Go to Appendix 10 
Go to Index 
Go to Appendix 12These computer exercises are small variants of those used at the teaching of Fortran at Linköping University, both for the students of technology and for the students in the mathematics and science program. Students of technology are supposed to solve the exercises 2 to 5, while mathematicians are supposed to solve the exercises 1, 2, 3, 4a and 5.
The Pascal program below solves an ordinary differential equation with the Runge-Kutta method. The assignment is to translate it from Pascal to Fortran. The output is required to include not only the numeric values but also suitable explanatory text. It is also valuable if the program can use other input values than those given in the Pascal code, 1 and 2, respectively.
For those who do not know Pascal the recommended alternative is to get an elementary textbook on numerical methods and code Runge-Kutta directly in Fortran.
program RK1;
(* A simple program in Pascal for Runge-Kutta's method for a
first order differential equation.
dy/dx = x^2 + sin(xy)
y(1) = 2 *)
var number, i : integer;
h, k1, k2, k3, k4, x, y : real;
function f(x,y : real) : real;
begin
f := x*x + sin(x*y)
end;
begin
number := 1;
while number > 0 do
begin
x := 1.0;
y := 2.0;
writeln(' Give the number of steps ');
read(number);
if number >= 1 then
begin
writeln(' Give the step length ');
read(h);
writeln(' x y');
writeln(x, y);
for i := 1 to number do
begin
k1 := h*f(x,y);
k2 := h*f(x+0.5*h,y+0.5*k1);
k3 := h*f(x+0.5*h,y+0.5*k2);
k4 := h*f(x+h,y+k3);
x := x + h;
y := y + (k1+2*k2+2*k3+k4)/6;
writeln(x, y);
end;
end;
end;
end.
If you wish to run this program you have to replace the first line
program RK1;
with the following extended line
program RK1(INPUT, OUTPUT);
when running on a DEC UNIX system, but not on a Sun Solaris system.
The program below is in Fortran 77 (or fix form Fortran 90). The assignment is first to correct the errors in the program. There are no errors in the numerical algorithm (Horner's scheme) or in the comments in the program. But some programming errors are included in the presented program, several of these are based on a mixture of Fortran and Pascal.
When the program is believed to be correct test with the equation x*x + 2 = 0.
Note that the program uses complex numbers, and that such values are input as pairs within parenthesis.
******************************************************************************
** This program calculates all roots (real and/or complex)
to an **
** N:th-degree polynomial with complex coefficients. (N
<= 10) **
**
**
**
n n-1
**
** P(z) = a z + a z
+ ... + a z + a
**
** n
n-1
1
0
**
**
**
** The execution terminates if
**
**
**
**
1) Abs (Z1-Z0) < EPS ==> Root found
= Z1 **
**
2) ITER > MAX
==>
Slow convergence **
**
**
** The program sets EPS to 1.0E-7 and MAX to 30
**
**
**
** The NEWTON-RAPHSON method is used: z1 = z0 -
P(z0)/P'(z0) **
** The values of P(z0) and P'(z0) are calculated with
HORNER'S SCHEME. **
**
**
** The array A(0:10) contains the complex coefficients of
the **
** polynomial P(z).
**
** The array B(1:10) contains the complex coefficients of
the **
** polynomial Q(z),
**
**
where P(Z) = (z-z0)*Q(z) + P(z0)
**
** The coefficients to Q(z) are calculated with HORNER'S
SCHEME. **
**
**
** When the first root has been obtained with the
NEWTON-RAPHSON **
** method, it is divided away (deflation).
**
** The quotient polynomial is Q(z).
**
** The process is repeated, now using the coefficients of
Q(z) as **
** input data.
**
** As a starting approximation the value STARTV = 1+i is
used **
** in all cases.
**
** Z0 is the previous approximation to the root.
**
** Z1 is the latest approximation to the root.
**
** F0 = P(Z0)
**
** FPRIM0 = P'(Z0)
**
******************************************************************************
COMPLEX A(0:10), B(1:10), Z0,
Z1, STARTV
INTEGER N, I, ITER, MAX
REAL EPS
DATA EPS/1E-7/, MAX /30/, STARTV /(1,1)/
******************************************************************************
20 WRITE(6,*) 'Give the degree of the polynomial'
READ (5,*) N
IF (N .GT. 10) THEN
WRITE(6,*) 'The polynomial degree is
maximum 10'
GOTO 20
WRITE (6,*) 'Give the coefficients of the polynomial,',
, ' as complex constants'
WRITE (6,*) 'Highest degree coefficient first'
DO 30 I = N, 0, -1
WRITE (6,*) 'A(' , I, ') = '
READ (5,*) A(I)
30 CONTINUE
WRITE (5,*) ' The roots are','
Number of iterations'
******************************************************************************
40 IF (N GT 0) THEN
C ****** Find the next root
******
Z0 = (0,0)
ITER = 0
Z1 = STARTV
50 IF (ABS(Z1-Z0) .GE. EPS) THEN
C ++++++ Continue the iteration until two
estimates ++++++
C ++++++ are sufficiently close.
++++++
ITER = ITER
+ 1
IF (ITER
.GT. MAX) THEN
C ------ Too many
iterations ==> TERMINATE
------
WRITE (6,*) 'Too many iterations.'
WRITE (6,*) 'The latest approximation to the root is ',Z1
GOTO 200
ENDIF
Z0 = Z1
HORNER (N,
A, B, Z0, F0, FPRIM0)
C ++++++
NEWTON-RAPHSON'S METHOD
++++++
Z1 = Z0 -
F0/FPRIM0
GOTO 50
ENDIF
C
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
100 WRITE (6,*) Z1, '
',Iter
C ****** The root is found. Divide it away
and seek the next one *****
N = N - 1
FOR I = 0 TO N DO
A(I) =
B(I+1)
GOTO 40
ENDIF
200 END
SUBROUTINE HORNER(N, A, B, Z, F, FPRIM)
****** For the meaning of the parameters - see the main program
******
****** BI and CI are temporary variable.
******
INTEGER N, I
COMPLEX A(1:10), B(0:10), Z, F, FPRIM, BI, CI
BI = A(N)
B(N) = BI
CI = BI
DO 60 I = N-1, 1, -1
BI = A(I) + Z*BI
CI = BI + Z*CI
B(I) = BI
C ++++++ BI = B(I) for
the calculation of P(Z)
++++++
C ++++++ CI for
the calculation of P'(Z)
++++++
60 CONTINUE
F = A(0) + Z*BI
FPRIM = CI
RETURN
END
****** END HORNER'S SCHEME
******
****** This program is composed by Ulla Ouchterlony 1984
******
****** The program is translated by Bo Einarsson 1995
******
Comments. I recommend the following process at the solution of this exercise.
f77 -u lab2_e.f
a.out
Using a modern system, for example Fortran 90, you put the statement IMPLICIT NONE first in booth the main program and the subroutine.
f77 -C lab2_e.f
a.out
The explicit DO-loop below will write seven lines, each one with one value of X
DO I = 1, 13, 2 WRITE(*,*) X(I) END DO
The implicit DO-loop below will write only one line, but with seven values of X
WRITE(*,*) ( X(I), I = 1, 13, 2)
This is computer exercise number three, and it consists of writing two small programs in the programming language Fortran 77.
Write a function in Fortran 77 with a main program for evaluation of the factorial function. Use only integers. Write the main program in such a way that it asks the user for an integer, for which the factorial is to be calculated. Make a test run and evaluate 10!
Write a program in Fortran 77 for tabulating the Bessel function J0(x). Use for example the NAG Fortran 77 Library, or the NAG Fortran 90 Library, or a similar library, for obtaining values of this function. Write the main program in such a way that it asks for an interval for which the function values are to be evaluated.
Make a test run and print a table with x and J0(x) for x = 0.0, 0.1, ... 1.0. Make the output to look nice!
At the use of the NAG Fortran 77 library the NAG User Notes are very useful. They describe the system specific parts of the library.
The required Bessel function in the NAG library is S17AEF and has the arguments X and IFAIL and in this order, where X is the argument for the Bessel function in single or double precision, and where IFAIL is an error parameter (integer). The error parameter is given the value 1 before entry, and is checked at exit. If it is 0 at exit, everything is OK, if it is 1 at exit, the magnitude of the argument is too large.
If the NAG library is installed in the normal way it is linked with the command
f77 prog.f -lnag
where prog.f is your program.
It is also permitted to use the new NAG Fortran 90 library. Please note that the functions have different names in this library (our Bessel function is nag_bessel_j0), and is linked with
f90 prog.f90 -lnagfl90
If you use Fortran 90 for your own program but has the NAG library available only in Fortran 77 special care is necessary, see chapter 15, Method 2.
On Sun you compile and link with
f90 prog.f -lnag -lF77
and on DEC ULTRIX with
f90 prog.f -lnag -lfor -lutil -li -lots
On DEC UNIX -lfor -lutil -lots are available.
It is very important to note in which precision the NAG library is made available on your system. Using the wrong precision will usually give completely wrong results!
Write a program in Fortran 77 for tabulating the Bosse function Bo(x). Write the main program in such a way that it asks for an interval for which the function values are to be evaluated.
Make a test run and print a table with x and Bo(x) for x = 0.0, 0.1, ... 1.0. Make the output to look nice! Investigate also what happens with the arguments +800 and -900!
The Bosse function BO is in the library libbosse.a on the course directory and has the arguments X and IFAIL and in this order, where X is the argument for the Bosse function in single or double precision, and where IFAIL is an error parameter (integer). The error parameter is given the value 1 before entry, and is checked at exit. If it is 0 at exit, everything is OK, if it is 1 at exit, the argument is too large, if it is 2 at exit, the argument is too small. If the error parameter is zero on entry and an error occurs the program will stop with an error message from the library error handling subroutine, and not return to the user's program. The error handling is similar to the one in the NAG Fortran 77 library.
If the Bosse library is installed in the normal way it is linked with the command
f77 prog.f -lbosse -L$KURSBIB
where prog.f is your program.
If you use Fortran 90 for your own program but you still have the Bosse library available only in Fortran 77 special care is necessary, see chapter 15, Method 2, see also the discussion above in exercise 3b.
To make life simpler we have however made also a Fortran 90 version of the Bosse library, which is used with
f90 prog.f90 -lbosse90 -L$KURSBIB
It is very important to note in which precision the Bosse library is made available on your system. Using the wrong precision will usually give completely wrong results!
This is computer exercise number four, and means full transition from Fortran 77 to Fortran 90. Free form of the source code is required!
Write a recursive function in Fortran 90 with a main program for evaluation of the factorial function. Use only integers. Write the main program in such a way that it asks the user for an integer, for which the factorial is to be calculated. Make a test run and evaluate 10!
Compare with Exercise 3 a.
Write a program in Fortran 90 for the numerical solution of a system of ordinary differential equations using the classical Runge-Kutta method. The following system is given
y'(t) = z(t) y(0) = 1
z'(t) = 2*y*(1+y^2) + t*sin(z) z(0) = 2
Calculate an approximation to y(0.2) using the step size 0.04. Repeat the calculation with the step sizes 0.02 and 0.01. The functions representing the differential functions are to be given as internal functions. It is therefore not permitted (in this exercise) to use the usual external functions.
Starting values of t, y and z are to be given interactively, as well as the step size h.
Note that the requirement not to use external functions makes it a little more difficult to use a subprogram for the Runge-Kutta steps (in this case also the subprogram has to be internal). The aim of this requirement is to give the student an opportunity to use the new internal functions of Fortran 90.
This is computer exercise number five. It consists of writing in Fortran 90 one main program and a number of subprograms, all in double precision, for the solution of a common numerical problem. The problem is to solve a linear system of equations Ax = b, using an available routine for the numerical task. At least four hours are suggested for this exercise.
The first subprogram LASMAT is used to interactively input the floating point values of a quadratic matrix. The subprogram is required to ask for the dimension and give the user the choice of providing from the keyboard either all the values of the matrix, or only those which are different from zero, The matrix shall then be stored in a file, using maximal accuracy and minimal storage space. The possible sparsity of the matrix is however not permitted to be exploited at this storage process. The subprogram shall ask for the name of the file, but the file type (extension) will be .mat. The user is not permitted to give the file type.
The second subprogram LASVEK is used to interactively input the floating point values of a vector. The subprogram is required to ask for the dimension and give the user the choice of providing from the keyboard either all the values of the vector, or only those which are different from zero, The vector shall then be stored in a file, using maximal accuracy and minimal storage space. The possible sparsity of the vector is however not permitted to be exploited at this storage process. The subprogram shall ask for the name of the file, but the file type (extension) will be .vek. The user is not permitted to give the file type.
The third subprogram MATIN reads a matrix from the file with the given name and stores it as an array (matrix) in the subprogram. The user is not permitted to give the file type.
The fourth subprogram VEKIN reads a vector from the file with the given name and stores it as an array (vector) in the subprogram. The user is not permitted to give the file type.
The fifth subprogram MATUT prints a matrix on paper, with rows and columns in the normal way if the dimension is at most 10, and in an easily understandable way if the dimension is larger than 10. The available 132 positions of a typical line printer are to be utilized as well as possible. Output on paper is under UNIX not done directly, so you will have to first write to a file, which is then moved to paper with the UNIX print command lpr or preferably fpr or asa, see the end of Appendix 2. In order to get 132 positions you can usually add -w132 to the print command.
The sixth subprogram VEKUT prints a vector on paper, preferably in the transposed form (line-wise).
The seventh subprogram LOS solves the system of equations through a call to the provided routine SOLVE_LINEAR_SYSTEM, which is given in double precision. No changes are permitted in that routine! If the dimension of the matrix and the vector do not agree, then the subprogram LOS or the main program is required to give an error message. An error message is also required if the routine SOLVE_LINEAR_SYSTEM detects that the matrix is singular.
The main program HUVUD utilizes LASMAT, LASVEK, MATIN and VEKIN in order to get the matrix A and the vector b, calls the solver subprogram LOS and prints the matrix A , the right hand side b and the solution x with the subprograms MATUT and VEKUT.
The dimension in all the programs shall use the dynamic possibilities of Fortran 90. One alternative is to use pointers at the specification, see chapter 12. Another possibility, suggested by Arie ten Cate is to introduce SAVEd arrays in a MODULE. A simple example of this is available.
The program is required to return to the starting point after each calculation, and ask for a new matrix and/or vector. It has to be possible to use a stored set of matrix and vector, without manual input of a lot of values.
In the exercise it is included the essential task of performing any additional specifications, test the reasonableness of the given values, and to construct suitable test matrices and vectors.
A compulsory test example follows
33 16 72 -359 A = -24 -10 -57 b = 281 -8 -4 -17 85
The students are required to hand in a complete program listing, and results from actual runs with matrices of the order 3, 10, and 12, including input of a sparse matrix.
I assume that the main program is on the file huvud.f90, the subroutine LASMAT on the file lasmat.f90, and so on, and that a possible complete program is on the file labb5.f90.
f90 huvud.f90 lasmat.f90 lasvek.f90 ... los.f90 solve.f90
Execute with
a.out
f90 -c huvud.f90
f90 -c lasmat.f90
f90 -c lasvek.f90
...
f90 -c los.f90
f90 -c solve.f90
f90 huvud.o lasmat.o lasvek.o ... los.o solve.o
Execute with
a.out
When one routine has been changed, now only that one has to be recompiled, followed by linking of all the routines. This method is much faster!
f90 labb5.f90
Execute with
a.out
This method is very slow!
labb5: huvud.o lasmat.o lasvek.o matin.o vekin.o \
matut.o vekut.o los.o solve.o
f90 -o labb5 huvud.o lasmat.o lasvek.o matin.o vekin.o \
matut.o vekut.o los.o solve.o
huvud.o: huvud.f90
f90 -c huvud.f90
lasmat.o: lasmat.f90
f90 -c lasmat.f90
lasvek.o: lasvek.f90
f90 -c lasvek.f90
matin.o: matin.f90
f90 -c matin.f90
vekin.o: vekin.f90
f90 -c vekin.f90
matut.o: matut.f90
f90 -c matut.f90
vekut.o: vekut.f90
f90 -c vekut.f90
los.o: los.f90
f90 -c los.f90
solve.o: solve.f90
f90 -c solve.f90
The above is used by moving the file makefile to your directory and when you wish to compile you write only make in the terminal window. Then only those files that have been changed as compared with the previous make command are recompiled (or all files if it is the first time), then all the object modules are linked to a program, ready to run with the command labb5.
If you have used other file names you have to make the appropriate modifications to the file makefile.
In principle make works in such a way that if an item which is after a colon : has a later time than that before the colon, the command on the next line is performed.
Remark. The backslash \ indicates a continuation line in UNIX.
The routine SOLVE_LINEAR_SYSTEM is available in single precision in the course directory with the name solve1.f90 and in double precision with the name solve.f90, and looks as follows. The single precision version is discussed at length in the Solution to Exercise (11.1).
SUBROUTINE SOLVE_LINEAR_SYSTEM(A, X, B, ERROR)
IMPLICIT NONE
! Array specifications
DOUBLE PRECISION, DIMENSION (:, :), INTENT (IN) :: A
DOUBLE PRECISION, DIMENSION (:), INTENT (OUT) :: X
DOUBLE PRECISION, DIMENSION (:), INTENT (IN) :: B
LOGICAL, INTENT (OUT) :: ERROR
! The work area M is A extended with B
DOUBLE PRECISION, DIMENSION (SIZE (B), SIZE (B) + 1) :: M
INTEGER, DIMENSION (1) :: MAX_LOC
DOUBLE PRECISION, DIMENSION (SIZE (B) + 1) :: TEMP_ROW
INTEGER :: N, K
! Initiate M
N = SIZE (B)
M (1:N, 1:N) = A
M (1:N, N+1) = B
! Triangularization phase
ERROR = .FALSE.
TRIANG_LOOP: DO K = 1, N - 1
! Pivotering
MAX_LOC = MAXLOC (ABS (M (K:N, K)))
IF ( MAX_LOC(1) /= 1 ) THEN
TEMP_ROW (K:N+1 ) = M (K, K:N+1)
M (K, K:N+1) = M (K-1+MAX_LOC(1), K:N+1)
M (K-1+MAX_LOC(1), K:N+1) = TEMP_ROW (K:N+1)
END IF
IF (M (K, K) == 0.0D0) THEN
ERROR = .TRUE. ! Singular matrix A
EXIT TRIANG_LOOP
ELSE
TEMP_ROW (K+1:N) = M (K+1:N, K) / M (K, K)
M (K+1:N, K+1:N+1) = M (K+1:N, K+1:N+1) &
- SPREAD( TEMP_ROW (K+1:N), 2, N-K+1) &
* SPREAD( M (K, K+1:N+1), 1, N-K)
M (K+1:N, K) = 0 ! These values are not used
END IF
END DO TRIANG_LOOP
IF (M (N, N) == 0.0D0) ERROR = .TRUE. ! Singular matrix A
! Back substitution
IF (ERROR) THEN
X = 0.0D0
ELSE
DO K = N, 1, -1
X (K) = M (K, N+1) - SUM (M (K, K+1:N) * X (K+1:N))
X (K) = X (K) / M (K, K)
END DO
END IF
END SUBROUTINE SOLVE_LINEAR_SYSTEM
This is the Exercise 5 as described above, but an additional requirement is to use pointers at the specification of the array to store the matrix, using the method in chapter 12.
This is the Exercise 5 as described above, but an additional requirement is to use a module and an allocatable and saved array at the specification of the array to store the matrix, using the alternative method in chapter 12 and which is illustrated in an example.
This is the Exercise 5 as described above, but instead of using the given routine solve.f90 for the solution of the system of equations, you have to select and use a suitable routine from the NAG library. Complete documentation is available in printed documents from NAG, via the NAG Web-site NAG Fortran Library, or NAG fl90 and FL90plus Libraries, respectively.
It is permitted to either use the "old" NAG library in Fortran 77 or the new library in Fortran 90. They are linked with
f90 prog.f90 -lnag
or
f90 prog.f90 -lnagfl90
respectively.
In the first case above, with the NAG library compiled with Fortran 77, it may be necessary to use method 2 from chapter 15, that is you compile and link the necessary libraries with an implementation dependent command.
It is also helpful to use the command naghelp when available.
At the the specification of the array to store the matrix, you may chose either the method in a) or b) above, or perhaps invent a third method.
If the makefile is used, the reference to solve.f90 must be changed.
Go to Appendix 10
Go to Index Go to Appendix 12