Allocating Storage for 1-Dimensional Arrays

Transcription

1 Allocating Storage for 1-Dimensional Arrays Recall that if we know beforehand what size we want an array to be, then we allocate storage in the declaration statement, e.g., real, dimension (100 ) :: temperatures However, sometimes we have to perform some calculations or have some values read in before we know an array s size Fortran 90 allows us to allocate storage during run time. In Fortran 77 if we didn t know the size of the array we had to estimate the size (and usually make it a little larger for safety). We still must declare the array as real, integer, etc. and we must tell the compiler that we are going to allocate its size during execution. Also we must declare the shape of the array. The declaration statement now becomes

2 real, dimension (:), allocatable :: temperatures The specifier allocatable tells the compiler we will be setting aside memory for this array during execution. The specifier dimension (:) tells the compiler that this is going to be a one-dimensional array, i.e., it gives the shape of the array. Later we will be looking at two-dimensional and higher arrays. Being able to allocate and release memory during execution is an extremely valuable tool because it allows us to save on storage. How do we allocate the memory during execution? through the use of the allocate statement How do we release memory during execution? through the use of the deallocate statement

3 The allocate/deallocate statements When we declare an array as allocatable, then we are telling the compiler that we will be allocating space during execution. Suppose that we have done some calculations and have determined that the entries of our array temperatures should range from 1 to n where n is the result of the calculations. Then to allocate the array we use allocate ( temperatures (n) ) or equivalently allocate ( temperatures (1:n) ) If we want to allocate more than one array we can put them in separate statements or we can put them in one allocate statement by making a list such as allocate (a(1:n), b(1:n) ) Each of these arrays must be initially declared as allocatable and we must

4 use the dimension specifier dimension (:) an error message.. Otherwise you will receive Where do we put the allocate statement? Basically, anyplace in the executable statements before we need to access the array. Since storage is an issue when we do large scale computations, it is good programming practice to release the storage of an array when we no longer need it. If we don t do this, the storage will be used until the execution of the program is terminated. To do this, we use the deallocate statement. For example, deallocate ( temperatures, a, b ) deallocates the 3 arrays listed. Note that we do not have to indicate their size when we deallocate them.

5 Our Fibonacci example revisited Here we are going to rewrite our example where now we are going to allocate storage for the sequence of numbers during execution. program fibonacci implicit none integer, dimension (:), allocatable :: fib sequence integer :: k,n terms print *, " enter number of terms in Fibonacci sequence" read *, n terms allocate ( fib sequence(1:n terms) )

6 fib sequence(1) = 0 if ( n terms > 1 ) then fib sequence(2) = 1 do k = 3, n terms fib sequence(k) = fib sequence(k-1) + fib sequence(k-2) end do else print *, "number of terms in sequence set to 1" end if deallocate ( fib sequence)

7 end program fibonacci

8 Operations on Vectors Addition and subtraction of vectors. This is done by adding/subtracting the corresponding components, i.e., for a, b R n c = a + b = (a 1 + b 1, a 2 + b 2,, a n + b n ) In fortran we can write c = a + b where the arrays have been dimensioned by real, dimension(n) :: a, b, c However, I prefer the following so that we are explicit that these are arrays and which entries we are modifying. c(1:n) = a(1:n) + b(1:n)

9 If a,b,c are vectors of length n and we just want to add the first m components with m < n then we can write c(1:m) = a(1:m) + b(1:m) Multiplication by a scalar To multiply the vector a by a scalar k, we simply multiply each component by k, i.e., In fortran we write c = k*a or more explicitly c(1:n) = k*a(1:n) c = k a = (ka 1, ka 2,, ka n ) Here k needs to be declared as a real number since the array a was declared as real (we don t use mixed arithmetic).

10 The dot product of two vectors Recall that the dot or scalar product of two vectors is a scalar which can be found by summing the product of their components, i.e., n c = a b = a i b i = a 1 b 1 + a 2 b a n b n i=1 This dot product is also useful when we want to determine the Euclidean length of a vector, i.e., a measure of its size a a We will be especially interested in this length if a represents a vector whose components represent components of the error we are making. Fortran has an intrinsic subprogram for computing the dot product of two one-dimensional arrays of the same length. c = dot product (a, b ) If arrays a and b are not of the same length, then you will get an error message. If arrays a and b are not one-dimensional arrays you will get an error.

11 Other Intrinsic Array-Processing Subprograms Fortran 90 has several other intrinsic functions for processing one-dimensional arrays. maxval (a ) This returns the maximum (not in absolute value) value in the given array. For example a= (/ 1.0, 3.0, -2.0,-5.0, 3.0 /) value = maxval (a ) returns the value 3.0 maxloc (a ) This returns a one-dimensional array containing one element whose value is the position of the first occurrence of the maximum value in a. This is an integer. For example

12 integer, dimension(1) :: location = maxloc (a ) location returns the integer location 2 since this is the first occurrence of the max value 3.0 minval(a), minloc (a) are defined analogously. product (a) i.e., returns the product (a scalar) of the elements of the array, a 1 a 2 a n size (a) returns the number of elements in the array. This can be useful because it can give us the option of not passing the size of a one-dimensional array to a subprogram. sum (a) returns the sum (a scalar) of the entries of the array, i.e., a 1 + a a n

13 Example - Finding the mean of a list Suppose we have a 1-dimensional array of real numbers which represents scores throughout the semester. For example, integer, parameter :: n = 9 real, dimension(n) :: scores scores=(\ 75, 92, 81, 68, 85, 89, 82, 77, 75 \) We can average the scores with the sum intrinsic function. average = sum(scores)/(float(n) 100)

14 Other Operations on Vectors For vectors in R 3 we can take the cross product which results in another vector c = a b = (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ) Fortran does not have an intrinsic routine for the cross product. We can find the angle, θ, between two vectors from the formula so in fortran we could write cos θ = a b a b theta = acos( dot product(a, b)/ & ( sqrt(dot product(a, a)) sqrt(dot product(b, b)) ) ) where recall that acos is the arccosine; e.g., arccosine(1) = 0 that is, the angle whose cosine is one is zero degrees.

15 There are a myriad of vector identities that can also be determined. We have seen that we can find the Euclidean length of a vector by taking the square root of the dot product of the vector with itself. There are other ways to quantify the length of a vector.

16 Classwork Write a program test vectors.f90 to set n as an integer parameter, say n = 4 and dimension real arrays a, b by n. Set the values of array a to be (1., 2., 3., 5.) by using the syntax a = (/ /) Print out all of a using the format 4f12.4 Print out only the first 2 components of a in an unformatted write. Set the one-dimensional array b to zero. Then reset the second element to -2 and the fourth entry to one and print out. Declare a new 1-dimensional array c as allocatable. Then read in the value 4 for the dimension of the array and then allocate it. Add code to set c = 2 a 1 2 b and print out c.

17 Add code to modify the vector a so that its first three components are multiplied by 2. Print out a to verify that the change has been properly made. Redefine c as c = a b and print out c. Add code to compute the length of c = c c and print the length.

18 Review of Vectors We first stated that vectors or one-dimensional arrays will be a useful data structure. All entries of the array must be of the same type; e.g., integer, real, etc. Syntax for setting aside memory for our one-dimensional array: Compile time example real, dimension(n) :: where n has been defined. Run time example real, allocatable, dimension(:) :: a allocate ( a (n) ) deallocate ( a ) We accessed a particular element in our array by, for example a

19 print *, a(5) or a section of an array print *, a(10:20) We can set the values of an array in several ways. We can read them from a file, hard wire them (as a(1)=1.2 ) or assign all entries using format a = (/ 1.0, 2.0, -3.0 /) Next we became familiar with some simple operations with vectors. c(1:n) = a(1:n) + b(1:n) c(1:n) = k * a(1:n) We also looked at some intrinsic subprograms in fortran whose arguments are one-dimensional arrays. These include dot product( a(1:n), b(1:n) maxval(a) size(a)

20 Vector Norms An important part of scientific computing is validation that your program is working properly. One way to do this is to compare your computational errors to expected theoretical errors or rates of convergence. In the last section for nonlinear equations we looked at the rate of convergence of a method and could compare this with the theoretical rates. In many cases we will have an error vector instead of a single scalar error (which we had in our nonlinear problems). From this vector we would like to obtain a single non-negative number. There are many ways to do this and which measure of the error you use can depend on what error you are trying to control. A vector norm will be a quantification of a vector in some sense.

21 How will we denote a norm? Commonly we use the notation a A subscript is usually added to indicate which norm, e.g., a 2. What properties do we want a norm to possess? Since the norm will be some measure of the length of a vector, the norm should always be 0 and only = 0 if the vector is the zero vector. The norm of a scalar times a vector, i.e., k a, must satisfy k a = k a So if we take a vector and multiply it by -3 then its length should change by 3. The triangle inequality a + b a + b This inequality derives its name from vectors in Euclidean space where in R 2 if a and b are the sides of a triangle, the length of the hypotenuse is less than the sum of the lengths of the two sides.

22 Three Common Vector Norms 1. Probably the most commonly used norm is the standard Euclidean length (sometimes called the l 2 norm ) a 2 = n a 2 i = ( a a a 2 n) 1/2 i=1 When the vector represents an error and we don t normalize this by the norm of the solution, then we must normalize by its length, i.e., a 2 = 1 n a 2 i n Why do we do this? Suppose we have a vector which represents components of our error and all of its components are exactly 0.1; whether the vector has i=1

23 length 10 or 100 we would like this measure of the error, i.e., e 2 to be the same. This will be the case if we normalize as above. For example, for a vector of length 10 ( ) 1/2 = (.1) 1/2 = ( 1 ) 1/2 10 ( ) = 0.1 and for a vector of length 100 ( 1 ) 1/2 ( ) 1/2 = (.1) 1/2 = ( ) = 0.1 However, if we always measure the error in a relative sense by normalizing by the norm of the solution, then this is not necessary. 2. Another commonly used measure is the max norm or the infinity norm a = max i=1,...,n a i

24 3. A third commonly used measure is the one norm a 1 = n a i = i=1 ( ) a 1 + a a n

25 Example Consider the vector v = Then the Euclidean norm, called the l 2 norm and denoted by 2 is given by v 2 = ( 5) 2 = = 30 = The infinity or max norm, denoted is given by v = max i=1,2,3 v i = max{2, 1, 5 } = 5 The one-norm, denoted 1 is given by v 1 = = 8.

26 Passing Arrays to a Subprogram Now we want to write functions to calculate the various norms of a vector. Suppose we want to write a routine to calculate the infinity norm of a vector. Recall that we can not use the intrinsic function maxval because this returns the maximum value but not in absolute value. Consequently, we have to write our own subprogram to do this. If we pass an array to a subprogram then the array must be defined in the calling unit and in the subprogram. When the subprogram is referenced, the first element of the actual array in the calling program is associated with the first element of the corresponding array in the subprogram. Successive array elements are then associated. Of course like variables, the name in the calling program doesn t have to match the array name in the subprogram.

27 To write a routine to calculate the maximum norm, we need to pass the vector whose norm we are going to compute. What about the length of the vector? We could always write function max norm ( x, n) result (norm) integer :: n real, dimension(n) :: x where our declaration statement for the vector x is the usual compile-time declaration. It doesn t matter whether the vector was an allocatable array or not in the calling program. However, Fortran allows us an alternative to passing the size of a one-dimensional array argument to a subprogram. We can simply use an assumed-shape array. For our problem this would be function max norm ( x ) result(norm) integer :: n real, dimension(:), intent(in) :: x

28 Warning: This works for one-dimensional arrays; for higher dimensional arrays we must specify the actual size (more precisely, we can only use an assumed shape for the last dimension) So the safest thing is to send the dimension of the array into the subprogram. Warning: If you need the length of a vector for a do loop then you can not use the size function to get this if you have dimensioned the array in your subprogram with the assumed shape (:). Also when you use intrinsics like dot product with two vectors defined through the assumed shape you must write the command as dot product ( a(1:n), b(1:n) ) instead of dot product (a, b) since the subprogram does not have the explicit value of n. The same is true using commands like maxval, etc. So the safest thing is to send the dimension of the array into the subprogram.

29 Classwork Now return to your code from last time test vectors.f90. First we want to write a function which returns the Euclidean length of a vector. Pass both the vector and its length as arguments to the function. Add this to the program test vector.f90 and test it with the vector c. Add print statements in the main program to output c 2. We want to write a function which computes the max or infinity norm of a vector. Pass both the vector and its length as arguments to the function. Add this to the program test vector.f90 and test it with the vector c.add print statements in the main program to output c. Remove the two functions from the code and put them in a module called norms. Modify your program test vectors.f90 to have access to this module and then compute c 2 and c.

30 Multi-dimensional Arrays In mathematics, we call 2-dimensional arrays as matrices. For example 1 4 A = is 3 by 2 matrix because it has 3 rows and 2 columns. To denote an element of the matrix we write A ij, e.g., A 2,1 = 2 An array in fortran can hold the entries of a matrix or it can just be used to store data. For example, suppose a firm has 10 employees each of whom work part time during the 5-day work week. We could represent their hours worked by a 10 5 array where each row of the array is for a different employee and each column is the number of hours worked; the first column Monday, second

31 column Tuesday, etc How do we dimension a higher dimensional array? Analogous to what we did for a 1-dimensional array. For example, for a 10 5 real array named hours real, dimension(10, 5) :: hours which has 10 5 = 50 storage locations. We can also allocate the storage in real time. real, allocatable, dimension(:, :) :: hours We can also set up integer arrays allocate(hours(10, 5)) integer, dimension(5, 5) :: a sets up an integer array a which has 25 integer storage locations.

32 We can also set up character or logical arrays. character(len = 10), dimension(5, 5) :: employees We can also set up storage arrays that have more than two dimensions, e.g., real, dimension(5, 3, 2) :: value sets up a real 3-dimensional array with = 30 storage locations. Fortran allows the use of up to 7-dimensional arrays but these are rarely used. You can access part of a matrix, just like you accessed part of a vector. For example, if we want to print out the 10th row of a 2-dimensional array we use and if we want the second column print, a(10, :) print, a(:, 2)

33 Example: Creating the Hilbert Matrix The Hilbert matrix is a notoriously bad matrix with very undesirable properties. However, its entries are easy to define and the matrix does appear in applications. If H is the Hilbert matrix then its (i, j) is defined by 1 H ij = i + j 1 For example, a 4 4 Hilbert matrix is Now suppose we want to write a program to create an n n Hilbert matrix. We query the user to input n which means that the array must be allocated during

34 run time. Our code might look like this. Here I have omitted the program, implicit none, and common statements for brevity. real, allocatable, dimension(:,: ) :: h integer :: i,j integer :: n print *, "enter the size of the desired Hilbert matrix" read *, n write(*,110) n,n allocate ( h(1:n, 1:n) ) do i = 1, n do j = 1,n h (i,j) = one / float(i+j-1)

35 end do end do print *, do i = 1, n write(*,120) h(i,:)! print a row at a time end do 110 format ("We are creating a ", I3, " by", I3, " Hilbert matrix" 120 format ( 10 (f8.5,2x) )

36 and if we set n = 6 we get We are creating a 6 by 6 Hilbert matrix

37 Mean Time to Failure - an example of handling arrays This example is taken from your suggested text Fortran 90 for Scientists and Engineers. One important statistic used in measuring the reliability of a component of a circuit is the mean time to failure which can be used to predict the circuit s lifetime. This is especially important in situations where repair is difficult such as for a space satellite. If only one circuit is tested, then we read in each failure time, add it to the previous failure time until all the times have been summed. We then average the failure times and get the result. We wouldn t have to use a 1-dimensional array with this approach. If we choose to use a 1-dimensional array then we simply read in all the failure times at once and store them in a vector and then we calculate the mean.

38 However, now assume that we have tested a number of circuits in a satellite. For each circuit we have tested it several times and recorded the time it failed. Now we want to find the mean time of failure for each circuit and then print out the circuit number with the smallest mean time to failure. Assume that 5 circuits were tested 10 times each and the following failures times is in the file failures.txt. The 10 failures times for circuit #1 are on the first line of the file, 10 failure times for circuit # 2 are on the second line of the file, etc. We first want to read these values into a 2-dimensional array of size We will use an unformatted read and use a do loop. We will output the failure times to check to make sure they are read in correctly. integer, parameter :: n_circuit = 5 integer, parameter :: n_test = 10 real, dimension (n_circuit, n_test) :: times real, dimension (n_circuit) :: mean real :: sum_failure, mean_small integer :: i, circuit_no

39 open ( unit=15, file="failures.txt")! Read in the failure times do i=1, n_circuit read (15,*) times(i,:) write(*,120) times(i,:) end do

40 Now the next thing to do is to calculate the average of the failure times for each circuit. Clearly we need a loop over the number of circuits. To calculate the sum of all the failure times for each circuit we could add another do loop but, if you remember, fortran has an intrinsic function sum which sums the entires in a vector. But we want to sum a row of a matrix but we can access it like a vector. For example, sum (a(1,:) sums the entries in the first row of an array a.! Calculate the mean for each circuit and store in mean do i = 1, n_circuit sum_failure = sum (times(i,:) ) mean (i ) = sum_failure / float(n_test) write(*,125) i, mean(i) end do mean time to failure for circuit number 1 is

41 mean time to failure for circuit number 2 is mean time to failure for circuit number 3 is mean time to failure for circuit number 4 is mean time to failure for circuit number 5 is Lastly we want the computer to find the circuit which has the shortest mean time to failure. From our output, this is clearly circuit # 2 but we want the computer to determine this for us because in practice we might have thousands of circuits. How do we do this? We could just set a variable mean small to a large number (since we are looking for the smallest) and loop over the calculated means and check to see if it is smaller than the current value of mean small. If it is, replace mean small by mean(i). Also we want to keep track of the circuit number.

42 ! Find the circuit with the shortest mean failure time mean_small = do i = 1, n_circuit if (mean(i) < mean_small ) then circuit_no = i mean_small = mean(i) end if end do write(*,130) circuit_no, mean_small Circuit number 2 has shortest mean time to failure of

43 Classwork 1. The file populations.txt contains the populations of 6 cities in the U.S. Each line of the file gives the census in 2010 and in Write a code to read in these populations using an unformatted read and store in a 2-dimensional real array. 2. The name of the cities are listed below the populations in the same file. Dimension a character array and then add code to read in the names of the cities. There is one on each line. Use an unformatted read. We need the names of the cities for printing. For example, if n cities has been defined as the number of cities we have character(len = 12), dimension(n cities) :: city names 3. Add code to calculate the percent increase in population each city has. Store this percent in an array and print out the information using the name of the city.

44 4. Add code to find the city with the largest increase in population over the four year period. Output your result using the name of the city. 5. Add code to find the city with the smallest increase in population over the four year period. Output your result using the name of the city.