Computational modeling

Transcription

1 Computational modeling Lecture 1 : Linear algebra - Matrix operations Examination next week: How to get prepared Theory and programming: Matrix operations Instructor : Cedric Weber Course : 4CCP1

2 Schedule Class/Week Chapter Topic Milestones 1 Monte Carlo UNIX system / Fortran 2 Monte Carlo Fibonacci sequence 3 Monte Carlo Random variables 4 Monte Carlo Central Limit Theorem 5 Monte Carlo Monte Carlo integration Milestone 1 6 Differential equations The Pendulum 7 Differential equations The Taylor s method 8 Differential equations A Quantum Particle in a box 9 Differential equations The Tacoma bridge Milestone 2 1 Linear Algebra Matrix operations Milestone 3 2

3 Where are we going? 1% Lecture 1-5 : Statistics and integrals 9% 8% 7% Lecture 6-9 : Differential equations Lecture 1 : Matrix operations 6% 5% 4% 3% 2% 1% % Milestone 3 : Matrix operations week 1 week 2 week 3 week 4 week 5 week 6 week 7 week 8 week 9 week 1 3

4 Examination Examination: Sit- in : - December 12 th & December 13 th (depending on your group) - you can use all your programs, bring all your notes, books, - you will have to solve problems during the 3h à detailed guidelines how to write your reports will be posted on KEATS on Dec 13 th Report : - Deadline end of January - 1 pages max (including figures) - discuss and complete the problems given during the examination (detailed guidelines will be posted on KEATS) 4

5 Examination : Dec. 12 th & 13 th Problem 1 : Monte Carlo calculation Problem 2 : Differential Equation Problem 3 : Matrix operations and manipulations 5

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7 Fortran : defining a matrix as a variable A matrix is a table (two dimensions) In Fortran, to define a variable as a matrix, we use the following syntax in the variable declaration section : real(8) :: matrixa ( 5, 5 ) The first number 5 is the number of lines, the second number 5 is the number of columns Reminder, for a vector it would be : real(8) :: vectora( 5 ) 7

8 Defining a matrix in Fortran Once the matrix is declared as a variable, we can first set all its elements to zero with a global operation : matrixa =. To enter elements into the matrix : matrixa(1,1) = - c matrixa(3,2) = - 1 Circle the elements in the table above that we just entered into the matrixa array. 8

9 Matrix/vector multiplication In Fortran, product of matrix and vectors are readily available, with the internal function MATMUL : In Fortran translates into : y = Ax y = MATMUL ( A, x) Where, for example, y, A, x are defined in Fortran as : real(8) :: y(1), A(1,1), x(1) 9

10 Sum of the elements of a vector/matrix c = N=2 v i & c = N=2 a ij i=1 i,j=1 program matrix real(8) :: v(2), a(2,2), c v(1)= 1 ; v(2)= - 1 c = SUM ( v ) a= a(1,1)=3 ; a(2,1)=-1 c = SUM ( a ) end program 1 SUM is an internal function of Fortran (available without the need of external libraries) It is a FUNCTION so it can be used as part of a calculation It returns the sum of the vector elements or the sum of the matrix elements What is the value of c in this example?

11 Matrix- vector product w = Av program matrix real(8) :: A(3,3),v(3),w(3) w = MATMUL ( A, v ) end program MATMUL is an internal function of Fortran (available without the need of external libraries) It is a FUNCTION so it can be used as part of a calculation 11

12 Matrix- Matrix product C = AB program matrix real(8) :: A(3,3), B(3,3), C(3,3) C = MATMUL ( A, B ) end program MATMUL can be used for both Matrix- Matrix or Matrix- Vector products 12

13 Matrix operations D =(A + B)C program matrix real(8) :: A(3,3),B(3,3),C(3,3),D(3,3) D = MATMUL ( (A+B), C ) end program 13 MATMUL can be used in combination with other matrix operations, such as additions Note: here the addition A+B is performed first, then the result is sent as an argument to the function MATMUL, which returns the product of (A+B) and C The final result is stored in the matrix D

14 Matrix- scalar product B = λa program matrix real(8) :: A(2,2), B(2,2), lambda B = lambda * A end program 14 B(1, 1) = λa(1, 1) B(1, 2) = λa(1, 2) B(2, 1) = λa(2, 1) B(2, 2) = λa(2, 2) A matrix can be multiplied by a scalar (real(8) simple variable) The obtained matrix B is the matrix obtained by the product of each matrix element of A by lambda

15 Rectangular Matrices? C = AB program matrix real(8) :: A(5,3),B(?,?),C(5,5) C = MATMUL ( A, B ) end program MATMUL can be used for rectangular matrices What should be the dimension of the B matrix here? C = A. B {N1 x N2} = {N1 x M}. { M x N2} ( ) = ( M )( ) M Dimension of B in example above? Fill in 15

16 Matrix transpose A = {a ij } A T = {a ji } program matrix real(8) :: A(3,3),C(3,3) C = transpose ( A ) end program Matrix transpose : permutation of rows and columns Only for square matrices It is a function, part of an equation/operation 16

17 Largest/Smallest matrix elements program matrix real(8) :: A(2,2) A = A(2,1)=3 write(*,*) MAXVAL(A) end program MAXVAL returns the value of the largest element of the matrix 17 MINVAL returns the value of the smallest matrix element

18 Writing a matrix to the terminal The index of the do loop ( i ) runs over the rows of the matrix A We write the matrix line by line program matrix real(8) :: A(2,2) integer :: i, j do i=1,2 write(*,*) A( i, 1 ), A( i,, 2 ) end do end program 18

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20 1. program multiply 2. real(8) :: mata(5,5), x(5), b(5) 3. mata=. 4. mata(1,1)=1. 5. mata(3,1)=5. Example 6. b=. 7. b(3)=1. A X = B 8. x = MATMUL ( A, B ) 9. end program 2

21 Symmetric matrix A matrix is symmetric if : A = A T By using what we just learned, how can we test if a matrix is symmetric in Fortran, with one line of code? Any idea? Hint - what can you say about : A A T program matrix real(8) :: A(2,2) A(1,1)= if( [ FILL IN ] ) then write(*,*) ʻMatrix is symmetric!ʼ endif end program 21

22 Symmetric matrix : test A matrix is symmetric if : program matrix real(8) :: A(2,2) A = A T A A T if( write(*,*) ʻMatrix is symmetric!ʼ endif end program MAXVAL ( ABS ( A TRANSPOSE(A) ) ) ==. ) then 22

23 Problems today Problem 1 : Matrix / vector operations Problem 2 : Matrix / Matrix operations 23