sessionInfo()## R version 4.1.1 (2021-08-10)
## Platform: aarch64-apple-darwin20 (64-bit)
## Running under: macOS Big Sur 11.5.2
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## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib
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## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
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## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## loaded via a namespace (and not attached):
## [1] digest_0.6.27 R6_2.5.0 jsonlite_1.7.2 magrittr_2.0.1
## [5] evaluate_0.14 rlang_0.4.11 stringi_1.7.3 jquerylib_0.1.4
## [9] bslib_0.2.5.1 rmarkdown_2.10 tools_4.1.1 stringr_1.4.0
## [13] xfun_0.25 yaml_2.2.1 compiler_4.1.1 htmltools_0.5.1.1
## [17] knitr_1.33 sass_0.4.0Question 1 What type do the above vectors hold?
a <- c(1L, "This is a character")
str(a)## chr [1:2] "1" "This is a character"b <- c(TRUE, "Hello World")
str(b)## chr [1:2] "TRUE" "Hello World"c <- c(FALSE, 2) # what is wrong here?
str(c)## num [1:2] 0 2In R, integers are specified by the suffix L (e.g. 1L is an integer 1). All other numbers are of class numeric. Note the function is.integer does not test whether a given variable has an integer value, but whether it belongs to the class integer.
So that integer or logical elements are coerced into characters to make a and b to be atomic vectors of characters.
2 here generates a numeric (double) instead of integer (without the suffix L). Logical element is coerced to numeric and c is an atomic vector of numeric.
# using the same objects a, b, c from the above question
a.logical <- as.logical(a)
a.integer <- as.integer(a)## Warning: NAs introduced by coerciona.numeric <- as.numeric(a)## Warning: NAs introduced by coercionb.logical <- as.logical(b)
b.integer <- as.integer(b)## Warning: NAs introduced by coercionb.numeric <- as.numeric(b)## Warning: NAs introduced by coercionc.logical <- as.logical(c)
c.integer <- as.integer(c)
c.numeric <- as.numeric(c)
c.character <- as.character(c)
d <- -5:5
d.logical <- as.logical(d)Question 2 What do you get after the coercions? Does any one suprise you? Can you figure out why?
Vectors a, b, and c already went through coercions before we explicitly coerce them. Therefore, when we explicitly coerce them, a and b are character vectors and c is numeric vector.
Some surprises:
Although as.logical(1L) = TRUE. as.logical("1") = FALSE.
Only 0L and 0 are coerced into FALSE by as.logical(). All others are coerced to TRUE.
a## [1] "1" "This is a character"str(a.logical)## logi [1:2] NA NAstr(a.integer)## int [1:2] 1 NAstr(a.numeric)## num [1:2] 1 NAb## [1] "TRUE" "Hello World"str(b.logical)## logi [1:2] TRUE NAstr(b.integer)## int [1:2] NA NAstr(b.numeric)## num [1:2] NA NAc## [1] 0 2str(c.logical)## logi [1:2] FALSE TRUEstr(c.integer)## int [1:2] 0 2str(c.numeric)## num [1:2] 0 2Question 3 Explain what you found.
First let’s create a vector \(\mathbf{v} = (969, 971, 972, \dots, 1022, 1023)\) of 54 elements
# finish the code below
v <- c(969, 971:1023)
# or
v <- 969:1023
v <- v[-2]
length(v)## [1] 54Then, let’s compute the sum \(\sum_{i=1}^{54}2^{v_i}\).
# finish the code below
v.power.sum <- sum(2^v)
v.power.sum## [1] 1.797693e+308well, the above sum returned Inf as of last year. Clearly, some update of R has pushed this limit.
Let’s take a look over \(\mathbf{w} = (968, 969, 971, 972, \dots, 1022, 1023)\)
w <- c(968:969, 971:1023)
w## [1] 968 969 971 972 973 974 975 976 977 978 979 980 981 982 983
## [16] 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998
## [31] 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013
## [46] 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023(w.power.sum <- sum(2^w))## [1] InfHow about only sum over 53 elements \(\sum_{i=2}^{54}2^{v_i}\) (note that the sum starts from \(v_2\)).
# finish the code below
v.power.sum.53 <- sum(2^v[-1])
v.power.sum.53## [1] 1.797693e+308How about only sum over 54 elements \[\sum_{i=2}^{55}2^{w_i}\]
w.power.sum.54 <- sum(2^w[-1])
w.power.sum.54## [1] 1.797693e+308Now let’s try putting the first element back
v.power.sum.second <- 2^v[1] + sum(2^v[-1])
v.power.sum.second## [1] 1.797693e+308and for w too.
w.power.sum.second <- 2^w[1] + sum(2^w[-1])
w.power.sum.second## [1] 1.797693e+308Reference - this should be the bigget number
biggest <- .Machine$double.xmax
biggest## [1] 1.797693e+308Findings:
The sum \(\sum_{i=1}^{55}2^{w_i}\) creates an overflow error for double
\(\sum_{i=2}^{55}2^{w_i}\) does not create overflow error
2^w[1] + sum(2^w[-1]) actually equals sum(2^w[-1]). This is because adding two numbers of dramatically different magnitudes, we get a + b = b! It is referred to as Catastrophic cancellation caused by the finite-precision arithmetic. You can read more here.