I am trying to use PPVAL function from the IMSL library, to evaluate a piecewise polynomial, similar to MATLAB's ppval function.
Sadly, I could not understand well the documentation on how to properly use the function.
I tried to evaluate a simple polynomial.
The polynomials p(x) = 1 and p(x) = x were ok. What I don't understand is when I try to get higher degree polynomials.
The program below tries to evaluate the polynomial p(x)=x^4:
program main
include 'LINK_FNL_SHARED.h'
USE PPVAL_INT
implicit none
double precision :: x(11), y(11)
double precision :: BREAK(2),PPCOEF(5,1)
integer :: ii
break = [0.0d0,10.0d0] ! breakpoints in increasing order
PPCOEF(:,1) = [0.0d0,0.0d0,0.0d0,0.0d0,1.0d0] ! polynomial coefficients
x = dble([0,1,2,3,4,5,6,7,8,9,10])
do ii=1,11
y(ii) = PPVAL(x(ii),break,PPCOEF)
end do
print *, 'x = ', x
print *, 'y = ', y
pause
end program main
But the function returns the polynomial p(x) = x^4/4!.
For degrees 2,3 and 5, the return is always, p(x) = x^2/2!, p(x)=x^3/3!, p(x)=x^5/5!. Why does this factor appear in the ppval function? Why can't I just supply the polynomial coefficients and evaluate it?
Is there another simpler function to evaluate polynomials in Fortran, like MATLAB polyval?
Related
In Newton's method, to solve a nonlinear system of equations we need to find the Jacobian matrix and the determinant of the inverse of the Jacobian matrix.
Here are my component functions,
real function f1(x,y)
parameter (pi = 3.141592653589793)
f1 = log(abs(x-y**2)) - sin(x*y) - sin(pi)
end function f1
real function f2(x,y)
f2 = exp(x*y) + cos(x-y) - 2
end function f2
For the 2x2 case I am computing the Jacobian matrix and determinant of the inverse of Jacobian matrix like this,
x = [2,2]
h = 0.00001
.
.
! calculate approximate partial derivative
! you can make it more accurate by reducing the
! value of h
j11 = (f1(x(1)+h,x(2))-f1(x(1),x(2)))/h
j12 = (f1(x(1),x(2)+h)-f1(x(1),x(2)))/h
j21 = (f2(x(1)+h,x(2))-f2(x(1),x(2)))/h
j22 = (f2(x(1),x(2)+h)-f2(x(1),x(2)))/h
! calculate the Jacobian
J(1,:) = [j11,j12]
J(2,:) = [j21,j22]
! calculate inverse Jacobian
inv_J(1,:) = [J(2,2),-J(1,2)]
inv_J(2,:) = [-J(2,1),J(1,1)]
DET=J(1,1)*J(2,2) - J(1,2)*J(2,1)
inv_J = inv_J/DET
.
.
How do I in Fortran extend this to evaluate a Jacobian for m functions evaluated at n points?
Here is a more flexible jacobian calculator.
The results with the 2×2 test case are what you expect
arguments (x)
2.00000000000000
2.00000000000000
values (y)
1.44994967586787
53.5981500331442
Jacobian
0.807287239448229 3.30728724371454
109.196300248300 109.196300248300
I check the results against a symbolic calculation for the given inputs of
Console.f90
program Console1
use ISO_FORTRAN_ENV
implicit none
! Variables
integer, parameter :: wp = real64
real(wp), parameter :: pi = 3.141592653589793d0
! Interfaces
interface
function fun(x,n,m) result(y)
import
integer, intent(in) :: n,m
real(wp), intent(in) :: x(m)
real(wp) :: y(n)
end function
end interface
real(wp) :: h
real(wp), allocatable :: x(:), y(:), J(:,:)
! Body of Console1
x = [2d0, 2d0]
h = 0.0001d0
print *, "arguments"
print *, x(1)
print *, x(2)
y = test(x,2,2)
print *, "values"
print *, y(1)
print *, y(2)
J = jacobian(test,x,2,h)
print *, "Jacobian"
print *, J(1,:)
print *, J(2,:)
contains
function test(x,n,m) result(y)
! Test case per original question
integer, intent(in) :: n,m
real(wp), intent(in) :: x(m)
real(wp) :: y(n)
y(1) = log(abs(x(1)-x(2)**2)) - sin(x(1)*x(2)) - sin(pi)
y(2) = exp(x(1)*x(2)) + cos(x(1)-x(2)) - 2
end function
function jacobian(f,x,n,h) result(u)
procedure(fun), pointer, intent(in) :: f
real(wp), allocatable, intent(in) :: x(:)
integer, intent(in) :: n
real(wp), intent(in) :: h
real(wp), allocatable :: u(:,:)
integer :: j, m
real(wp), allocatable :: y1(:), y2(:), e(:)
m = size(x)
allocate(u(n,m))
do j=1, m
e = element(j, m) ! Get kronecker delta for j-th value
y1 = f(x-e*h/2,n,m)
y2 = f(x+e*h/2,n,m)
u(:,j) = (y2-y1)/h ! Finite difference for each column
end do
end function
function element(i,n) result(e)
! Kronecker delta vector. All zeros, except the i-th value.
integer, intent(in) :: i, n
real(wp) :: e(n)
e(:) = 0d0
e(i) = 1d0
end function
end program Console1
I will answer about evaluation in different points. This is quite simple. You just need an array of points, and if the points are in some regular grid, you may not even need that.
You may have an array of xs and array of ys or you can have an array of derived datatype with x and y components.
For the former:
real, allocatable :: x(:), y(:)
x = [... !probably read from some data file
y = [...
do i = 1, size(x)
J(i) = Jacobian(f, x(i), y(i))
end do
If you want to have many functions at the same time, the problem is always in representing functions. Even if you have an array of function pointers, you need to code them manually. A different approach is to have a full algebra module, where you enter some string representing a function and you can evaluate such function and even compute derivatives symbolically. That requires a parser, an evaluator, it is a large task. There are libraries for this. Evaluation of such a derivative will be slow unless further optimizing steps (compiling to machine code) are undertaken.
Numerical evaluation of the derivative is certainly possible. It will slow the convergence somewhat, depending on the order of the approximation of the derivative. You do a difference of two points for the numerical derivative. You can make an interpolating polynomial from values in multiple points to get a higher-order approximation (finite difference approximations), but that costs machine cycles.
Normally we can use auto difference tools as #John Alexiou mentioned. However in practise I prefer using MATLAB to analytically solve out the Jacobian and then use its build-in function fortran() to convert the result to a f90 file. Take your function as an example. Just type these into MATLAB
syms x y
Fval=sym(zeros(2,1));
Fval(1)=log(abs(x-y^2)) - sin(x*y) - sin(pi);
Fval(2)=exp(x*y) + cos(x-y) - 2;
X=[x;y];
Fjac=jacobian(Fval,X);
fortran(Fjac)
which will yield
Fjac(1,1) = -y*cos(x*y)-((-(x-y**2)/abs(-x+y**2)))/abs(-x+y**2)
Fjac(1,2) = -x*cos(x*y)+(y*((-(x-y**2)/abs(-x+y**2)))*2.0D0)/abs(-
&x+y**2)
Fjac(2,1) = -sin(x-y)+y*exp(x*y)
Fjac(2,2) = sin(x-y)+x*exp(x*y)
to you. You just get an analytical Jacobian fortran function.
Meanwhile, it is impossible to solve the inverse of a mxn matrix because of rank mismatching. You should simplify the system of equations to get a nxn Jacobin.
Additionally, when we use Newton-Raphson's method we do not solve the inverse of the Jacobin which is time-consuming and inaccurate for a large system. An easy way is to use dgesv in LAPACK for dense Jacobin. As we only need to solve the vector x from system of linear equations
Jx=-F
dgesv use LU decomposition and Gaussian elimination to solve above system of equations which is extremely faster than solving inverse matrix.
If the system of equations is large, you can use UMFPACK and its fortran interface module mUMFPACK to solve the system of equations in which J is a sparse matrix. Or use subroutine ILUD and LUSOL in a wide-spread sparse matrix library SPARSEKIT2.
In addition to these, there are tons of other methods which try to solve the Jx=-F faster and more accurate such as Generalized Minimal Residual (GMRES) and Stabilized Bi-Conjugate Gradient (BICGSTAB) which is a strand of literature.
I understand Monte carlo simulation is for estimating area by plotting random points and calculating the ration between the points outside the curve and inside the curve.
I have well calculated the value of pi assuming radius of curve to be unity.
Here is the code
program pi
implicit none
integer :: count, n, i
real :: r, x, y
count = 0
n=500
CALL RANDOM_SEED
DO i = 1, n
CALL RANDOM_NUMBER(x)
CALL RANDOM_NUMBER(y)
IF (x*x + y*Y <1.0) count = count + 1
END DO
r = 4 * REAL(count)/n
print *, r
end program pi
But to find integration , Textbook says to apply same idea. But I'm lost on How to write a code if I want to find the integration of
f(x)=sqrt(1+x**2) over a = 1 and b = 5
Before when radius was one, I did assume point falling inside by condition x*2+y**2 but How can I solve above one?
Any help is extremely helpful
I will write the code first and then explain:
Program integral
implicit none
real f
integer, parameter:: a=1, b=5, Nmc=10000000 !a the lower bound, b the upper bound, Nmc the size of the sampling (the higher, the more accurate the result)
real:: x, SUM=0
do i=1,Nmc !Starting MC sampling
call RANDOM_NUMBER(x) !generating random number x in range [0,1]
x=a+x*(b-a) !converting x to be in range [a,b]
SUM=SUM+f(x) !summing all values of f(x). EDIT: SUM is also an instrinsic function in Fortran so don't call your variable this, I named it so, to illustrate its purpose
enddo
print*, (b-a)*(SUM/Nmc) !final result of your integral
end program integral
function f(x) !defining your function
implicit none
real, intent(in):: x
real:: f
f=sqrt(1+x**2)
end function f
So what's happening:
The integral can be written as
. where:
(this g(x) is a uniform probability distribution of the variable x in [a,b]). And we can write the integral as:
where .
So, finally, we get that the integral should be:
So, all you have to do is generate a random number in the range [a,b] and then calcualte the value of your function for this x. Then do this lots of times (Nmc times), and calculate the sum. Then just divide with Nmc, to find the average and then multiply with (b-a). And this is what the code does.
There's lots of stuff on the internet for this. here's one example that visualizes it pretty nice
EDIT: Second way, that is the same as the Pi method:
Nin=0 !Number of points inside the function (under the curve)
do i=1,Nmc
call random_number(x)
call random_number(y)
x=a+x*(b-a)
y=f_min+y(f_max-f_min)
if (f(x)<y) Nin=Nin+1
enddo
print*, (f_max-f_min)*(b-a)*(real(Nin)/Nmc)
All of this, you could then enclose it in an outer do loop summing the (f_max-f_min)(b-a)(real(Nin)/Nmc) and in the end printing its average.
For this example, what you do is essentially creating an enclosing box from a to b (x dimension) and from f_min to f_max (y dimension) and then doing a sampling of points inside this area and counting the points that are in the function (Nin).Obviously you will have to know the minimum (f_min) and maximum (f_max) value of your function in the range [a,b]. Alternatively you could use arbitrarily low/high values for your f_min f_max but then you will be wasting a lot of points and your error will be bigger.
I'm attempting to write a program that calculates the value of sin(0.75) using the Taylor series, until the absolute difference between the calculated value and FORTRAN'S intrinsic sin function is less then 1E-6, printing each iteration to the screen. (it also needs to print the absolute difference at each step, but I haven't gotten round to implementing that yet!). Here's my code:
program taylor
implicit none
real :: x = 0.75
if (abs(x - sin(0.75)) > 10.00**(-7)) then
print *, x
x = x - ((x**3)/6)
x = x + ((x**5)/120)
x = x - ((x**7)/5040)
end if
end program taylor
This just prints out 0.75 to the terminal, why is this? How can I fix it so that it does what I specified above?
Thanks, any help is appreciated.
I am trying to write a program to solve for pipe diameter for a pump system I've designed. I've done this on paper and understand the mechanics of the equations. I would appreciate any guidance.
EDIT: I have updated the code with some suggestions from users, still seeing quick divergence. The guesses in there are way too high. If I figure this out I will update it to working.
MODULE Sec
CONTAINS
SUBROUTINE Secant(fx,xold,xnew,xolder)
IMPLICIT NONE
INTEGER,PARAMETER::DP=selected_real_kind(15)
REAL(DP), PARAMETER:: gamma=62.4
REAL(DP)::z,phead,hf,L,Q,mu,rho,rough,eff,pump,nu,ppow,fric,pres,xnew,xold,xolder,D
INTEGER::I,maxit
INTERFACE
omitted
END INTERFACE
Q=0.0353196
Pres=-3600.0
z=-10.0
L=50.0
mu=0.0000273
rho=1.940
nu=0.5
rough=0.000005
ppow=412.50
xold=1.0
xolder=0.90
D=11.0
phead = (pres/gamma)
pump = (nu*ppow)/(gamma*Q)
hf = phead + z + pump
maxit=10
I = 1
DO
xnew=xold-((fx(xold,L,Q,hf,rho,mu,rough)*(xold-xolder))/ &
(fx(xold,L,Q,hf,rho,mu,rough)-fx(xolder,L,Q,hf,rho,mu,rough)))
xolder = xold
xold = xnew
I=I+1
WRITE(*,*) "Diameter = ", xnew
IF (ABS(fx(xnew,L,Q,hf,rho,mu,rough)) <= 1.0d-10) THEN
EXIT
END IF
IF (I >= maxit) THEN
EXIT
END IF
END DO
RETURN
END SUBROUTINE Secant
END MODULE Sec
PROGRAM Pipes
USE Sec
IMPLICIT NONE
INTEGER,PARAMETER::DP=selected_real_kind(15)
REAL(DP)::xold,xolder,xnew
INTERFACE
omitted
END INTERFACE
CALL Secant(f,xold,xnew,xolder)
END PROGRAM Pipes
FUNCTION f(D,L,Q,hf,rho,mu,rough)
IMPLICIT NONE
INTEGER,PARAMETER::DP=selected_real_kind(15)
REAL(DP), PARAMETER::pi=3.14159265d0, g=9.81d0
REAL(DP), INTENT(IN)::L,Q,rough,rho,mu,hf,D
REAL(DP)::f, fric, reynold, coef
fric=(hf/((L/D)*(((4.0*Q)/(pi*D**2))/2*g)))
reynold=((rho*(4.0*Q/pi*D**2)*D)/mu)
coef=(rough/(3.7d0*D))
f=(1/SQRT(fric))+2.0d0*log10(coef+(2.51d0/(reynold*SQRT(fric))))
END FUNCTION
You very clearly declare the function in the interface (and the implementation) as
FUNCTION f(L,D,Q,hf,rho,mu,rough)
IMPLICIT NONE
INTEGER,PARAMETER::DP=selected_real_kind(15)
REAL(DP), PARAMETER::pi=3.14159265, g=9.81
REAL(DP), INTENT(IN)::L,Q,rough,rho,mu,hf,D
REAL(DP)::fx
END FUNCTION
So you need to pass 7 arguments to it. And none of them are optional.
But when you call it, you call it as
xnew=xold-fx(xold)*((xolder-xold)/(fx(xolder)-fx(xold))
supplying a single argument to it. When you try to compile it with gfortran for example, the compiler will complain for not getting any argument for D (the second dummy argument), because it stops with the first error.
It seems that the initial values for xold and xolder are too far from the solution. If we change them as
xold = 3.0d-5
xolder = 9.0d-5
and changing the threshold for convergence more tightly as
IF (ABS(fx(xnew,L,Q,hf,rho,mu,rough)) <= 1.0d-10) THEN
then we get
...
Diameter = 7.8306011049894322E-005
Diameter = 7.4533171406818087E-005
Diameter = 7.2580746283970710E-005
Diameter = 7.2653611474296094E-005
Diameter = 7.2652684750264582E-005
Diameter = 7.2652684291155581E-005
Here, we note that the function f(x) is defined as
FUNCTION f(D,L,Q,hf,rho,mu,rough)
...
f = (1/(hf/((L/D)*((4*Q)/pi*D)))) !! (1)
+ 2.0 * log( (rough/(3.7*D)) + (2.51/(((rho*((4*Q)/pi*D))/mu) !! (2)
* (hf/((L/D)*((4*Q)/pi*D))))) !! (3)
)
END FUNCTION
where terms in Lines (1) and (3) are both constant, while terms in Line (2) are some constants over D. So, we see that f(D) = c1 - 2.0 * log( D / c2 ), so we can obtain the solution analytically as D = c2 * exp(c1/2.0) = 7.26526809959e-5, which agrees well with the numerical solution above. To get a rough idea of where the solution is, it is useful to plot f(D) as a function of D, e.g. using Gnuplot.
But I am afraid that the expression for f(D) itself (given in the Fortran code) might include some typo due to many parentheses. To avoid such issues, it is always useful to first arrange the expression for f(D) as simplest as possible before making a program.
(One TIP is to extract constant factors outside and pre-calculate them.)
Also, for debugging purposes it is sometimes useful to check the consistency of physical dimensions and physical units of various terms. Indeed, if the magnitude of the obtained solution is too large or too small, there might be some problem of conversion factors for physical units, for example.
I'm relatively new to Fortran and I have an assignment to find quadrature weights and points where the points are the zeros of the nth legendre polynomial (found using Newton's method); I made functions to find the value of Pn(x) and P'n(x) to sub into Newton's method.
However when actually using the functions in my quadrature subroutine it comes back with:
Coursework2a.f90:44.3:
x = x - P(n,x)/dP(n,x)
1
Error: Unclassifiable statement at (1)
Does anybody know any reasons why this statement could be classed as unclassifiable?
subroutine Quadrature(n)
implicit none
integer, parameter :: dpr = selected_real_kind(15) !Double precision
real(dpr) :: P, dP, x, x_new, error = 1, tolerance = 1.0E-6, Pi = 3.141592 !Define Variables
integer, intent(in) :: n
integer :: i
!Next, find n roots. Start with first guess then iterate until error is greater than some tolerance.
do i = 1,n
x = -cos(((2.0*real(i)-1.0)/2.0*real(n))*Pi)
do while (error > tolerance)
x_new = x
x = x - P(n,x)/dP(n,x)
error = abs(x_new-x)
end do
print *, x
end do
end subroutine Quadrature
The line
x = -cos(((2.0*real(i)-1.0)/2.0*real(n))*Pi)
is likely missing a set of brackets around the denominator. As it is, the line divides (2.0*real(i)-1.0) by 2.0, then multiplies the whole thing by real(n). This may be why you get the same root for each loop.
real function p(n,x)
real::n,x
p=2*x**3 !or put in the function given to you.
end function
real function dp(n,x)
real::n,x
dp=6*x**2 !you mean derivative of polynomial p, I guess.
end function
Define function like this separately outside the main program. Inside the main program declare the functions like:
real,external::p, dp