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The simplest tool you can use to estimate code performance is
the `time`

function.
This function measures the running time of your code snippet. The `time`

function does not
count the time spent outside of MuPAD^{®} processes: external function
calls or processes running in the background do not affect the results
of `time`

. The
function returns results measured in milliseconds.

The following example demonstrates different algorithms implemented
for the same task. The task is to check whether each integer from
1 to 1000 appears in a 1000×1000 matrix
of random integers. To create a 1000×1000 matrix
of random integers, use `linalg::randomMatrix`

:

matrixSize := 1000: M := linalg::randomMatrix(matrixSize, matrixSize, Dom::Integer):

The direct approach is to write a procedure that checks every element of a matrix proceeding row by row and column by column:

f := proc(M, n, x) begin for j from 1 to n do for k from 1 to n do if M[j, k] = x then return(TRUE) end_if end_for end_for; return(FALSE) end_proc:

Call the procedure `f`

1000 times to check
if each number from 1 to 1000 appears in that matrix:

g := proc() begin f(M, matrixSize, i) $ i = 1..1000 end_proc:

This algorithm is very inefficient for the specified task. The
function `f`

performs 10^{4} computation
steps to find out that an integer does not occur in the matrix `M`

.
Since you call the function `f`

1000 times, executing
this algorithm takes a long time:

time(g())

62435.902

In this example, the bottleneck of the chosen approach is obviously
the algorithm that accesses each matrix element. To accelerate the
computation, rewrite the procedure `f`

using the
bisection method. Before using this method, convert the matrix `M`

to
a list and sort the list. Then select the first and last elements
of the sorted list as initial points. Each step in this algorithm
divides the list of elements at the midpoint:

f := proc(M, n, x) begin if (M[1] - x)*(M[n] - x) > 0 then return(FALSE) elif (M[1] - x)*(M[n] - x) = 0 then return(TRUE); else a := 1: b := n: while (b - a > 1) do if is(b - a, Type::Odd) then c := a + (b - a + 1)/2 else c := a + (b - a)/2 end_if; if M[c] - x = 0 then return(TRUE) elif (M[a] - x)*(M[c] - x) < 0 then b := c: else a := c: end_if; end_while; end_if; return(FALSE) end_proc:

Use the `op`

function
to access all elements of the matrix `M`

. This function
returns a sequence of elements. Use brackets to convert this sequence
to a list. Then use the `sort`

function
to sort the list in ascending order. Finally, call the procedure `f`

for
each integer from 1 to 1000:

g := proc() local M1; begin M1 := sort([op(M)]): f(M1, matrixSize^2, i) $ i = 1..1000 end_proc:

Using the bisection method instead of accessing each matrix element significantly improves the performance of the example:

time(g())

3724.233

Typically, the best approach is to use the appropriate MuPAD functions
whenever possible. For example, to improve performance further, rewrite
the code using the MuPAD function `has`

. Also, converting a matrix to a set
can reduce the number of elements. (MuPAD removes duplicate elements
of a set.) In addition to speed up, this approach makes your code
significantly shorter and easier to read:

g := proc() local M1; begin M1 := {op(M)}: has(M1, i) $ i = 1..1000 end_proc:

In this case, execution time is even shorter than for the code that implements the bisectional method:

time(g())

1508.094

Results returned by the `time`

function exclude the time spent
on calls to external programs. If your code uses external programs,
you can measure the total time spent by that code, including calls
to external processes. To measure the total time, use the `rtime`

function instead
of `time`

. For
example, the function call `rtime()`

returns the
elapsed time of the current MuPAD session. This time includes
idle time of the current session:

t := rtime(): print(Unquoted, "This session runtime is ".stringlib::formatTime(t))

This session runtime is 5 minutes, 25.579 seconds

When measuring code performance using `rtime`

, avoid running other processes
in the background. Also ensure that enough memory is available. The `rtime`

function counts
the total time, including idle time during which some other process
uses the CPU or your computer swaps data between different types of
memory.