# "Power-Method"
powerMethod <- function(A, r0 = rep(1, nrow(A)), treshold = 10^(-10))
{
  r_alt <- r0
  matExp <- A
  for (i in 2:10000)
  {
    matExp <- matExp%*%A
    r <- matExp%*%r0/norm(matExp%*%r0,'F')
    #print(r)
    if (norm(r-r_alt,'F') < treshold)
    {
      print(paste0("Found solution after ", i, " iterations."))
      break
    }
    r_alt <- r
  }
  return(r[,1])
}

# Type 0: A with a_ij equals 1 for win, 0.5 for draw and 0 for loss
# Type 1: S_ij: team i scored s_ij points versus team j
# Type 2: A with a_ij = (S_ij+1)/(S_ij+S_ji+2)
generateA <- function(dataSet, type = 0)
{
  teams <- as.character(unique(dataSet$HomeTeam))
  A <- matrix(0, nrow=length(teams), ncol=length(teams))
  rownames(A) <- teams
  colnames(A) <- teams
  for (i in 1:nrow(dataSet))
  {
    homeTeam <- as.character(dataSet$HomeTeam[i])
    awayTeam <- as.character(dataSet$AwayTeam[i])
    if (type == 0)
    {
      gd <- dataSet$FTHG[i] - dataSet$FTAG[i]
      if (gd > 0)
      {
        A[homeTeam,awayTeam] <- A[homeTeam,awayTeam] + 1
      }
      else if (gd < 0)
      {
        A[awayTeam,homeTeam] <- A[awayTeam,homeTeam] + 1
      }
      else
      {
        A[homeTeam,awayTeam] <- A[homeTeam,awayTeam] + 0.5
        A[awayTeam,homeTeam] <- A[awayTeam,homeTeam] + 0.5
      }   
    }
    else if (type == 1)
    {
      A[homeTeam,awayTeam] <- A[homeTeam,awayTeam] + dataSet$FTHG[i]
      A[awayTeam,homeTeam] <- A[awayTeam,homeTeam] + dataSet$FTAG[i]
    }
    else if (type == 2)
    {
      A[homeTeam,awayTeam] <- A[homeTeam,awayTeam] + (dataSet$FTHG[i]+1)/(dataSet$FTHG[i]+dataSet$FTAG[i]+2)
      A[awayTeam,homeTeam] <- A[awayTeam,homeTeam] + (dataSet$FTAG[i]+1)/(dataSet$FTHG[i]+dataSet$FTAG[i]+2)
    }
  }
  return(A)
}

generateS <- function(dataSet)
{
  return(generateA(dataSet, type = 1))
}

generateB <- function(dataSet)
{
  S <- generateS(dataSet)
  B <- matrix(0, nrow=nrow(S), ncol=ncol(S))
  rownames(B) <- rownames(S)
  colnames(B) <- colnames(S)
  for (i in 1:nrow(S))
  {
    for (j in 1:ncol(S))
    {
      if (i != j)
      {
        B[i,j] <- -S[i,j]*S[j,i]
      }
      else # (i == j)
      {
        sum <- 0
        for (k in 1:nrow(S))
        {
          #sum <- sum + S[i,k]^2
          sum <- sum + S[k,i]^2
        }
        B[i,j] <- sum
      }
    }
  }
  return(B)
}

minL <- function(B)
{
  l <- 0
  for (i in 1:nrow(B))
  {
    mrsum <- 0
    for (j in 1:ncol(B))
    {
      if (i != j)
      {
        mrsum <- mrsum + abs(B[i,j])
      }
      else # (i == j)
      {
        mrsum <- mrsum - abs(B[i,j])
      }
    }
    if ((mrsum > 0) & (mrsum > l))
    {
      l <- mrsum
    }
  }
  return(l)
}