Optimizing a Solution to an Overdetermined System Hearthstone is a collectible card video game similar to Magic: The Gathering, where players play cards with certain values of "attack" and "health" for certain "mana" costs. After being introduced to some of the basics of R, I decided that I would find some of the "values" of these characteristics by finding the best solutions to an overdetermined system, as demonstrated here.

Let a card's mana cost, attack, health, and card value* be represented as the vector (m,a,h,v). Three cards are (1,1,2,1), (2,3,2,1), (4,4,5,1), and (7,9,5,1). This corresponds to the overdetermined system a+2h+v=1, 2a+3h+v=2, 4a+5h+v=4, 9a+5h+v=7. The below code would find values of a, h, and v that minimize the sum of the squared differences between the LHS and RHS of the four equations (more cards are shown than the example).

*Card value is the inherent value in having a card to play, as opposed to not having a card to play

numberOfCards = 9
numberOfEffects = 3 # Must be less than or equal to numberOfCards

# Mana, Attack, Health, Card Value

allCards = matrix(
    c(1,2,1,1),                       # Murloc Raider
    ncol = numberOfEffects + 1,
    byrow = TRUE
    )
allCards = rbind(allCards,c(2,2,3,1)) # River Crocolisk
allCards = rbind(allCards,c(2,3,2,1)) # Bloodfen Raptor
allCards = rbind(allCards,c(3,5,1,1)) # Magma Rager
allCards = rbind(allCards,c(4,2,7,1)) # Oasis Snapjaw
allCards = rbind(allCards,c(4,4,5,1)) # Chillwind Yeti
allCards = rbind(allCards,c(6,6,7,1)) # Boulderfist Ogre
allCards = rbind(allCards,c(7,7,7,1)) # War Golem
allCards = rbind(allCards,c(7,9,5,1)) # Core Hound

coefficientsMatrix = matrix(
    NA,
    nrow = numberOfEffects,
    ncol = numberOfEffects,
    byrow = TRUE)
for(i in 1:numberOfEffects) {
    for(j in 1:numberOfEffects) {
        coefficientsMatrix[i,j] = sum(allCards[,i+1] * allCards[,j+1])
    }
}

constantsVector = vector(mode = "integer", length = numberOfEffects)
for(i in 1:numberOfEffects) {
    constantsVector[i] = sum(allCards[,1] * allCards[,i+1])
}

effectValues = solve(coefficientsMatrix, constantsVector, tol = 1e-9)
cat(effectValues, "\n\n") # For demonstration of values of each effect

cardValueMatrix = allCards[,-1]

cardValues = vector(mode = "integer", length = numberOfCards)
for(i in 1:numberOfCards) {
    cardValues[i] = round(sum(cardValueMatrix[i,] * effectValues), digits = 3)
    cat("Card ", i, ": ", cardValues[i], "\n", sep = "")
}


I know of a couple things that could be improved in this code. The most glaring improvement that needs to be made is the creation of coefficientsMatrix; I've read that making a matrix of the correct size and filling it with NA only to fill in its values later is very memory-intensive. Additionally, I am unsure if the way I create the allCards matrix with rbind is the best way to do this.
Here is how you can completely vectorize your code, making use of matrix functions such as crossprod, colSums, and %*%. Without for loops, your code will be more concise and faster to execute. 

The construction of allCards was indeed a problem. Incrementally growing an object like you did is very memory intensive: each time you add a row, a new object is built from scratch. On the other hand, your comment that I've read that making a matrix of the correct size and filling it with NA only to fill in its values later is very memory-intensive is not accurate. Here the object is created once and filled in-place by the for loops.

To summarize: growing an object iteratively is the worst. Initializing an object to its final size and filling it in place via for loops is preferable. Using a vectorized function to create the object in one shot is the best. Hope it helps!

numberOfCards &lt;- 9
numberOfEffects &lt;- 3

allCards &lt;- rbind(
  c(1,2,1,1), # Murloc Raider
  c(2,2,3,1), # River Crocolisk
  c(2,3,2,1), # Bloodfen Raptor
  c(3,5,1,1), # Magma Rager
  c(4,2,7,1), # Oasis Snapjaw
  c(4,4,5,1), # Chillwind Yeti
  c(6,6,7,1), # Boulderfist Ogre
  c(7,7,7,1), # War Golem
  c(7,9,5,1)  # Core Hound
)

cardMana        &lt;- allCards[, 1]
cardValueMatrix &lt;- allCards[, -1]
coefficientsMatrix &lt;- crossprod(cardValueMatrix)
constantsVector &lt;- colSums(cardMana * cardValueMatrix)
effectValues &lt;- solve(coefficientsMatrix, constantsVector, tol = 1e-9)
cat(effectValues, "\n\n")
cardValues &lt;- cardValueMatrix %*% effectValues
cat(sprintf("Card %i: %.3f", seq(numberOfCards), cardValues), sep = "\n")