Expressions :: Matrices and Arrays (MATLAB®)

Expressions

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Variables Numbers Operators Functions Examples of Expressions

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Variables

Like most other programming languages, the MATLAB language provides mathematical expressions, but unlike most programming languages, these expressions involve entire matrices.

MATLAB does not require any type declarations or dimension statements. When MATLAB encounters a new variable name, it automatically creates the variable and allocates the appropriate amount of storage. If the variable already exists, MATLAB changes its contents and, if necessary, allocates new storage. For example,

num_students = 25

creates a 1-by-1 matrix named num_students and stores the value 25 in its single element. To view the matrix assigned to any variable, simply enter the variable name.

Variable names consist of a letter, followed by any number of letters, digits, or underscores. MATLAB is case sensitive; it distinguishes between uppercase and lowercase letters. A and a are not the same variable.

Although variable names can be of any length, MATLAB uses only the first N characters of the name, (where N is the number returned by the function namelengthmax), and ignores the rest. Hence, it is important to make each variable name unique in the first N characters to enable MATLAB to distinguish variables.

N = namelengthmax
N =
    63

The genvarname function can be useful in creating variable names that are both valid and unique.

Numbers

MATLAB uses conventional decimal notation, with an optional decimal point and leading plus or minus sign, for numbers. Scientific notation uses the letter e to specify a power-of-ten scale factor. Imaginary numbers use either i or j as a suffix. Some examples of legal numbers are

3              -99            0.0001
9.6397238      1.60210e-20    6.02252e23
1i             -3.14159j      3e5i

MATLAB stores all numbers internally using the long format specified by the IEEE^® floating-point standard. Floating-point numbers have a finite precision of roughly 16 significant decimal digits and a finite range of roughly 10^-308 to 10⁺³⁰⁸.

Numbers represented in the double format have a maximum precision of 52 bits. Any double requiring more bits than 52 loses some precision. For example, the following code shows two unequal values to be equal because they are both truncated:

x = 36028797018963968;
y = 36028797018963972;
x == y
ans =
      1

Integers have available precisions of 8-bit, 16-bit, 32-bit, and 64-bit. Storing the same numbers as 64-bit integers preserves precision:

x = uint64(36028797018963968);
y = uint64(36028797018963972);
x == y
ans =
      0

The section Avoiding Common Problems with Floating-Point Arithmetic gives a few of the examples showing how IEEE floating-point arithmetic affects computations in MATLAB. For more examples and information, see Technical Note 1108 — Common Problems with Floating-Point Arithmetic.

MATLAB software stores the real and imaginary parts of a complex number. It handles the magnitude of the parts in different ways depending on the context. For instance, the sort function sorts based on magnitude and resolves ties by phase angle.

sort([3+4i, 4+3i])
ans =
   4.0000 + 3.0000i   3.0000 + 4.0000i

This is because of the phase angle:

angle(3+4i)
ans =
    0.9273
angle(4+3i)
ans =
    0.6435

The "equal to" relational operator == requires both the real and imaginary parts to be equal. The other binary relational operators > <, >=, and <= ignore the imaginary part of the number and consider the real part only.

Operators

Expressions use familiar arithmetic operators and precedence rules.

`+`	Addition
-	Subtraction
`*`	Multiplication
`/`	Division
`\`	Left division (described in Linear Algebra in the MATLAB documentation)
`^`	Power
`'`	Complex conjugate transpose
`( )`	Specify evaluation order

Functions

MATLAB provides a large number of standard elementary mathematical functions, including abs, sqrt, exp, and sin. Taking the square root or logarithm of a negative number is not an error; the appropriate complex result is produced automatically. MATLAB also provides many more advanced mathematical functions, including Bessel and gamma functions. Most of these functions accept complex arguments. For a list of the elementary mathematical functions, type

help elfun

For a list of more advanced mathematical and matrix functions, type

help specfun
help elmat

Some of the functions, like sqrt and sin, are built in. Built-in functions are part of the MATLAB core so they are very efficient, but the computational details are not readily accessible. Other functions are implemented in the MATLAB programing language, so their computational details are accessible.

There are some differences between built-in functions and other functions. For example, for built-in functions, you cannot see the code. For other functions, you can see the code and even modify it if you want.

Several special functions provide values of useful constants.

`pi`	3.14159265...
`i`	Imaginary unit,
`j`	Same as `i`
`eps`	Floating-point relative precision,
`realmin`	Smallest floating-point number,
`realmax`	Largest floating-point number,
`Inf`	Infinity
`NaN`	Not-a-number

Infinity is generated by dividing a nonzero value by zero, or by evaluating well defined mathematical expressions that overflow, i.e., exceed realmax. Not-a-number is generated by trying to evaluate expressions like 0/0 or Inf-Inf that do not have well defined mathematical values.

The function names are not reserved. It is possible to overwrite any of them with a new variable, such as

eps = 1.e-6

and then use that value in subsequent calculations. The original function can be restored with

clear eps

Examples of Expressions

You have already seen several examples of MATLAB expressions. Here are a few more examples, and the resulting values:

rho = (1+sqrt(5))/2
rho =
    1.6180

a = abs(3+4i)
a =
     5

z = sqrt(besselk(4/3,rho-i))
z =
   0.3730+ 0.3214i

huge = exp(log(realmax))
huge =
  1.7977e+308

toobig = pi*huge
toobig =
   Inf