Arithmetic Ops Group Methods for sprays
Ops.spray.RdAllows arithmetic operators to be used for spray calculations, such as addition, multiplication, division, integer powers, etc. Objects of class spray are interpreted as sparse multivariate polynomials.
Usage
# S3 method for class 'spray'
Ops(e1, e2 = NULL)
spray_negative(S)
spray_times_spray(S1,S2)
spray_times_scalar(S,x)
spray_plus_spray(S1,S2)
spray_plus_scalar(S,x)
spray_power_scalar(S,n)
spray_power_scalar_stla(S,n)
spray_eq_spray(S1,S2)
spray_eq_numeric(S1,x)Details
The function Ops.spray() passes unary and binary arithmetic
operators (“+”, “-”, “*”,
“/”,“==”, and “^”) to the
appropriate specialist function.
The most interesting operators are “*” and
“+” which execute multivariate polynomial multiplication
and addition respectively.
Testing for equality uses spray_eq_spray(). Note that
spray_eq_spray(S1,S2) is algebraically equivalent to
is.zero(S1-S2), but faster (FALSE is returned as soon as
a mismatch is found).
Function spray_power_scalar() is the functional representation
for powers: spray_power_scalar(X,n) is the same as X^n.
Function spray_power_scalar_stla() is an experimental
replacement for spray_power_scalar() that may offer speed
advantages.
Examples
M <- matrix(sample(0:3,21,replace=TRUE),ncol=3)
a <- spray(M,sample(7))
b <- homog(3,4)
# arithmetic operators mostly work as expected:
a + 2*b
#> val
#> 1 2 1 = 2
#> 0 2 2 = 2
#> 1 1 2 = 2
#> 1 0 3 = 2
#> 0 3 1 = 2
#> 0 0 4 = 2
#> 2 2 2 = 6
#> 1 0 0 = 3
#> 2 0 2 = 2
#> 0 4 0 = 2
#> 4 0 0 = 2
#> 3 0 1 = 2
#> 1 3 0 = 2
#> 3 2 1 = 1
#> 2 2 0 = 2
#> 3 1 0 = 2
#> 0 1 3 = 2
#> 1 1 3 = 4
#> 0 0 0 = 7
#> 2 1 1 = 7
#> 3 3 3 = 2
a - a*b^2/4
#> val
#> 5 4 4 = -8.00
#> 3 8 1 = -2.50
#> 8 2 3 = -2.00
#> 3 4 2 = -6.75
#> 3 3 7 = -9.00
#> 6 4 2 = -10.00
#> 9 1 2 = -2.50
#> 7 2 4 = -4.00
#> 7 3 3 = -3.00
#> 3 4 7 = -9.75
#> 2 4 6 = -5.00
#> 5 3 6 = -13.50
#> 4 2 7 = -8.00
#> 1 1 11 = -1.00
#> 7 4 1 = -5.00
#> 3 10 4 = -1.00
#> 7 0 1 = -3.50
#> 4 4 6 = -15.00
#> 5 1 7 = -5.00
#> 4 8 2 = -5.50
#> 3 2 9 = -3.25
#> 8 2 4 = -5.50
#> 0 8 0 = -1.75
#> 2 0 6 = -5.25
#> 4 0 5 = -3.00
#> 5 4 3 = -12.50
#> 4 7 6 = -4.00
#> 2 6 0 = -5.25
#> 1 7 5 = -3.00
#> 4 1 4 = -6.00
#> 3 3 2 = -17.50
#> 2 7 4 = -4.00
#> 4 2 3 = -7.50
#> 6 1 5 = -6.25
#> 5 3 9 = -1.50
#> 4 2 6 = -7.50
#> 4 6 4 = -15.50
#> 2 4 7 = -8.00
#> 6 1 6 = -4.00
#> 2 5 5 = -6.25
#> 3 5 9 = -1.50
#> 2 9 1 = -1.25
#> 6 3 4 = -6.00
#> 7 1 5 = -3.00
#> 2 2 8 = -2.50
#> 5 1 2 = -10.50
#> 4 5 0 = -3.00
#> 2 7 0 = -1.50
#> 2 1 9 = -1.25
#> 5 3 1 = -6.00
#> 1 8 0 = -0.75
#> 7 6 4 = -4.00
#> 7 2 0 = -2.25
#> 8 1 3 = -3.75
#> 11 2 1 = -0.25
#> 6 0 2 = -5.25
#> 5 5 4 = -17.50
#> 6 3 8 = -2.00
#> 2 2 4 = -15.75
#> 2 6 6 = -7.50
#> 2 7 3 = -3.75
#> 2 8 4 = -4.50
#> 6 4 3 = -4.00
#> 2 3 7 = -3.75
#> 1 2 6 = -2.25
#> 3 6 3 = -7.50
#> 2 2 5 = -4.50
#> 0 3 5 = -7.00
#> 1 1 3 = 4.00
#> 5 1 6 = -5.00
#> 1 3 9 = -3.00
#> 2 5 2 = -4.50
#> 3 1 4 = -14.00
#> 3 7 3 = -3.00
#> 2 5 1 = -10.50
#> 1 6 1 = -7.00
#> 2 2 2 = 6.00
#> 3 7 2 = -5.00
#> 0 6 2 = -5.25
#> 2 9 3 = -3.00
#> 4 3 6 = -10.00
#> 2 5 7 = -6.00
#> 5 3 0 = -7.00
#> 1 4 8 = -4.00
#> 5 7 2 = -7.50
#> 1 7 0 = -3.50
#> 7 1 4 = -5.00
#> 4 7 3 = -10.50
#> 6 6 5 = -5.00
#> 2 1 6 = -3.00
#> 3 5 6 = -13.00
#> 9 4 1 = -0.75
#> 7 4 3 = -11.25
#> 6 2 5 = -6.00
#> 4 3 5 = -11.25
#> 1 1 7 = -1.50
#> 9 3 5 = -1.50
#> 7 3 7 = -2.50
#> 4 5 8 = -3.00
#> 3 3 6 = -7.50
#> 5 7 5 = -4.50
#> 3 4 6 = -10.00
#> 4 2 8 = -5.00
#> 3 2 4 = -6.75
#> 4 4 9 = -2.00
#> 2 6 4 = -5.00
#> 3 6 8 = -2.00
#> 2 2 10 = -1.50
#> 2 10 2 = -1.50
#> 3 3 11 = -0.50
#> 5 5 2 = -10.00
#> 7 5 2 = -8.00
#> 7 1 0 = -3.50
#> 6 5 1 = -6.25
#> 6 1 1 = -7.00
#> 2 3 9 = -3.00
#> 2 7 5 = -6.00
#> 3 3 3 = -5.50
#> 5 8 1 = -0.75
#> 5 3 5 = -9.00
#> 5 2 2 = -6.75
#> 4 1 8 = -4.00
#> 1 8 4 = -2.00
#> 6 1 2 = -4.50
#> 1 0 8 = -0.75
#> 3 7 4 = -10.00
#> 4 2 2 = -15.75
#> 8 6 3 = -2.00
#> 9 5 3 = -1.50
#> 5 4 5 = -17.25
#> 4 1 3 = -14.00
#> 5 2 5 = -10.00
#> 6 4 7 = -4.00
#> 0 0 8 = -1.75
#> 10 3 4 = -1.00
#> 3 7 7 = -2.50
#> 8 2 2 = -5.00
#> 6 6 2 = -9.50
#> 6 2 0 = -5.25
#> 3 9 2 = -3.50
#> 3 11 3 = -0.50
#> 11 3 3 = -0.50
#> 7 3 4 = -11.00
#> 10 1 1 = -1.25
#> 5 4 8 = -3.00
#> 2 5 6 = -8.00
#> 1 5 7 = -5.00
#> 2 1 10 = -2.00
#> 5 0 4 = -3.75
#> 4 3 7 = -10.00
#> 3 1 8 = -2.50
#> 7 7 3 = -2.50
#> 6 3 0 = -3.00
#> 1 7 1 = -1.50
#> 6 0 3 = -3.00
#> 4 8 5 = -3.00
#> 8 0 0 = -1.75
#> 3 1 9 = -3.00
#> 8 4 2 = -6.00
#> 3 4 1 = -14.00
#> 3 3 8 = -6.50
#> 1 3 5 = -3.00
#> 5 2 6 = -8.00
#> 4 5 5 = -17.00
#> 1 3 4 = -14.00
#> 4 9 1 = -0.50
#> 2 3 3 = -17.50
#> 2 1 1 = 5.00
#> 3 6 0 = -2.25
#> 2 6 1 = -3.00
#> 8 1 4 = -2.00
#> 7 6 1 = -1.25
#> 6 2 6 = -8.50
#> 2 1 5 = -10.50
#> 1 1 6 = -7.00
#> 3 5 5 = -9.00
#> 4 3 10 = -1.00
#> 7 3 2 = -7.50
#> 4 9 4 = -2.00
#> 5 2 7 = -6.75
#> 4 1 7 = -3.75
#> 0 4 4 = -8.75
#> 9 2 3 = -3.75
#> 0 7 1 = -3.50
#> 5 3 4 = -12.50
#> 4 6 7 = -4.00
#> 3 0 5 = -7.00
#> 0 0 0 = 7.00
#> 4 4 1 = -6.00
#> 8 4 5 = -3.00
#> 1 0 0 = 3.00
#> 7 2 3 = -7.50
#> 9 2 1 = -2.50
#> 2 3 4 = -6.00
#> 5 6 6 = -5.00
#> 4 5 4 = -8.00
#> 3 2 1 = 1.00
#> 6 2 1 = -4.50
#> 4 3 1 = -14.00
#> 4 7 1 = -3.75
#> 6 7 4 = -4.00
#> 5 2 1 = -10.50
#> 2 3 8 = -6.00
#> 6 7 1 = -1.00
#> 4 4 0 = -8.75
#> 5 6 1 = -5.00
#> 2 4 2 = -15.75
#> 3 5 1 = -4.50
#> 4 3 2 = -7.50
#> 1 5 2 = -10.50
#> 6 2 4 = -10.00
#> 5 0 3 = -7.00
#> 5 5 7 = -4.50
#> 3 2 3 = -17.50
#> 7 2 5 = -7.25
#> 1 6 2 = -2.25
#> 6 3 5 = -14.00
#> 10 2 2 = -2.00
#> 8 0 1 = -1.50
#> 4 10 3 = -1.00
#> 4 4 5 = -10.00
#> 3 6 4 = -6.00
#> 1 2 5 = -10.50
#> 9 3 2 = -4.00
#> 3 4 5 = -10.00
#> 6 5 3 = -14.50
#> 1 4 3 = -14.00
#> 0 2 6 = -5.25
#> 10 3 1 = -0.50
#> 1 0 7 = -3.50
#> 3 2 8 = -6.00
#> 8 3 1 = -3.75
#> 10 4 3 = -1.00
#> 5 1 3 = -6.00
#> 2 6 5 = -6.00
#> 3 5 4 = -10.00
#> 3 1 5 = -4.50
#> 3 10 1 = -0.25
#> 3 8 6 = -2.00
#> 3 8 3 = -6.75
#> 6 3 3 = -11.25
#> 1 6 6 = -4.00
#> 1 2 10 = -2.00
#> 3 2 7 = -5.00
#> 2 2 9 = -4.00
#> 5 5 3 = -5.00
#> 8 5 1 = -1.00
#> 3 0 6 = -2.25
#> 8 1 0 = -1.50
#> 8 5 4 = -3.00
#> 1 9 3 = -1.00
#> 8 3 3 = -7.50
#> 8 3 6 = -2.00
#> 7 5 5 = -4.50
#> 7 1 1 = -3.00
#> 3 9 5 = -1.50
#> 7 0 2 = -2.25
#> 4 6 2 = -7.50
#> 5 4 0 = -3.75
#> 9 1 3 = -1.00
#> 0 5 3 = -7.00
#> 4 6 3 = -4.00
#> 2 4 8 = -4.50
#> 2 8 2 = -2.50
#> 0 1 7 = -3.50
#> 1 5 3 = -3.00
#> 5 8 4 = -3.00
#> 6 5 6 = -5.00
#> 1 4 4 = -3.75
#> 2 4 3 = -6.00
#> 4 5 3 = -11.25
#> 5 6 3 = -14.25
#> 6 8 3 = -2.00
#> 5 9 3 = -1.50
#> 3 4 10 = -1.00
#> 6 4 4 = -16.00
#> 2 8 3 = -2.00
#> 3 6 5 = -13.25
#> 4 4 4 = -12.50
#> 4 0 4 = -8.75
#> 2 0 7 = -1.50
#> 9 4 4 = -2.00
#> 7 4 6 = -4.00
#> 9 0 0 = -0.75
#> 3 5 0 = -7.00
a+b
#> val
#> 3 1 0 = 1
#> 2 2 0 = 1
#> 1 2 1 = 1
#> 0 2 2 = 1
#> 1 3 0 = 1
#> 0 0 4 = 1
#> 2 2 2 = 6
#> 1 0 0 = 3
#> 1 1 2 = 1
#> 1 0 3 = 1
#> 0 3 1 = 1
#> 2 0 2 = 1
#> 0 4 0 = 1
#> 4 0 0 = 1
#> 3 0 1 = 1
#> 3 2 1 = 1
#> 1 1 3 = 4
#> 0 0 0 = 7
#> 0 1 3 = 1
#> 2 1 1 = 6
#> 3 3 3 = 2
S1 <- spray(partitions::compositions(4,3))
S2 <- spray(diag(3)) # S2 = x+y+z
stopifnot( (S1+S2)^3 == S1^3 + 3*S1^2*S2 + 3*S1*S2^2 + S2^3 )