# Mutable arithmetic

The high level interface can be combined with the low level wrapper to allow for efficient computations using mutable arithmetic.

In the future it would be nice to have an interface to MutableArithmetics.jl, see #118.

The following methods are useful for mutating part of a value

Arblib.radrefFunction
radref(x::ArbLike, prec = precision(x))

Return a MagRef referencing the radius of x.

Arblib.midrefFunction
midref(x::ArbLike, prec = precision(x))

Return an ArfRef referencing the midpoint of x.

Arblib.realrefFunction
realref(z::AcbLike, prec = precision(z))

Return an ArbRef referencing the real part of x.

Arblib.imagrefFunction
imagref(z::AcbLike, prec = precision(z))

Return an ArbRef referencing the imaginary part of x.

Arblib.refFunction
ref(v::Union{ArbVectorOrRef,AcbVectorOrRef}, i)

Similar to v[i] but instead of an Arb or Acb returns an ArbRef or AcbRef which still shares the memory with the i-th entry of v.

ref(A::Union{ArbMatrixOrRef,AcbMatrixOrRef}, i, j)

Similar to A[i,j] but instead of an Arb or Acb returns an ArbRef or AcbRef which still shares the memory with the (i,j)-th entry of A.

ref(p::Union{ArbPoly,ArbSeries,AcbPoly,AcbSeries}, i)

Similar to p[i] but instead of an Arb or Acb returns an ArbRef or AcbRef which shares the memory with the i-th coefficient of p.

Note

Using ref for reading coefficients is always safe, but if the coefficient is mutated then care has to be taken. See the comment further down for how to handle mutation.

It only allows accessing coefficients that are allocated. For ArbPoly and AcbPoly this is typically all coefficients up to the degree of the polynomial, but can be higher if for example Arblib.fit_length! is used. For ArbSeries and AcbSeries all coefficients up to the degree of the series are guaranteed to be allocated, even if the underlying polynomial has a lower degree.

If the coefficient is mutated in a way that the degree of the polynomial changes then one needs to also update the internally stored length of the polynomial.

• If the new degree is the same or lower this can be achieved with
Arblib.normalise!(p)
• If the new degree is higher you need to manually set the length. This can be achieved with
Arblib.set_length!(p, len)
Arblib.normalise!(p)
where len is one higher than (an upper bound of) the new degree.

See the extended help for more details.

Extended help

Here is an example were the leading coefficient is mutated so that the degree is lowered.

julia> p = ArbPoly([1, 2], prec = 64) # Polynomial of degree 1
1.0 + 2.0⋅x

julia> Arblib.zero!(Arblib.ref(p, 1)) # Set leading coefficient to 0
0

julia> Arblib.degree(p) # The degree is still reported as 1
1

julia> length(p) # And the length as 2
2

julia> p # Printing gives weird results
1.0 +

julia> Arblib.normalise!(p) # Normalising the polynomial updates the degree
1.0

julia> Arblib.degree(p) # This is now correct
0

julia> p # And so is printing
1.0

Here is an example when a coefficient above the degree is mutated.

julia> q = ArbSeries([1, 2, 0], prec = 64) # Series of degree 3 with leading coefficient zero
1.0 + 2.0⋅x + 𝒪(x^3)

julia> Arblib.one!(Arblib.ref(q, 2)) # Set the leading coefficient to 1
1.0

julia> q # The leading coefficient is not picked up
1.0 + 2.0⋅x + 𝒪(x^3)

julia> Arblib.degree(q.poly) # The degree of the underlying polynomial is still 1
1

julia> Arblib.set_length!(q, 3) # After explicitly setting the length to 3 it is ok
1.0 + 2.0⋅x + 1.0⋅x^2 + 𝒪(x^3)

## Examples

Compare computing $\sqrt{x^2 + y^2}$ using mutable arithmetic with the default.

julia> using Arblib, BenchmarkToolsjulia> x = Arb(1 // 3)[0.33333333333333333333333333333333333333333333333333333333333333333333333333333 +/- 4.78e-78]julia> y = Arb(1 // 5)[0.20000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 3.89e-78]julia> res = zero(x)0julia> f(x, y) = sqrt(x^2 + y^2)f (generic function with 1 method)julia> f!(res, x, y) = begin
Arblib.sqr!(res, x)
Arblib.fma!(res, res, y, y)
return Arblib.sqrt!(res, res)
endf! (generic function with 1 method)julia> @benchmark f($x,$y) samples=10000 evals=500BenchmarkTools.Trial: 8714 samples with 500 evaluations.
Range (min … max):  608.042 ns … 394.946 μs  ┊ GC (min … max):  0.00% … 82.40%
Time  (median):     625.255 ns               ┊ GC (median):     0.00%
Time  (mean ± σ):     1.149 μs ±   9.007 μs  ┊ GC (mean ± σ):  21.81% ±  2.84%

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608 ns        Histogram: log(frequency) by time       1.09 μs <

Memory estimate: 448 bytes, allocs estimate: 8.julia> @benchmark f!($res,$x, $y) samples=10000 evals=500BenchmarkTools.Trial: 10000 samples with 500 evaluations. Range (min … max): 367.680 ns … 1.072 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 369.692 ns ┊ GC (median): 0.00% Time (mean ± σ): 389.196 ns ± 63.030 ns ┊ GC (mean ± σ): 0.00% ± 0.00% █ ▂▂ ▁▁ ▁ █▄██▇█████▇▇▆▇▇▅▅▄▅▄▅▆▅▆▆▆▆▇▇▇▇██▇▇▇▇▆▆▆▆▅▆▄▅▅▄▄▃▃▄▄▃▂▃▄▆▆▄▄ █ 368 ns Histogram: log(frequency) by time 640 ns < Memory estimate: 0 bytes, allocs estimate: 0. Set the radius of the real part of an Acb. julia> using Arblibjulia> z = Acb(1, 2)1.0 + 2.0imjulia> Arblib.set!(Arblib.radref(Arblib.realref(z)), 1e-10)1.0e-10julia> z[1.000000000 +/- 1.01e-10] + 2.0im Compare a naive implementation of polynomial evaluation using mutable arithmetic with one not using using it. julia> using Arblib, BenchmarkToolsjulia> p = ArbPoly(1:10)1.0 + 2.0⋅x + 3.0⋅x^2 + 4.0⋅x^3 + 5.0⋅x^4 + 6.0⋅x^5 + 7.0⋅x^6 + 8.0⋅x^7 + 9.0⋅x^8 + 10.0⋅x^9julia> x = Arb(1 // 3)[0.33333333333333333333333333333333333333333333333333333333333333333333333333333 +/- 4.78e-78]julia> res = zero(x)0julia> function eval(p, x) res = zero(x) xi = one(x) for i in 0:Arblib.degree(p) res += p[i] * xi xi *= x end return res endeval (generic function with 2 methods)julia> function eval!(res, p, x) Arblib.zero!(res) xi = one(x) for i in 0:Arblib.degree(p) Arblib.addmul!(res, Arblib.ref(p, i), xi) Arblib.mul!(xi, xi, x) end return res endeval! (generic function with 1 method)julia> @benchmark eval($p, $x) samples = 10000 evals = 30BenchmarkTools.Trial: 10000 samples with 30 evaluations. Range (min … max): 2.806 μs … 6.760 ms ┊ GC (min … max): 0.00% … 81.85% Time (median): 2.881 μs ┊ GC (median): 0.00% Time (mean ± σ): 8.280 μs ± 131.701 μs ┊ GC (mean ± σ): 33.79% ± 2.18% ██▆▅▄▄▄▃▃▃▂▂▁▁▁▂▁▁ ▁▂▃▃▂▂▂▂▁▁▁ ▂ ██████████████████████████████████▇▇██▇▆▇▆▆▆▆▆▅▆▆▅▄▆▆▅▅▁▁▄▃ █ 2.81 μs Histogram: log(frequency) by time 5.19 μs < Memory estimate: 3.94 KiB, allocs estimate: 70.julia> @benchmark eval!($res, $p,$x) samples = 10000 evals = 30BenchmarkTools.Trial: 10000 samples with 30 evaluations.
Range (min … max):  1.134 μs … 994.812 μs  ┊ GC (min … max): 0.00% … 46.78%
Time  (median):     1.156 μs               ┊ GC (median):    0.00%
Time  (mean ± σ):   1.278 μs ±   9.937 μs  ┊ GC (mean ± σ):  3.64% ±  0.47%

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1.13 μs         Histogram: frequency by time        1.78 μs <

Memory estimate: 160 bytes, allocs estimate: 3.julia> @benchmark $p($x) samples = 10000 evals = 30 # Arb implementation for referenceBenchmarkTools.Trial: 10000 samples with 30 evaluations.
Range (min … max):  786.300 ns … 987.704 μs  ┊ GC (min … max): 0.00% … 45.16%
Time  (median):     800.167 ns               ┊ GC (median):    0.00%
Time  (mean ± σ):   921.684 ns ±   9.869 μs  ┊ GC (mean ± σ):  4.84% ±  0.45%

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786 ns        Histogram: log(frequency) by time       1.16 μs <

Memory estimate: 160 bytes, allocs estimate: 3.