IntervalQuestionStat (version 0.2.0)

arithmetic: Interval arithmetic operations

Description

These commands apply basic interval-valued arithmetic operations.

Usage

# S4 method for numeric,IntervalData
+(e1, e2)
# S4 method for IntervalData,numeric
+(e1, e2)
# S4 method for IntervalData,IntervalData
+(e1, e2)
# S4 method for numeric,IntervalData
-(e1, e2)
# S4 method for IntervalData,numeric
-(e1, e2)
# S4 method for IntervalData,IntervalData
-(e1, e2)
# S4 method for IntervalData,ANY
-(e1, e2) # -e1
# S4 method for numeric,IntervalData
*(e1, e2)
# S4 method for IntervalData,numeric
*(e1, e2)
# S4 method for numeric,IntervalList
+(e1, e2)
# S4 method for IntervalList,numeric
+(e1, e2)
# S4 method for IntervalData,IntervalList
+(e1, e2)
# S4 method for IntervalList,IntervalData
+(e1, e2)
# S4 method for IntervalList,IntervalList
+(e1, e2)
# S4 method for numeric,IntervalList
-(e1, e2)
# S4 method for IntervalList,numeric
-(e1, e2)
# S4 method for IntervalData,IntervalList
-(e1, e2)
# S4 method for IntervalList,IntervalData
-(e1, e2)
# S4 method for IntervalList,IntervalList
-(e1, e2)
# S4 method for IntervalList,ANY
-(e1, e2) # -e1
# S4 method for numeric,IntervalList
*(e1, e2)
# S4 method for IntervalList,numeric
*(e1, e2)
# S4 method for numeric,IntervalMatrix
+(e1, e2)
# S4 method for IntervalMatrix,numeric
+(e1, e2)
# S4 method for matrix,IntervalMatrix
+(e1, e2)
# S4 method for IntervalMatrix,matrix
+(e1, e2)
# S4 method for IntervalData,IntervalMatrix
+(e1, e2)
# S4 method for IntervalMatrix,IntervalData
+(e1, e2)
# S4 method for IntervalList,IntervalMatrix
+(e1, e2)
# S4 method for IntervalMatrix,IntervalList
+(e1, e2)
# S4 method for IntervalMatrix,IntervalMatrix
+(e1, e2)
# S4 method for numeric,IntervalMatrix
-(e1, e2)
# S4 method for IntervalMatrix,numeric
-(e1, e2)
# S4 method for matrix,IntervalMatrix
-(e1, e2)
# S4 method for IntervalMatrix,matrix
-(e1, e2)
# S4 method for IntervalData,IntervalMatrix
-(e1, e2)
# S4 method for IntervalMatrix,IntervalData
-(e1, e2)
# S4 method for IntervalList,IntervalMatrix
-(e1, e2)
# S4 method for IntervalMatrix,IntervalList
-(e1, e2)
# S4 method for IntervalMatrix,IntervalMatrix
-(e1, e2)
# S4 method for IntervalMatrix,ANY
-(e1, e2) # -e1
# S4 method for numeric,IntervalMatrix
*(e1, e2)
# S4 method for IntervalMatrix,numeric
*(e1, e2)
# S4 method for matrix,IntervalMatrix
*(e1, e2)
# S4 method for IntervalMatrix,matrix
*(e1, e2)

Value

This function returns a single interval or a list or matrix of intervals, that is, an IntervalData object or an IntervalList or IntervalMatrix instance, containing the result of the involved operation. When these binary operators are used with IntervalList and IntervalMatrix objects, they return the result of the element by element operations recycling the elements of the object with the shortest dimensions if necessary, showing a warning when they are recycled only fractionally.

Arguments

e1

A single numeric value, a vector, a matrix, a single interval, or a list or a matrix of intervals saved as a numeric, matrix, IntervalData, IntervalList, or IntervalMatrix object, respectively.

e2

A single numeric value, a vector, a matrix, a single interval, or a list or a matrix of intervals saved as a numeric, matrix, IntervalData, IntervalList, or IntervalMatrix object, respectively.

Author

José García-García

Details

Implementation of basic interval arithmetic calculations (see, for example, Moore et al., 2009) through Minkowski's sum (see Minkowski, 1903) and a product by a scalar operation. In particular, +, -, and * operators allow to carry out these computations. Using mid/spr-characterization, these operations can be settled for any two interval-valued data \(A\) and \(B\) and a real number \(\gamma\) as follows: $$A + B = [(\mathrm{mid}~A +\mathrm{mid}~B) \mp (\mathrm{spr}~A + \mathrm{spr}~B)]$$ and $$\gamma \cdot A = \left\{ \begin{array}{ll} \left[\gamma\cdot \mathrm{mid}~A\mp\gamma\cdot\mathrm{spr}~A\right] & \hbox{if $\gamma \geq 0$,}\\[1.5ex] \left[\gamma\cdot \mathrm{mid}~A\pm\gamma\cdot\mathrm{spr}~A\right] & \hbox{if $\gamma < 0$.}\end{array}\right.$$

References

• Minkowski, H. (1903). Volumen und oberfläche. Mathematische Annalen, 57:447-495.

• Moore, R.E.; Kearfott, R.B.; Cloud, M.J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, USA. tools:::Rd_expr_doi("10.1137/1.9780898717716").

Examples

## Some basic arithmetic interval operations

## IntervalData
i1 <- IntervalData(0, 1)
i2 <- IntervalData(2, 3)

i1 + i2   ## Sum of two intervals
i1 + 1    ## Sum of an interval and a real number
1 + i1    ## Sum of a real number and an interval 

i1 - i2   ## Subtraction of two intervals
i1 - i1   ## Note that i1 - i1 is not {0}
i1 - 1    ## Subtraction of an interval and a real number
1 - i1    ## Subtraction of a real number and an interval
- i1      

2 * i1    ## Product between a scalar and an interval
-2 * i1   ## Product between a scalar and an interval
i1 * 2    ## Product between an interval and a scalar
i1 * (-2) ## Product between an interval and a scalar

## IntervalList
list1 <- IntervalList(c(0, 3, 2, 5), c(4, 5, 4, 8))
list2 <- IntervalList(c(3, 0, 3, 1), c(7, 4, 6, 2))

list1 + list2   ## Sum of two list of intervals
list1 + 1       ## Sum of a list of intervals and a real number
1 + list1       ## Sum of a real number and a list of intervals
1:4 + list1     ## Sum of a vector and a list of intervals
list1 + 1:4     ## Sum of a list of intervals and a vector

list1 - list2   ## Subtraction of two lists of intervals
list1 - 1       ## Subtraction of a list of intervals and a real number
1 - list1       ## Subtraction of a real number and a list of intervals
1:4 - list1     ## Subtraction of a vector and a list of intervals
list1 - 1:4     ## Subtraction of a list of intervals and a vector
- list1

2 * list1       ## Product between a scalar and a list of intervals
-2 * list1      ## Product between a scalar and a list of intervals
list1 * 2       ## Product between a list of intervals and a scalar
list1 * (-2)    ## Product between a list of intervals and a scalar
1:4 * list1     ## Product between a vector and a list of intervals
list1 * 1:4     ## Product between a list of intervals and vector

## IntervalMatrix
matrix1 <- IntervalMatrix(matrix(c(0, 1, 1, 2, 2, 3, 3, 4), 2, 4))
matrix2 <- IntervalMatrix(matrix(c(4, 5, 5, 6, 6, 7, 7, 8), 2, 4))
m <- matrix(1:4, 2, 2)

matrix1 + matrix2  ## Sum of two matrices of intervals
matrix1 + 1        ## Sum of a matrix of intervals and a scalar
1 + matrix1        ## sum of a scalar and a matrix of intervals
matrix1 + m        ## Sum of a matrix of intervals and a matrix
m + matrix1        ## Sum of a matrix and a matrix of intervals
matrix1 + i1       ## Sum of a matrix of intervals and an interval
i1 + matrix1       ## Sum of an interval and a matrix of intervals
matrix1 + list1    ## Sum of a matrix and a list of intervals
list1 + matrix1    ## Sum of a list and a matrix of intervals

matrix1 - matrix2  ## Subtraction of two matrices of intervals
matrix1 - 1        ## Subtraction of a matrix of intervals and a scalar
1 - matrix1        ## Subtraction of a scalar and a matrix of intervals
matrix1 - m        ## Subtraction of a matrix of intervals and a matrix
m - matrix1        ## Subtraction of a matrix and a matrix of intervals
matrix1 - i1       ## Subtraction of a matrix of intervals and an interval
i1 - matrix1       ## Subtraction of an interval and a matrix of intervals
matrix1 - list1    ## Subtraction of a matrix and a list of intervals 
list1 - matrix1    ## Subtraction of a list and a matrix of intervals 
- matrix1

matrix1 * 2        ## Product between a matrix of intervals and a scalar
2 *  matrix1       ## Product between a scalar and a matrix of intervals
matrix1 * m        ## Product between a matrix of intervals and a matrix
m * matrix1        ## Product between a matrix and a matrix of intervals