discrete derivative

source("findPoints.R")
Y <- genAllPoints(M=2)
#Y

U<-array(dim = c(dim(Y)[1], 2, 2) )
for( i in c(0,1)){
  for( j in c(0,1)){
    Uij<-genUijNew(Y,i,j)
    for (k in c(1:dim(Y)[1])){
      U[k, i+1,j+1]<-Uij[k,k]
    }
  }
}
# for (k in c(1:dim(Y)[1])){
#   print(Y[k,])
#   print(U[k,,])
# }

library("Matrix")
library('rARPACK')

Lmu<- genLmuNew(Y,mu=1)
N<-dim(Lmu)[1]
lam<-eigen(Lmu)$values
# L<- Lmu %o% matrix(c(1, 0, 0, 1), 2, 2, byrow=TRUE)
# Lmu
plot(x=c(1:N) , y=Re(lam))

Lmu[c(1,12,32),c(1,12,32)]

##            [,1]       [,2]       [,3]
## [1,] 0-1.27324i 0+1.27324i 0+0.00000i
## [2,] 0+0.00000i 0-1.27324i 0+1.27324i
## [3,] 0+0.00000i 0-1.27324i 0+1.27324i

derivative

library(ggplot2)
library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

Ns<-c(10,30,50,70,90,200)
l<-list()
df<-data.frame("id"=c(0), "l"=c(0), "N"=c(0), "type"=c(0))
df<-df[-1,]
for (N in Ns){ 
  a<-matrix(rep(0,N*N), nrow=N,ncol=N)
  for(i in c(1:N)){
    for(j in c(1:N)){
      if(j==(i+1)%%N)  a[i,j]<-  1
      if(j==(i-1)%%N)  a[i,j]<- 0
      if(j==N && i==1) a[i,j]<- 0
      if(j==N && i==N-1) a[i,j]<-1
      a[i,i]<- -1
    }
  }
  a<--1i*a/2

  a2<- (a+Conj(t(a)))
  a2<- a2%*%a2
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a+ad)",N) ))
  
  a2<- a%*%Conj(t(a))
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a*ad)",N) ))
}
gg<-ggplot(df)+theme_bw()
gg<- gg+ geom_point(aes(x=id, y=l, shape=as.factor(type), color=as.factor(N) ))
gg<- gg+scale_shape_manual(values=c(5,3))
fig<-ggplotly(gg)
fig

#print(htmltools::tagList(fig))

Ns<-c(10,30,50,70,90,200)
l<-list()
count<-1
df<-data.frame("id"=c(0), "l"=c(0), "N"=c(0))
df<- df[-1,]
for (N in Ns){ 
  a<-matrix(rep(0,N*N), nrow=N,ncol=N)
  for(i in c(1:N)){
    for(j in c(1:N)){
      if(j==(i+1)%%N)  a[i,j]<-  1
      if(j==(i-1)%%N)  a[i,j]<- -1
      if(j==N && i==1) a[i,j]<- -1
      if(j==N && i==N-1) a[i,j]<- 1
    }
  }
  
  
  # a-Conj(t(a))
  # print(a)
  a<- -1i*a/2
  a2<- a%*%a
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N) ))
  # plot(x=c(1:N) , y=l[[count]])
  # count<- count+1
}
gg<-ggplot(df)+theme_bw()
gg<- gg+ geom_point(aes(x=id, y=l, color=as.factor(N), shape=as.factor(N) ))
gg<- gg#+xlim(0,40)

fig<-ggplotly(gg)
fig

Mixing bakward and forward derivative randomly

Here we want to check what happen to the eigenvalues of \(\hat P^2=(-id/dx)\) when we discretise the derivative mixing randomly forward and backward derivatives at each point

library(ggplot2)
library(plotly)

forward_deriv_row<-function(i,N){
  row<-rep(0,N)
  for(j in c(1:N)){
      if(j==(i+1)%%N)  row[j]<-   1
      if(j==(i-1)%%N)  row[j]<-   0
      if(j==N && i==1) row[j]<-   0
      if(j==N && i==N-1) row[j]<- 1
  }
  row[i]<- -1
  return(row)
}

backward_deriv_row<-function(i,N){
  row<-rep(0,N)
  for(j in c(1:N)){
      if(j==(i+1)%%N)  row[j]<-   0
      if(j==(i-1)%%N)  row[j]<-   -1
      if(j==N && i==1) row[j]<-   -1
      if(j==N && i==N-1) row[j]<- 0
  }
  row[i]<- -1
  return(row)
}

Ns<-c(10,30,50,70,90,200)
l<-list()
df<-data.frame("id"=c(0), "l"=c(0), "N"=c(0), "type"=c(0))
df<-df[-1,]
for (N in Ns){ 
  a<-matrix(rep(0,N*N), nrow=N,ncol=N)
  for(i in c(1:N)){
    r<-runif(1)
    if(r<0.5) a[i,]<-forward_deriv_row(i,N)
    else        a[i,]<-backward_deriv_row(i,N)
  }
  a<- -1i*a/2

  a2<- a+Conj(t(a))
  a2<- a2%*%a2
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a+ad)",N) ))
  
  a2<- a%*%Conj(t(a))
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a*ad)",N) ))
}
gg<-ggplot(df)+theme_bw()
gg<- gg+ geom_point(aes(x=id, y=l, shape=as.factor(type), color=as.factor(N) ),
                    )
gg<- gg+scale_shape_manual(values=c(5,3))
fig<-ggplotly(gg)
fig

Alternating bakward and forward derivative randomly

Here we choose: - forward derivative for the \(x \mod 2=1\) - backward derivative for the \(x \mod 2=0\)

Ns<-c(10,30,50,70,90,200)
l<-list()
df<-data.frame("id"=c(0), "l"=c(0), "N"=c(0), "type"=c(0))
df<-df[-1,]
for (N in Ns){ 
  a<-matrix(rep(0,N*N), nrow=N,ncol=N)
  for(i in c(1:N)){
    if(i%%2==1) a[i,]<-forward_deriv_row(i,N)
    else        a[i,]<-backward_deriv_row(i,N)
  }
  a<- -1i*a/2

  a2<- a+Conj(t(a))
  a2<- a2%*%a2
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a+ad)",N) ))
  
  a2<- a%*%Conj(t(a))
  l<-eigen(a2)$values[N:1]
  df<-rbind(df,data.frame("id"=c(1:N), "l"=l, "N"=rep(N,N), "type"=rep("(a*ad)",N) ))
}
gg<-ggplot(df)+theme_bw()
gg<- gg+ geom_point(aes(x=id, y=l, shape=as.factor(type), color=as.factor(N) ),
                    )
gg<- gg+scale_shape_manual(values=c(5,3))
fig<-ggplotly(gg)
fig