MIT6_094IAP10_lec03

MIT6_094IAP10_lec03 - 6.094 Introduction to programming in...

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6.094 Introduction to programming in MATLAB Danilo Š ć epanovi ć IAP 2008 Lecture 3 : Solving Equations and Curve Fitting
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Homework 2 Recap • How long did it take? • Using min with matrices: » a=[3 7 5;1 9 10; 30 -1 2]; » b=min(a); % returns the min of each column » m=min(b); % returns min of entire a matrix » m=min(min(a)); % same as above » m=min(a(:)); % makes a a vector, then gets min •C o m m o n m i s t a k e : » [m,n]=find(min(a)); % think about what happens • How to make and run a function: save the file, then call it from the command window like any other function. No need to 'compile' or make it official in any other way
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Outline (1) Linear Algebra (2) Polynomials (3) Optimization (4) Differentiation/Integration (5) Differential Equations
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Systems of Linear Equations • Given a system of linear equations ¾ x+2y-3z=5 ¾ -3x-y+z=-8 ¾ x-y+z=0 • Construct matrices so the system is described by Ax=b » A=[1 2 -3;-3 -1 1;1 -1 1]; » b=[5;-8;0]; • And solve with a single line of code! » x=A\b; ¾ x is a 3x1 vector containing the values of x, y, and z • The \ will work with square or rectangular systems. • Gives least squares solution for rectangular systems. Solution depends on whether the system is over or underdetermined. MATLAB makes linear algebra fun!
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More Linear Algebra • Given a matrix » mat=[1 2 -3;-3 -1 1;1 -1 1]; • Calculate the rank of a matrix » r=rank(mat); ¾ the number of linearly independent rows or columns • Calculate the determinant » d=det(mat); ¾ mat must be square ¾ if determinant is nonzero, matrix is invertible • Get the matrix inverse » E=inv(mat); ¾ if an equation is of the form A*x=b with A a square matrix, x=A\b is the same as x=inv(A)*b
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Matrix Decompositions • MATLAB has built-in matrix decomposition methods • The most common ones are » [V,D]=eig(X) ¾ Eigenvalue decomposition » [U,S,V]=svd(X) ¾ Singular value decomposition » [Q,R]=qr(X) ¾ QR decomposition
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Exercise: Linear Algebra • Solve the following systems of equations: ¾ System 1: ¾ System 2: 43 4 32 xy += −+ = 22 4 3 34 2 −= =
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Exercise: Linear Algebra • Solve the following systems of equations: ¾ System 1: ¾ System 2: 43 4 32 xy += −+ = 22 4 3 34 2 −= = » A=[1 4;-3 1]; » b=[34;2]; » rank(A) » x=inv(A)*b; » A=[2 -2;-1 1;3 4]; » b=[4;3;2]; » rank(A) ¾ rectangular matrix » x1=A\b; ¾ gives least squares solution » error=abs(A*x1-b)
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Outline (1) Linear Algebra (2) Polynomials (3) Optimization (4) Differentiation/Integration (5) Differential Equations
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Polynomials • Many functions can be well described by a high-order polynomial • MATLAB represents a polynomials by a vector of coefficients ¾ if vector P describes a polynomial ax 3 +bx 2 +cx+d • P=[1 0 -2] represents the polynomial x 2 -2 • P=[2 0 0 0] represents the polynomial 2x 3 P(1) P(2) P(3) P(4)
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Polynomial Operations • P is a vector of length N+1 describing an N-th order polynomial • To get the roots of a polynomial » r=roots(P) ¾ r is a vector of length N • Can also get the polynomial from the roots » P=poly(r) ¾ r is a vector length N • To evaluate a polynomial at a point » y0=polyval(P,x0) ¾
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MIT6_094IAP10_lec03 - 6.094 Introduction to programming in...

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