n <- 0:199
N <- length(n)
fib <- numeric(N)
pell <- numeric(N)
approx_sqrt2 <- rep(NA, N)
error <- rep(NA, N)
accuracy <- integer(N)
fib[1] <- 0
fib[2] <- 1
pell[1] <- 0
pell[2] <- 1
for (i in 3:N) {
fib[i] <- fib[i - 1] + fib[i - 2]
pell[i] <- 2 * pell[i - 1] + pell[i - 2]
}
sqrt2_true <- sqrt(2)
for (i in 2:(N - 1)) {
approx_sqrt2[i] <- pell[i + 1] / pell[i]
error[i] <- abs(approx_sqrt2[i] - sqrt2_true)
if (!is.na(error[i]) && error[i] == 0) {
accuracy[i] <- 16
} else if (!is.na(error[i])) {
accuracy[i] <- floor(-log10(error[i]))
}
}
pelldata <- data.frame(
Index = n,
Fib = fib,
Pell = pell,
Approx_Sqrt2 = approx_sqrt2,
Diff_from_sqrt2 = round(error, 5),
Decimal_Accuracy = accuracy
)
pelldata
n <- 0:199
N <- length(n)
fib <- numeric(N)
pell <- numeric(N)
approx_sqrt2 <- rep(NA, N)
error <- rep(NA, N)
accuracy <- integer(N)
fib[1] <- 0
fib[2] <- 1
pell[1] <- 0
pell[2] <- 1
for (i in 3:N) {
fib[i] <- fib[i - 1] + fib[i - 2]
pell[i] <- 2 * pell[i - 1] + pell[i - 2]
}
sqrt2_true <- sqrt(2)
for (i in 2:(N - 1)) {
approx_sqrt2[i] <- pell[i + 1] / pell[i]
error[i] <- abs(approx_sqrt2[i] - sqrt2_true)
if (!is.na(error[i]) && error[i] == 0) {
accuracy[i] <- 16
} else if (!is.na(error[i])) {
accuracy[i] <- floor(-log10(error[i]))
}
}
pelldata <- data.frame(
Index = n,
Fib = fib,
Pell = pell,
Approx_Sqrt2 = approx_sqrt2,
Diff_from_sqrt2 = round(error, 5),
Decimal_Accuracy = accuracy
)
pelldata
View(pelldata)