vo - xdd[2]; // substrate2
+    rhs[3] = mu_x10 * en * s1 * s2 * go  - en * (fs1 + fs2) / vo - xdd[3]; // cell volume
+    rhs[4] = fs1 + fs2 - zw - xdd[4]; // fermentation volume
+    rhs[5] = - gamma_xg * en * s1 * s2 * go - go * (fs1 + fs2) / vo - xdd[5]; // goodies
+    rhs[6] = zw * pr + (fs1 + fs2 - zw) * pr + vo * (mu_p10 * en * s1 * s2 - pr * (fs1 + fs2)/vo) - xdd[6]; // product accumulated
+    rhs[7] = fs1 * s11 - xdd[7]; // substrate1 accumulated
+    rhs[8] = fs2 * s22 - xdd[8]; // substrate2 accumulated
+    //rhs[6] =  vo * mu_p10 * en * s1 * s2 - xdd[6];
+}
+
+
+static void mfcn(double *ts, double *sd, double *sa,
+		 double *p, double *pr, double *mval, long *dpnd, InfoPtr *info)
+{
+    if (*dpnd) {*dpnd = MFCN_DPND(0, *sd, 0, 0, 0);	return;}
+/*
+    double p_sub1, p_sub2; // price
+    double a_pro, a_sub1, a_sub2; // amount
+    p_sub1 = 1.0;
+    p_sub2 = 0.5;
+    a_pro  = sd[6];
+    a_sub1 = sd[7];
+    a_sub2 = sd[8];
+    // variable costs:
+    *mval = (p_sub1 * a_sub1 + p_sub2 * a_sub2) / a_pro;*/
+    
+    *mval = (sd[7] + 0.5*sd[8])/sd[6];
+}
+
+static void rdfcnR(double *ts, double *sd, double *sa, double *u, double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+    if (*dpnd) {*dpnd = RFCN_DPND(0, *sd, 0, 0, 0, 0); return;}
+    
+    res[0] = 0.04 - sd[1];
+}
+
+/* main function */
+extern "C" void def_model(void){
+    def_mdims(NMOS, NP, NRC, NRCE);
+    def_mstage(0,	NXD, NXA, NU,	mfcn, NULL,	0, 0, 0, NULL, ffcn, NULL,	NULL, NULL	);
+    //def_imstage( 0, NXD, NXA, NU, 0, 0, 0, 0, mfcn, 2, iffcn, NULL,NULL, NULL, NULL, NULL);
+    //def_mpc( 0, "I", NPR, 1, 0, rdfcnR, NULL );
+    //def_mpc( 0, "E", NPR, 1, 0, rdfcnR, NULL );
+}
+
+
+
+/* unsued */
+static void rdfcn_i(double *ts, double *sd, double *sa, double *u, double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+  if (*dpnd) {*dpnd = RFCN_DPND(0, *sd, 0, 0, 0, 0); return;}
+  
+  res[0] =  0.1 - sd[0]; // lb critical 
+  res[1] =  0.06 - sd[1]; // ub critical (0.04 orig)
+  res[2] =  0.03 - sd[2];
+  res[3] =  0.1 - sd[3]; // Cell Volume
+  res[4] =  0.45 - sd[4]; // deterministic anyway
+  res[5] =  0.1 - sd[5];
+  res[6] =  0.05 - sd[6];
+  res[7] =  0.2 - sd[7];
+  res[8] =  0.025 - sd[8];
+}
+
--- a/examples/fermenter0_lin.dat
+++ b/examples/fermenter0_lin.dat
--- a/examples/fermenterAD.cpp
+++ b/examples/fermenterAD.cpp
--- a/examples/fermenterNISP.cpp
+++ b/examples/fermenterNISP.cpp
--- a/examples/fermenterNISP_lin.dat
+++ b/examples/fermenterNISP_lin.dat
--- a/examples/vdpol0.cpp
+++ b/examples/vdpol0.cpp
+/*
+ *
+ *  MUSCOD-II/MC2_TEST/SRC/vdpol.c
+ *  (c) Daniel B. Leineweber, 1995
+ *
+ *  van der Pol problem (Bock/Plitt 1984)
+ *
+ *  $Id: vdpol.cpp 725 2011-08-10 23:04:16Z ckirches $
+ *
+ */
+
+#include <cmath>
+#include "def_usrmod.hpp"
+
+#define  NMOS   1
+#define  NP     0
+#define  NRC    0
+#define  NRCE   0
+
+#define  NXD    2
+#define  NXA    0
+#define  NU     1
+#define  NPR    0
+
+#define  NRD_E  1
+#define  NRDE_E 1
+
+
+
+static void lfcn(double *t, double *xd, double *xa, double *u,
+  double *p, double *lval, double *rwh, long *iwh, InfoPtr *info)
+{
+  *lval = 0.5*(xd[0]*xd[0] + xd[1]*xd[1] + u[0]*u[0]);
+}
+
+static void ffcn(double *t, double *xd, double *xa, double *u,
+  double *p, double *rhs, double *rwh, long *iwh, InfoPtr *info)
+{
+  rhs[0] = xd[1];
+  rhs[1] = -xd[0] + (1.0 - xd[0]*xd[0])*xd[1] + u[0];
+}
+
+const double EPS = 0.0;
+static void rdfcn_e(double *ts, double *sd, double *sa, double *u,
+  double *p, double *pr, double *res, long *dpnd, InfoPtr *info)
+{
+  if (*dpnd) {
+    *dpnd = RFCN_DPND(0, *sd, 0, 0, 0, 0);
+    return;
+  }
+   res[0] = sd[1] - sd[0] - 1.0;
+   //	res[0] = sd[1] - sd[0] - 1.0 + EPS;
+   //	res[1] = -(sd[1] - sd[0] - 1.0) + EPS;
+}
+
+extern "C" void def_model(void)
+{
+  def_mdims(NMOS, NP, NRC, NRCE);
+  def_mstage(
+    0,
+    NXD, NXA, NU,
+    NULL, lfcn,
+    0, 0, 0, NULL, ffcn, NULL,
+    NULL, NULL
+  );
+  def_mpc(0, "End Point", NPR, 1,0, rdfcn_e, NULL);
+}
+
--- a/examples/vdpol0.dat
+++ b/examples/vdpol0.dat
+*
+*
+*  MUSCOD-II/MDAT/vdpol.dat
+*  (c) Daniel B. Leineweber, 1995
+*
+*  van der Pol problem (Bock/Plitt, 1984)
+*
+*  $Id: vdpol.dat 658 2009-06-24 14:16:16Z potschka $
+*
+*
+
+
+*  # of multiple shooting intervals on each model stage
+nshoot
+0: 20
+
+
+*  model stage duration start values, scale factors, and bounds
+h
+0: 5.0
+
+h_sca
+0: 5.0
+
+h_min
+0: 5.0
+
+h_max
+0: 5.0
+
+
+*  model stage duration fixed value flags
+h_fix
+0: 1
+
+
+*  differential state start values, scale factors, and bounds
+sd(*,*)
+0: 1.0
+1: 0.0
+
+sd_fix(0,0)
+ALL: 1
+
+sd_sca(*,*)
+ALL: 1
+
+sd_min(*,*)
+0: -10.0
+1: -10.0
+
+sd_max(*,*)
+0: 10.0
+1: 10.0
+
+*  control parameterization types 
+u_type(*)
+0: 0
+
+*  control start values, scale factors, and bounds
+u(*,*)
+0: 0.5
+
+u_sca(*,*)
+0: 1.0
+
+u_min(*,*)
+0: -2.0
+
+u_max(*,*)
+0: 2.0
+
+
+*  control slope start values, scale factors, and bounds
+udot(*,*)
+0: 0.0
+
+udot_sca(*,*)
+0: 1.0
+
+udot_min(*,*)
+0: -10.0
+
+udot_max(*,*)
+0: 10.0
+
+
+*  decoupled i.p.c. scale factors
+rd_sca(0,S)
+0: 1.0
+1: 1.0
+
+rd_sca(0,E)
+0: 1.0
+1: 1.0
+
+*  objective scale and expected range; # of values in history plot
+of_sca
+1.0
+
+of_min
+0.0
+
+of_max
+10.0
+
+nhist
+30
+
+
+
+**********************
+* Choosing libraries *
+**********************
+
+libmodel
+SRC/libvdpol0
+
+libhessian
+hess_update
+
+libsolve
+solve_tbox
+
+libcond
+cond_std
+
+libtchk
+tchk
+
+libmssqp
+mssqp_standard
+
+libeval
+eval_ind
+
+libind
+0: ind_rkfswt
+
+libqps
+qps_qpopt
+
+libplot
+plot_noplot
+
+**********************************
+* Setting algorithmic parameters *
+**********************************
+
+options_acc
+1e-6
+options_ftol
+-1.0
+options_itol
+-1.0
+options_rfac
+0.0
+options_levmar
+0.0
+options_qp_featol
+1.0e-8
+options_qp_relax
+1.1
+options_nhtopy
+0
+options_frstart
+0
+options_frmax
+0
+options_itmax
+100
+options_plevel_screen
+0
+options_plevel_file
+1
+options_plevel_matlab
+0
+options_bflag
+-1
+options_qp_itmax
+10000
+options_qp_expand
+99999999
+options_sflag
+0
+options_options_wflag
+0
+options_cflag
+0
+options_output_ps
+0
+options_output_gif
+0
+
--- a/examples/vdpolAD.cpp
+++ b/examples/vdpolAD.cpp
--- a/examples/vdpolNISP.cpp
+++ b/examples/vdpolNISP.cpp
--- a/examples/vdpolNISP.dat
+++ b/examples/vdpolNISP.dat
--- a/src-pce/hermitepoly.cpp
+++ b/src-pce/hermitepoly.cpp
--- a/src-pce/hermitepoly.h
+++ b/src-pce/hermitepoly.h
+extern const double C;
+extern const long T[];
+extern double W[],X[];
+
+void fnisp(int k, int NFUN, int M, double *W, double ** evalpoly, double **rhs, double *res);
+void dfnisp(int k, int d, int NFUN, int M, double * W, double ** evalpoly,  double **rhs,double *res);
+
+/* non-normalized version */
+void fnisp0(int k, int NFUN, int M, double *W, double ** evalpoly, double **rhs, double *res);
+void dfnisp0(int k, int d, int NFUN, int M, double * W, double ** evalpoly,  double **rhs,double *res);
+
+double evalpce(int ND, int k, double *x, double *xi, int NORM);
+double evaldpce(int ND, int k, double *x, double *xi, int NORM);
+double evald2pce(int ND, int k, double *x, double *xi, int NORM);
+
+double evalhermite(int d, int k, double *xi, int NORM);
+
+/* ND=1 */
+//void hermitequad(int k, int m, double x[], double w[]);
+double evalpce_ND1(int k, double *x, double *xi);
+double evaldpce_ND1(int k, double *x, double *xi);
+double evald2pce_ND1(int k, double *x, double *xi);
+double evalpcenorm_ND1(int k, double *x, double *xi);
+double evaldpcenorm_ND1(int k, double *x, double *xi);
+double evald2pcenorm_ND1(int k, double *x, double *xi);
+
+double evalhermite_ND1(int k, double *xi);
+double evalhermitenorm_ND1(int k, double *xi);
+double evaltrunchermite_ND1(int k, double *xi);
+double evaltruncpce_ND1(int k, double *x, double *xi);
+
+/* ND=2 */
+double evalpce_ND2(int k, double *x, double *xi);
+double evalpcenorm_ND2(int k, double *x, double *xi);
+double evaldpce_ND2(int k, double *x, double *xi);
+double evaldpcenorm_ND2(int k, double *x, double *xi);
+
+double evalhermite_ND2(int k, double *xi);
+double evalhermitenorm_ND2(int k, double *xi);
+double evaltrunchermite_ND2(int k, double *xi);
+double evaltruncpce_ND2(int k, double *x, double *xi);
+
+/* ND=3 */
+//double evalpce_ND3(int k, double *x, double *xi);
+double evalpcenorm_ND3(int k, double *x, double *xi);
+//double evaldpce_ND3(int k, double *x, double *xi);
+
+//double evalhermite_ND3(int k, double *xi);
+double evalhermitenorm_ND3(int k, double *xi);
+//double evaldpcenorm_ND3(int k, double *x, double *xi);
--- a/src-pce/legendrepoly.cpp
+++ b/src-pce/legendrepoly.cpp
--- a/src-pce/legendrepoly.h
+++ b/src-pce/legendrepoly.h
+extern double W[],X[],T[];
+
+void fnisp(int k, int NFUN, int M, double *W, double ** evalpoly, double **rhs, double *res);
+void dfnisp(int k, int d, int NFUN, int M, double * W, double ** evalpoly,  double **rhs,double *res);
+
+double evalpce(int ND, int k, double *x, double *xi, int NORM);
+double evaldpce(int ND, int k, double *x, double *xi, int NORM);
+//double evald2pce(int ND, int k, double *x, double *xi, int NORM)
+
+double evallegendre(int d, int k, double *xi, int NORM);
+
--- a/src-pce/normalrule.h
+++ b/src-pce/normalrule.h
+/******* ND=1 *******/
+//double X[1] = {0.0};
+
+//double X[5] = {-2.85697001387280558049, -1.35562617997426571037, -0.00000000000000003599, 1.35562617997426615446, 2.85697001387280558049};
+//double X[7] = {-3.75043971772574113999, -2.36675941073454065844, -1.15440539473996817144, -0.00000000000000008676, 1.15440539473996817144, 2.36675941073454021435, 3.75043971772574113999};
+//double X[12] = {-5.50090170446774529012, -4.27182584793228325992, -3.22370982877009648604, -2.25946445100079928991, -1.34037519715161645983, -0.44440300194413895341, 0.44440300194413884238, 1.34037519715161823619, 2.25946445100079840174, 3.22370982877009959466, 4.27182584793227970721, 5.50090170446774529012}; 
+//double  X[18] = {-7.13946484914648, -6.007745911359597, -5.05407268544274, -4.188020231629403, -3.374736535778091, -2.595833688911241, -1.839779921508647, -1.098395518091501, -0.3652457555076976, 0.3652457555076979, 1.098395518091501, 1.839779921508645, 2.59583368891124, 3.37473653577809, 4.188020231629404, 5.054072685442738, 6.007745911359601, 7.139464849146481};
+double X[20] = {-7.61904854167975464918, -6.51059015701365240147, -5.57873880589320503276, -4.73458133404605518990, -3.94396735065731762759, -3.18901481655339003041, -2.45866361117236653655, -1.74524732081412636830, -1.04294534880275069355, -0.34696415708135580624, 0.34696415708135608380, 1.04294534880275113764, 1.74524732081412614626, 2.45866361117236831291, 3.18901481655339091859, 3.94396735065731629533, 4.73458133404605430172, 5.57873880589320414458, 6.51059015701365151330, 7.61904854167975997825};
+/* N=30 */
+//double  X[28] = {-8.680837722732214, -7.825051744352809, -7.055396866960289, -6.339997686869598, -5.662381850082875, -5.012600596486519, -4.384020365898052, -3.771894423159235, -3.172634639420403, -2.583402100229275, -2.001858612956431, -1.426005658374114, -0.8540733517109721, -0.2844387607362086, 0.2844387607362088, 0.8540733517109731, 1.426005658374115, 2.001858612956432, 2.583402100229273, 3.172634639420402, 3.771894423159237, 4.384020365898054, 5.012600596486519, 5.662381850082874, 6.339997686869599, 7.055396866960293, 7.825051744352809, 8.680837722732212};
+//double  X[30] = {-9.706235997359524, -8.680837722732214, -7.825051744352809, -7.055396866960289, -6.339997686869598, -5.662381850082875, -5.012600596486519, -4.384020365898052, -3.771894423159235, -3.172634639420403, -2.583402100229275, -2.001858612956431, -1.426005658374114, -0.8540733517109721, -0.2844387607362086, 0.2844387607362088, 0.8540733517109731, 1.426005658374115, 2.001858612956432, 2.583402100229273, 3.172634639420402, 3.771894423159237, 4.384020365898054, 5.012600596486519, 5.662381850082874, 6.339997686869599, 7.055396866960293, 7.825051744352809, 8.680837722732212, 9.706235997359524};
+/* N=40 */
+//double X[34] = {-8.94950454385555538295, -8.27894062365947291937, -7.64616376454145907360, -7.04173840645383286585, -6.45942337758376794454, -5.89480567537201594064, -5.34460544572008444675, -4.80628719209387256228, -4.27782615636274776705, -3.75755977616898428906, -3.24408873299987066119, -2.73620834046543182083, -2.23285921863487191175, -1.73309059063172132831, -1.23603200479915797949, -0.74087072528593000964, -0.24683289602272429075, 0.24683289602272487362, 0.74087072528593178600, 1.23603200479916019994, 1.73309059063172243853, 2.23285921863487235584, 2.73620834046543048856, 3.24408873299987066119, 3.75755977616898562133, 4.27782615636274954340, 4.80628719209387256228, 5.34460544572008622310, 5.89480567537201860517, 6.45942337758376794454, 7.04173840645382842496, 7.64616376454145996178, 8.27894062365947469573, 8.94950454385555538295};
+
+
+/*normalized */
+const double C = 1.0;  // scaling factor for weights 
+const long int  T[32] = {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1};
+
+//double W[1] = {1.0};
+//double W[5] = {0.01125741132772069969, 0.22207592200561251938, 0.53333333333333321491, 0.22207592200561246387, 0.01125741132772068581};
+//double W[7] = {0.00054826885597221984, 0.03075712396758649436, 0.24012317860501264377, 0.45714285714285690654, 0.24012317860501275479, 0.03075712396758662967, 0.00054826885597221865};
+//double W[12] = {0.00000014999271676372, 0.00004837184922590639, 0.00220338068753319748, 0.02911668791236416212, 0.14696704804533000654, 0.32166436151283034350, 0.32166436151283034350, 0.14696704804532972899, 0.02911668791236421069, 0.00220338068753319532, 0.00004837184922590633, 0.00000014999271676372};
+//double W[18] = {0.00000000000441658877, 0.00000000590548847884, 0.00000102155239763698, 0.00005179896144116208, 0.00106548479629165068, 0.01051651775194133338, 0.05489663248022256387, 0.16068530389351259879, 0.27278323465428777617, 0.27278323465428755412, 0.16068530389351307064, 0.05489663248022253611, 0.01051651775194134032, 0.00106548479629165480, 0.00005179896144116197, 0.00000102155239763699, 0.00000000590548847884, 0.00000000000441658877};
+double W[20] = {0.00000000000012578007, 0.00000000024820623623, 0.00000006127490259983, 0.00000440212109023081, 0.00012882627996192966, 0.00183010313108048677, 0.01399783744710104451, 0.06150637206397697315, 0.16173933398400003325, 0.26079306344955488495, 0.26079306344955477392, 0.16173933398399958916, 0.06150637206397704254, 0.01399783744710100808, 0.00183010313108048677, 0.00012882627996192923, 0.00000440212109023086, 0.00000006127490259983, 0.00000000024820623623, 0.00000000000012578007};
+//double W[28] = {0.00000000000000001586, 0.00000000000001624080, 0.00000000000457342587, 0.00000000051784594672, 0.00000002882175154048, 0.00000089090888686211, 0.00001657998163067345, 0.00019651294398482469, 0.00154470733986608860, 0.00829574755772326082, 0.03111177018350123891, 0.08278683671562220292, 0.15804695320902073519, 0.21799997181557764780, 0.21799997181557731474, 0.15804695320902087396, 0.08278683671562188373, 0.03111177018350117993, 0.00829574755772333715, 0.00154470733986608903, 0.00019651294398482407, 0.00001657998163067352, 0.00000089090888686212, 0.00000002882175154048, 0.00000000051784594672, 0.00000000000457342587, 0.00000000000001624080, 0.00000000000000001586};
+//double W[30] = {0.00000000000000000000, 0.00000000000000001586, 0.00000000000001624080, 0.00000000000457342587, 0.00000000051784594672, 0.00000002882175154048, 0.00000089090888686211, 0.00001657998163067345, 0.00019651294398482469, 0.00154470733986608860, 0.00829574755772326082, 0.03111177018350123891, 0.08278683671562220292, 0.15804695320902073519, 0.21799997181557764780, 0.21799997181557731474, 0.15804695320902087396, 0.08278683671562188373, 0.03111177018350117993, 0.00829574755772333715, 0.00154470733986608903, 0.00019651294398482407, 0.00001657998163067352, 0.00000089090888686212, 0.00000002882175154048, 0.00000000051784594672, 0.00000000000457342587, 0.00000000000001624080, 0.00000000000000001586, 0.00000000000000000000};
+
+//N=40 points