Commentaries added/updated

R/lauricella.R

@@ -53,8 +53,8 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
   
   k <- 5
   
-  # # M: data.frame of the indices for the nested sums
-  # # (i.e. all arrangements of n elements from {0:k})
+  # M: data.frame of the indices for the nested sums
+  # (i.e. all arrangements of n elements from {0:k})
   M <- expand.grid(rep(list(0:k), n))
 
   # Sum of the indices
@@ -88,14 +88,17 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
   #   # sum(mapply(function(i, j) lnfact[i, j], ind+1, 1:n))
   #   sum(access(ind, lnfact))
   # }
+  # Table of all combinations of the log(m_i)
   gridlnfact <- expand.grid(as.data.frame(lnfact))
+  # Sum of the logarithms
   sumlnfact <- rowSums(gridlnfact)
   
-  # pochhammer(a, m_1+...+m_n) for m_1 = 0...k, ...,  m_p = 0...k
+  # Logarithms of pochhammer(a, m_1+...+m_n) for m_1 = 0...k, ...,  m_n = 0...k
   # lnapoch <- sapply(Msum, function(j) lnpochhammer(a, j))
   lnapoch <- sapply(Msumunique, function(j) lnpochhammer(a, j))
   names(lnapoch) <- Msumunique
 
+  # Logarithms of pochhammer(b_i, m_i) for m_1 = 0...k, ...,  m_n = 0...k
   lnbpoch <- as.data.frame(
     matrix(nrow = max(M) + 1, ncol = n, dimnames = list(Munique, 1:n)))
   for (i in Munique) for (j in 1:n) {
@@ -112,10 +115,12 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
   #   # sum(mapply(function(i, j) lnbpoch[i, j], ind+1, 1:n))
   #   sum(access(ind, lnbpoch))
   # }
+  # Table of all combinations of the log(pochhammer(b_i, m_i))
   gridlnbpoch <- expand.grid(as.data.frame(lnbpoch))
+  # Sum of the logarithms
   sumlnpochb <- rowSums(gridlnbpoch)
   
-  # pochhammer(c, m_1+...+m_n) for m_1 = 0...k, ...,  m_p = 0...k
+  # Logarithms of pochhammer(c, m_1+...+m_n) for m_1 = 0...k, ...,  m_n = 0...k
   # lngpoch <- sapply(Msum, function(j) lnpochhammer(g, j))
   lngpoch <- sapply(Msumunique, function(j) lnpochhammer(g, j))
   names(lngpoch) <- Msumunique
@@ -168,6 +173,8 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     #   }
     # }
     # M1 <- M
+    
+    # M: data.frame of the indices for the nested sums
     M <- expand.grid(rep(list(k1), n))
     if (n > 1) {
       for (i in 1:(n-1)) {
@@ -183,12 +190,13 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     }
     # M2 <- M
     
+    # Sum of the indices
     Msum <- rowSums(M)
 
     Munique <- (max(Munique)+1):max(M)
     Msumunique <- (max(Msumunique)+1):max(Msum)
     
-    # Product x^{m_1}/m_1! * ... * x^{m_n}/m_n! for m_1, ...,  m_n given by the rows of M
+    # Product x^{m_1} * ... * x^{m_n} for m_1, ...,  m_n given by the rows of M
     # xfact <- apply(M, 1, function(Mi) prod( x^Mi ))
     # lnfact <- apply(M, 1, function(Mi) sum(sapply(Mi, lnfactorial)))
     xfactsupp <- as.data.frame(
@@ -197,7 +205,6 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
       # Product pochhammer(b_1,m_1) * ...* pochhammer(b_n, m_n)
       xfactsupp[i, j] <- x[j]^Munique[i]
     }
-    
     gridxfact <- expand.grid(xfactsupp)
     for (i in 1:(n-1)) {
       indsupp <- combn(n, i)
@@ -212,10 +219,11 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     }
     prodxfact <- apply(gridxfact, 1, prod)
     
+    # Logarithm of the product m_1! * ... * m_n! for m_1, ...,  m_n given by the rows of M
+    # i.e. \sum_{i=0}^n{\log{m_i!}}
     lnfactsupp <- as.data.frame(
       matrix(sapply(Munique, lnfactorial), nrow = length(Munique),
              ncol = n, dimnames = list(Munique, 1:n)))
-    
     gridlnfact <- expand.grid(lnfactsupp)
     for (i in 1:(n-1)) {
       indsupp <- combn(n, i)
@@ -230,7 +238,7 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     }
     sumlnfact <- rowSums(gridlnfact)
     
-    # pochhammer(a, m_1+...+m_n) for m_1, ...,  m_n given by the rows of M
+    # Logarithms of pochhammer(a, m_1+...+m_n) for m_1, ...,  m_n given by the rows of M
     # lnapoch <- sapply(Msum, function(j) lnpochhammer(a, j))
     lnapochsupp <- sapply(Msumunique, function(j) lnpochhammer(a, j))
     names(lnapochsupp) <- Msumunique
@@ -241,13 +249,14 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     #   # Product pochhammer(b_1,m_1) * ...* pochhammer(b_n, m_n)
     #   lnbpoch[i] <- sum( sapply(1:n, function(j) lnpochhammer(b[j], M[i, j])) )
     # }
+    # Logarithms of pochhammer(b_i, m_i) for m_1, ...,  m_n given by the rows of M
     lnbpochsupp <- as.data.frame(
       matrix(nrow = length(Munique), ncol = n, dimnames = list(Munique, 1:n)))
     for (i in 1:length(Munique)) for (j in 1:n) {
       # Product pochhammer(b_1,m_1) * ...* pochhammer(b_n, m_n)
       lnbpochsupp[i, j] <- lnpochhammer(b[j], Munique[i])
     }
-    
+    # Table of all combinations of the log(pochhammer(b_i, m_i))
     gridlnbpoch <- expand.grid(lnbpochsupp)
     for (i in 1:(n-1)) {
       indsupp <- combn(n, i)
@@ -260,9 +269,10 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
         gridlnbpoch <- rbind(gridlnbpoch, expand.grid(lnbpochlist))
       }
     }
+    # Sum of the logarithms
     sumlnpochb <- rowSums(gridlnbpoch)
     
-    # pochhammer(c, m_1+...+m_n) for m_1 = 0...k, ...,  m_p = 0...k
+    # Logarithms of pochhammer(g, m_1+...+m_n) for m_1, ...,  m_n given by the rows of M
     # lngpoch <- sapply(Msum, function(j) lnpochhammer(g, j))
     lngpochsupp <- sapply(Msumunique, function(j) lnpochhammer(g, j))
     names(lngpochsupp) <- Msumunique
@@ -275,6 +285,7 @@ lauricella <- function(a, b, g, x, eps = 1e-06) {
     res1 <- lnapoch[Msum+1] + sumlnpochb - lngpoch[Msum+1] - sumlnfact
     res2 <- sum( prodxfact * exp(res1) )
     
+    # Add the new calculated values to the tables of former values
     xfact <- rbind(xfact, xfactsupp)
     lnfact <- rbind(lnfact, lnfactsupp)
     lnbpoch <- rbind(lnbpoch, lnbpochsupp)