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3.1 Introduction
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Lists are general objects that contain collections of other objects. In reading this chapter you will see that lists are used for a variety of applications. Therefore, Mathematica offers an extensive collection of list manipulation commands. The objects within a list are contained within curly brackets, {}. Alternatively, the List command may be used to define a list.
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List[elements] represents a list of objects. elements represents the members of the list separated by commas. List[elements] is equivalent to {elements}. 2, 3, 4} is a list of numbers. List[1, 2, 3, 4] represents the same list.
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EXAMPLE 1
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List[a, b, c, d]
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b, c, d}
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List[a, b, c, d] is equivalent to {a, b, c, d}.
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Lists can be given symbolic names so they can be easily referenced. Any operation performed on a list will be performed on each element of the list.
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EXAMPLE 2
list = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 1/list
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2 3 4 5 6 7 8 9 10
When executing these instructions, care must be taken to use a lowercase l in list to avoid conflict with the Mathematica command List.
list2 {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} list {1, 2, 3, 2, 5, 6, 7, 2 2, 3, 10}
If two or more lists contain the same number of elements, new lists can be created using standard operations.
EXAMPLE 3
list1 = {1, 2, 3, 4, 5}; list2 = {2, 3, 2, 3, 2};
list1 + list2 5, 5, 7, 7}
Lists
list1 * list2 {2, 6, 6, 12, 10} list1/list2
1 2 3 4 5 , , , , 2 3 2 3 2
list1list2 {1, 8, 9, 64, 25}
The following list commands are simple but extremely useful:
Total[list] gives the sum of the elements of list. Accumulate[list] returns a list having the same length as list containing the successive partial sums of list. Max[list] returns the largest number in list. Min[list] returns the smallest number in list.
EXAMPLE 4
list = {1, 2, 3, 4, 5} Total[list] 15 Accumulate[list] {1, 3, 6, 10, 15} Max[list] 5 Min[list] 1
SOLVED PROBLEMS
3.1 Construct a list of the factorials of the integers 1 through 10.
SOLUTION
list = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
list! 2, 6, 24, 120, 720, 5040, 40 320, 362 880, 3 628 800}
3.2 Construct a list of the first ten positive integer powers of 2.
SOLUTION
list = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}; 2list
4, 8, 16, 32, 64, 128, 256, 512, 1024}
3.3 Construct a list whose elements are the sum of the squares of the first five positive integers added to their respective cubes.
SOLUTION
list = (1, 2, 3, 4, 5} list2 + list3 or list^2 + list^3 {2, 12, 36, 80, 150}
Lists
3.4 Define list1 = {1, 3, 5, 7, 9} and list2 = {2, 4, 6, 8, 10}. Construct a list whose five elements are the products of the entries of the two lists.
SOLUTION
list1 = {1, 3, 5, 7, 9}; list2 = {2, 4, 6, 8, 10};
list1 * list2 12, 30, 56, 90}
3.2 Generating Lists
The most common lists are lists of equally spaced numbers. The Range command allows convenient construction. The values of m, n, and d in the following description need not be integer valued. Negative values are acceptable as well.
Range[n] generates a list of the first n consecutive integers. Range[m, n] generates a list of numbers from m to n in unit increments. Range[m, n, d] generates a list of numbers from m through n in increments of d.
EXAMPLE 5
Range[10] {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Range[5, 10] {5, 6, 7, 8, 9, 10} Range[25, 5, 2] {25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5} Range[1/3, 1, 1/12] 1, 5 , 1, 7 , 2, 3, 5, 11, 1 3 12 2 12 3 4 6 12 Range[1, 2, .1] {1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.}
Lists with more complicated structures can be constructed with the Table command. There are several different forms.
Table[expression, {n}] generates a list containing n copies of the object expression. Table[expression, {k, n}] generates a list of the values of expression as k varies from 1 to n. Table[expression, {k, m, n}] generates a list of the values of expression as k varies from m to n. Table[expression, {k, m, n, d}] generates a list of the values of expression as k varies from m to n in steps of d.
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