round

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round round utilities

Round Plot utility. 1 φ# x.1 x.2 d RPlot M 05 matrix : α 0.25 : δ 0.2 : 13 mat x x.1 x.2 range M M x d + x d + φ# . 14 blue 15 mat stack : for X x.1 α - (x.2 α + (x.1 α - δ + range : X φ# vectorize augment M 21 sys 31 line :

Utility shorthand. x U.\002B\½ x 0.5 x UnitsOf * + :

Utility shorthand. x U.\002D\½ x 0.5 x UnitsOf * - :

Truncation

In SMath we need to start with the trunc function, expanding it for taking complexes, and it's 'dual', the fractionary part

Trunc(x) truncate x to the next nearest integer towards  0. x Trunc x Re trunc i x Im trunc * + :

Frac(x) returns the fractional part of x. x Frac x x Trunc - :

ξ Trunc 3 - 3 0 RPlot ξ Frac 3 - 3 0 RPlot

For the floor and ceil functions SMath have the Custom Functions plugin by Davide Carpi. Those functions expands over matrices and complexes but can't be used for integrals. We redefine them for pedagogical purposes, but notice that our implementation fails for small values.

1020 -^Floor 0

1020 -^(- Floor 1 -

Correct answers given by Custom Functions plugin.

Floor(x) is the greatest integer 
less than or equal to x. x Floor x x Frac x UnitsOf + Frac - :

Ceil(x) is the smallest integer 
greater than or equal to x. x Ceil x x Frac x UnitsOf - Frac - :

ξ Floor 3 - 3 0 RPlot ξ Ceil 3 - 3 0 RPlot

1020 -^Floor 11020 -^*

1020 -^(- Floor 11020 -^* -

Wrong answers

Notice that

and Mathematica:

2.7 3.5 i * + Floor 23 i * +

but

2.7 3.5 i * + floor maple 31 i + (* ≡

Other possible definitions include

Floor x Floor x Trunc 1 x + x Trunc - Trunc + 1 - :

Ceil x Ceil x - Floor - :

Rounding

We can define different functions to calculate the nearest integer to a number according to which is the treatment of the numbers in the set M = {x : Floor(F(x)) = 1/2} = {x : x + 1/2 ∈ Z} since these numbers are equidistant from the nearest lower and upper integers.

We introduce the RoundUp and RoundDn functions as

RoundUp function. x R.\002B\\221E\ x U.\002B\½ Floor :

RoundDn function. x R.\002D\\221E\ x U.\002D\½ Ceil :

ξ R.\002B\\221E\ 3 - 3 0.5 RPlot ξ R.\002D\\221E\ 3 - 3 0.5 RPlot

Notice that we use d=0.5 for access to the points in the set M.

And RoundZero and RoundInf (avoiding the use of if)

RoundZero function. x R.0 x U.\002B\½ Floor x 0 < (* x U.\002D\½ Ceil x 0 ≥ (* + :

RoundInf function. x R.∞ x U.\002D\½ Ceil x 0 < (* x U.\002B\½ Floor x 0 ≥ (* + :

ξ R.0 3 - 3 0.5 RPlot ξ R.∞ 3 - 3 0.5 RPlot

The typical rounding function R(x) is actually equals to R.∞:

Round function. x R x U.\002D\½ Trunc x 0 < (* x U.\002B\½ Trunc x 0 ≥ (* + :

To avoid bias in the treatment of numbers, the rounding rule is usually used in practice known as "banking" or "Gaussian", R2, according to which values of numbers are always rounded to even numbers. In a symmetric way we can define a rule R1 that rounds to odd numbers, and again, avoiding the use of if statements.

RoundEven Function x R.2 m x Floor 0.5 x UnitsOf * + x ≡ (: 2 0.5 x U.\002B\½ * Floor * m * x U.\002B\½ Floor m ¬ (* + 21 line :

RoundOdd function. x R.1 m x Floor 0.5 x UnitsOf * + x ≡ (: 2 0.5 x U.\002D\½ * Floor * x UnitsOf + (m * x U.\002B\½ Floor m ¬ (* + 21 line :

ξ R.1 3 - 3 0.5 RPlot ξ R.2 3 - 3 0.5 RPlot

Digits Rounding

x n R.\002B\\221E\ 10 n^x * R.\002B\\221E\ 10 n -^* :

x n R.0 10 n^x * R.0 10 n -^* :

x n R.1 10 n^x * R.1 10 n -^* :

To round with n decimal digits we can define them according with the rule:

x n R.\002D\\221E\ 10 n^x * R.\002D\\221E\ 10 n -^* :

x n R.∞ 10 n^x * R.∞ 10 n -^* :

x n R.2 10 n^x * R.2 10 n -^* :

x f.n 10 n^x * f 10 n -^* ≡

We can apply this rule to all the above rounding functions: R.+∞, R.0 ... etc. For the usual round function it is

x n R 10 n^x * R 10 n -^* :

SMath have its own round function:

x n round 10 n^x * R.2 10 n -^* ≡

x n 1 round 10 n^x * R 10 n -^* ≡

and

x r x 0 round : x r 3 - 3 0.5 RPlot 2 1 line x r x 0 1 round : x r 3 - 3 0.5 RPlot 2 1 line

Mathcad extend round functions for two arguments

Ceil(x, y) returns the smallest 
multiple of y greater than or equal to x. x y Floor y x y / Floor * :

Floor(x, y) returns the greatest 
multiple of y less than or equal to x. x y Ceil y x y / Ceil * :

Round(x, y) Rounds x to the 
closest multiple of y. x y Round y x y / R * :

Trunc(x, y) Returns the value 
of trunc(x/y)*y. x y Trunc y x y / Trunc * :

Also have the function mantissa

x Mantissa x x Floor - :

ξ Mantissa 3 - 3 0.5 RPlot

Because it's extensive use in the practice, we show the MS Excel round functions and it's implementation in SMath

In Spanish:

F x n R.Excel 10 n^x * R 10 n -^* F ROUND ≡ 10 n^x * Floor x 0 < (* 10 n^x * Ceil x 0 ≥ (* + 10 n^/ F ROUNDUP ≡ 10 n^x * Trunc 10 n -^* F ROUNDDOWN ≡ 2 0.5 x * Floor x 0 < (* 0.5 x * Ceil x 0 ≥ (* + (* F EVEN ≡ 2 x 1 + 2 / Floor x 0 < (* x 1 - 2 / Ceil x 0 ≥ (* + (* F ODD ≡ x Floor F INT ≡ 10 n^x * Trunc 10 n -^* F TRUNC ≡ x n Floor F FLOOR ≡ x n Ceil F CEILING ≡ x n Round F MROUND ≡ #N/A cases :

REDONDEAR

REDONDEAR.MAS

REDONDEAR.MENOS

REDONDEA.PAR

REDONDEA.IMPAR

ENTERO

TRUNCAR

MULTIPLO.INFERIOR

MULTIPLO.SUPERIOR

REDOND.MULT

Significant Figures Rounding

To round with m significant figures we can define them according with the rule:

x fs.m f m 1 - x abs log10 Floor - el \0028\x\0029\ * ≡

For example

x m RS x m 1 - x abs log10 Floor - R :

92542 RS 9300

etc.

Alvaro