Complex Numbers and Basic Functionality

Description

Basic functions which support complex arithmetic in R, in addition to the arithmetic operators +, -, *, /, and ^.

Usage

complex(length.out = 0, real = numeric(), imaginary = numeric(),
        modulus = 1, argument = 0)
as.complex(x, ...)
is.complex(x)

Re(z)
Im(z)
Mod(z)
Arg(z)
Conj(z)

Arguments

Details

Complex vectors can be created with complex. The vector can be specified either by giving its length, its real and imaginary parts, or modulus and argument. (Giving just the length generates a vector of complex zeroes.)

as.complex attempts to coerce its argument to be of complex type: like as.vector it strips attributes including names. Since R version 4.4.0, as.complex(x) for “number-like” x, i.e., types "logical", "integer", and "double", will always keep imaginary part zero, now also for NA's. Up to R versions 3.2.x, all forms of NA and NaN were coerced to a complex NA, i.e., the NA_complex_ constant, for which both the real and imaginary parts are NA. Since R 3.3.0, typically only objects which are NA in parts are coerced to complex NA, but others with NaN parts, are not. As a consequence, complex arithmetic where only NaN's (but no NA's) are involved typically will not give complex NA but complex numbers with real or imaginary parts of NaN. All of these many different complex numbers fulfill is.na(.) but only one of them is identical to NA_complex_.

Note that is.complex and is.numeric are never both TRUE.

The functions Re, Im, Mod, Arg and Conj have their usual interpretation as returning the real part, imaginary part, modulus, argument and complex conjugate for complex values. The modulus and argument are also called the polar coordinates. If z = x + i y with real x and y, for r = Mod(z) = \sqrt{x^2 + y^2}, and \phi = Arg(z), x = r \cos(\phi) and y = r \sin(\phi). They are all internal generic primitive functions: methods can be defined for them individually or via the Complex group generic.

In addition to the arithmetic operators (see Arithmetic) +, -, *, /, and ^, the elementary trigonometric, logarithmic, exponential, square root and hyperbolic functions are implemented for complex values.

Matrix multiplications (%*%, crossprod, tcrossprod) are also defined for complex matrices (matrix), and so are solve, eigen or svd.

Internally, complex numbers are stored as a pair of double precision numbers, either or both of which can be NaN (including NA, see NA_complex_ and above) or plus or minus infinity.

S4 methods

as.complex is primitive and can have S4 methods set.

Re, Im, Mod, Arg and Conj constitute the S4 group generic Complex and so S4 methods can be set for them individually or via the group generic.

Note

Operations and functions involving complex NaN mostly rely on the C library's handling of ‘⁠double complex⁠’ arithmetic, which typically returns complex(re=NaN, im=NaN) (but we have not seen a guarantee for that). For + and -, R's own handling works strictly “coordinate wise”.

Operations involving complex NA, i.e., NA_complex_, return NA_complex_.

Only since R version 4.4.0, as.complex("1i") gives 1i, it returned NA_complex_ with a warning, previously.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See Also

Arithmetic; polyroot finds all n complex roots of a polynomial of degree n.

Examples

require(graphics)

0i ^ (-3:3)

matrix(1i^ (-6:5), nrow = 4) #- all columns are the same
0 ^ 1i # a complex NaN

## create a complex normal vector
z <- complex(real = stats::rnorm(100), imaginary = stats::rnorm(100))
## or also (less efficiently):
z2 <- 1:2 + 1i*(8:9)

## The Arg(.) is an angle:
zz <- (rep(1:4, length.out = 9) + 1i*(9:1))/10
zz.shift <- complex(modulus = Mod(zz), argument = Arg(zz) + pi)
plot(zz, xlim = c(-1,1), ylim = c(-1,1), col = "red", asp = 1,
     main = expression(paste("Rotation by "," ", pi == 180^o)))
abline(h = 0, v = 0, col = "blue", lty = 3)
points(zz.shift, col = "orange")

## as.complex(<some NA>): numbers keep Im = 0:
stopifnot(identical(as.complex(NA_real_), NA_real_ + 0i)) # has always been true
NAs <- vapply(list(NA, NA_integer_, NA_real_, NA_character_, NA_complex_),
              as.complex, 0+0i)
stopifnot(is.na(NAs), is.na(Re(NAs))) # has always been true
showC <- function(z) noquote(paste0("(", Re(z), ",", Im(z), ")"))
showC(NAs)
Im(NAs) # [0 0 0 NA NA]  \\ in R <= 4.3.x was [NA NA 0 NA NA]
stopifnot(Im(NAs)[1:3] == 0)


## The exact result of this *depends* on the platform, compiler, math-library:
(NpNA <- NaN + NA_complex_) ; str(NpNA) # *behaves* as 'cplx NA' ..
stopifnot(is.na(NpNA), is.na(NA_complex_), is.na(Re(NA_complex_)), is.na(Im(NA_complex_)))
showC(NpNA)# but does not always show '(NaN,NA)'
## and this is not TRUE everywhere:
identical(NpNA, NA_complex_)
showC(NA_complex_) # always == (NA,NA)