How to use cross apply instead of cursors in SQL Server

I recently ran into a stored procedure with cursor logic, I wanted to see what would be the performance gain if cursor logic is rewritten with set theory operations.

Here is a simplified description of the stored procedure. There is a table with five columns, one column is an identity and a primary, (let's call it the ID), three columns (x, y, z) are of type integers and the fifth column (a) is a computed column. The computation is quite complex so it cannot be declared as a computed column expression. For each row, the values of the three columns (x, y, z) are passed in as a parameters to a custom function (where the logic is encapsulated) which spits out a calculated value. Finally, for every row the column "a" is updated with the calculated value from the custom function. There are around 7000 rows in this table. 

I put this blogpost on code project as well. Check out the following link
http://www.codeproject.com/Tips/654894/How-to-use-cross-apply-instead-of-cursors-in-SQL-S

Let's create a table called test1 with five columns. For this example lets stick with a simple logic, the fifth column is a sum of cols x,y,z.   

Step 1: Create the test table  
create table test1 
(
    id int not null identity(1,1), 
    x int,
    y int,
    z int,
    a int null
)

Step 2: Insert dummy data    
insert into test1 values (5,5,5, NULL)
go 10
-- (I inserted 9132 rows, took five minutes for the code to execute) 

Step 3: Using cursors to update the column "a"  

declare @x1 int
declare @x2 int
declare @x3 int
declare @x4 int
declare @x5 int
declare c1 cursor local for  
select id, x, y, z, a from test1
open c1
    while (0=0)
        begin
        fetch next from c1 into 
        @x1, @x2,@x3,@x4,@x5
        
        if (@@FETCH_STATUS = -1)
            break
        
        -- your logic
        set @x5 = @x4 + @x2 + @x3
        
    
        update test1
        set test1.a = @x5
        from test1
        where id = @x1
        end
close c1
deallocate c1

-- Exexcution time 01:09

Step 4: Reset column "a" 

update test1 set a = null 

Step 5: Create a table valued function (tvf) shown below

create function dbo.fnsomelogic (@x int ,@y int, @z int)
returns @val table
(
    q int
)
as 
begin 
    declare @q int 
    set @q = @x + @y +@z    
    insert into @val (q) values (@q)
    return
end

-- tvf can be invoked as shown below
-- select * from dbo.fnsomelogic(1,2,3)

Step 6: use cross apply and the tvf  to update column a

update test1
set test1.a = c.q
from test1 b cross apply dbo.fnsomelogic(b.x,b.y,b.z) c
--Execution Time: (9132 row(s) affected) in less than a second.   

If you observe the messages, the cursor which is a row based operator, displays (1 row(s) affected) for every row it updated, unlike the cross apply which displays (9132 row(s) affected). Although the problem is screaming out use cursors, however with little observation a cross apply along with a table valued function can boost the performance significantly. Relational/Set theory concepts are deeply embedded within SQL Server.