CM2202: Scientific Computing and Multimedia Applications Lab Class Week 5. School of Computer Science & Informatics
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1 CM2202: Scientific Computing and Multimedia Applications Lab Class Week 5 School of Computer Science & Informatics
2 Vector operators Definition (Vector Addition, Subtraction and Scalar Multiplication in R n ) Let v and w be two vectors in R n and k a real number. The following rules are well-defined: v + w = (v 1 + w 1, v 2 + w 2,..., v n + w n ). v w = (v 1 w 1, v 2 w 2,..., v n w n ) kv = (kv 1, kv 2,..., kv n ). These rules coincide with the geometrical interpretation for two-dimensional vectors (see previous definitions). 2 / 24
3 Vector Addition, Subtraction and Scalar Multiplication in MATLAB MATLAB directly supports vector addition, subtraction and scalar multiplication: >> v=[1 2 5]; >> w=[3-1 1]; >> v+w ans = >> v-w ans = >> 3 * v ans = >> w*(-1) ans = / 24
4 Scalar product Definition (Scalar product) Given two vectors v and w in R n with components (v 1, v 2,..., v n ) and (w 1, w 2,..., w n ). We define the scalar product (or (standard) inner product, dot product) of v and w as v.w or v, w = n v i w i i=1 Note what the scalar product does: It takes two vectors and assigns them a real number. Problem (Scalar product) Work out the scalar product of vectors v = (1, 2) and w = (2, 3) Note the notations v.w and v, w are equivalent. We use the v.w notation. 4 / 24
5 Scalar product using MATLAB MATLAB provides a vector function dot that computes the dot product of two vectors (of any, identical dimension). >> v = [3 2-1] >> w = [2-1 1] >> dot(v, w) ans = 3 >> sum(v.*w) ans = 3 dot(v, w) is equivalent to sum(v.*w) note v.*w is an array multiplication that returns a vector of the same size. 5 / 24
6 Properties of scalar products Theorem (Cauchy-Schwarz inequality) Let v and w be vectors in R n Then they satisfy the Cauchy-Schwarz inequality v.w v w. Theorem (Angle Between Two Vectors) If n = 2, 3 we even have the relation v.w = v w cos(θ) We call θ the angle between v and w. 6 / 24
7 Euclidean norm of a vector Definition (Euclidean norm of a vector) For a vector v R n we define its norm as v = v.v This norm is called the Euclidean norm of the vector v. The Euclidean norm of a vector coincides with the length of the vector in R 2 and R 3. v2 (v 1, v 2) y v v v1 By Pythagoras Theorem, v = x v v 2 2 = v.v 7 / 24
8 Euclidean norm of a vector in MATLAB The default behaviour of MATLAB function norm for a given vector input is to return the Euclidean norm (also called 2-norm): >> v = [3 4] >> norm(v) ans = 5 >> sqrt(dot(v, v)) ans = 5 8 / 24
9 The Vector Cross Product Besides the scalar product that maps two vectors from R n to R we also need a product that maps two vectors from R n to a vector in R n. Definition (The vector cross product in R 2 ) We define the vector cross product of v, w R 2 as a mapping : R 2 R 2 R with v w = v 1 w 2 v 2 w 1 The vector product in R 2 is anti-symmetric, i.e. v w = w v 9 / 24
10 The Vector Cross Product (cont.) Definition (The vector cross product in R 3 ) We define the vector cross product of v, w R 3 as a mapping : R 3 R 3 R 3 with v 2 w 3 v 3 w 2 v w = v 3 w 1 v 1 w 3. v 1 w 2 v 2 w 1 The vector product in R 3 is also anti-symmetric, i.e. v w = w v The vector cross product has very useful properties, especially: for finding orthogonal vectors in R 3. for area and volume calculations in R 2 and R 3 respectively. 10 / 24
11 Vector cross products in MATLAB MATLAB provides a vector function cross to compute the cross product of two vectors in R 3 : >> v=[1 2 3]; >> w=[-1 1 2]; >> cross(v, w) ans = / 24
12 Differentiation in MATLAB Derivative of a poly() structure - polyder(), Poly Diff Eg.m >> a = [ ] ; % 3 x ˆ2 + 6 x + 9 >> k = p o l y d e r ( a ) k = 6 6 >>b = [ ] ; >> k = p o l y d e r ( b ) % x ˆ2 + 2 x k = 2 2 >> k = p o l y d e r ( a, b ) %(3x ˆ2 + 6 x + 9 ) ( x ˆ2 + 2 x ) k = MATLAB code for this at Poly Diff Eg.m 12 / 24
13 Symbolic Differentiation in MATLAB Symbolic Derivative via diff(), Symb Diff Eg.m >>syms x ; % D e c l a r e f ( x ) >>f = x ˆ2 + 2 x +1; % D i f f e r e n t i a t e f ( x ) >>d f = d i f f ( f ) ans = 2 x + 2 MATLAB code for this at Symb Diff Eg.m 13 / 24
14 Computing and Plotting Tangent to a Curve Symb Diff Eg.m continued % G r a d i a n t a t ( x0, y0 ) x0 = 1 ; y0= 5 ; % compute m + c ( y a x i s I n t e r c e p t ) m = s u b s ( df, x0 ) ; c = y0 m x0 ; %D e c l a r e t a n e q n = m x + c ; %P l o t f ( x ) and t a n g e n t a t ( x0, y0 ) e z p l o t ( tan eqn,[ ] ) ; h o l d on ; e z p l o t ( f,[ ] ) ; MATLAB code for this at Symb Diff Eg.m 14 / 24
15 Symb Diff Eg.m Output x x x 15 / 24
16 Stationary Points in MATLAB MATLAB Example:Stationary point eg.m syms x f = x ˆ3 3 x ˆ2 9 x + 2 ; d f = d i f f ( f ) ; % 1 s t d e r i v a t i v e d f 2 = d i f f ( d f ) ; %2nd d e r i v a t i v e % f i n d Turning p o i n t s % need d o u b l e ( ) from c o n v e r t s y m b o l i c v a r i a b l e t u r n p t s x = d o u b l e ( s o l v e ( d f ) ) ; t u r n p t s y = d o u b l e ( s u b s ( f, t u r n p t s x ) ) ; max min = d o u b l e ( s u b s ( df2, t u r n p t s x ) ) ;.... P l o t R e s u l t s ( s e e S t a t i o n a r y p o i n t e g.m f i l e ) Full MATLAB code at: Stationary point eg.m 16 / 24
17 Stationary point eg.m Output 600 x 3 3 x 2 9 x Max Min x 17 / 24
18 MATLAB Integration: poly() Integrals MATLAB lets you integrate poly() Polynomials: polyint() and trapz() >> p = [ ] % p = x ˆ2 % I n d e f i n i t e I n t e g r a l >> p o l y i n t ( p ) % Ans = x ˆ3/3 ans = % D e f i n i t e I n t e g r a l ( v i a n u m e r i c a l i n t e g r a t i o n ) >> t r a p z ( 6:6, p o l y v a l ( p, 6:6)) ans = / 24
19 MATALB Example: Area Under Graph (1) MATLAB Code to solve and plot above example: integral area.m % d e c l a r e f u n c t i o n and p l o t i t syms x p l o t r a n g e = [ 1 0, 1 0 ] ; f = x ˆ 2 ; e z p l o t ( f, p l o t r a n g e ) ; h o l d on ; % d e f i n e d e f i n i t e i n t e g r a l l i m i t s i n t l i m i t s = [ 6 6 ] ; % I n t e g r a t e i n t f = i n t ( f ) % I n d e f i n i t e I n t e g r a l % D e f i n i t e I n t e g r a l v i a I n d e f i n i t e r e s u l t i n t v a l i n d = i n t ( f, i n t l i m i t s ( 1 ), i n t l i m i t s ( 2 ) ) 19 / 24
20 MATALB Example: Area Under Graph (2) integral area.m Cont. % I d i o t Check! D e f i n i t e I n t e g r a l v i a I n d e f i n i t e r e s u l t s u b s ( i n t f, i n t l i m i t s ( 2 ) ) s u b s ( i n t f, i n t l i m i t s ( 1 ) ) % not r e a l l y a n e c e s s a r y b i t o f code as %Def. I n t i s t h e way to do i t! % Set up p l o t r a n g e = i n t l i m i t s ( 1 ) : 0. 1 : i n t l i m i t s ( 2 ) ; y = s u b s ( f, r a n g e ) ; %sample v a l u e s on c u r v e % Shade a r e a below t h e c u r v e a r e a ( range, y, FaceColor, [ 0, 0, 1 ], L i n e S t y l e, none ) ; Note: area()- useful plot function to shade areas of given graph values. MATLAB code for this at integral area.m 20 / 24
21 Example: Area Between Two Curves (1) MATLAB Code: integral area between.m % d e c l a r e f u n c t i o n s syms x p l o t r a n g e = [ 3, 3 ] ; f 1 = x ˆ2 + 6 ; f 2 = x ˆ2 2 x + 2 ; % Find P o i n t s o f I n t e r s e c t i o n o f c u r v e s r o o t s i n t e r s e c t = s o r t ( d o u b l e ( s o l v e ( f 1 f 2 ) ) ) ; % D e f i n i t e I n t e g r a l a r e a i n t e r s e c t = i n t ( f 1 f2, r o o t s i n t e r s e c t ( 1 ), r o o t s i n t e r s e c t ( 2 ) ) 21 / 24
22 Example: Area Between Two Curves (1) integral area between.m Cont. % Set up p l o t r a n g e = r o o t s i n t e r s e c t ( 1 ) : 0. 1 : r o o t s i n t e r s e c t ( 2 ) ; y = s u b s ( f1, r a n g e ) ; %sample v a l u e s on c u r v e f 1 % Shade Area f 1 a r e a ( range, y, FaceColor, [ 0, 0, 1 ], L i n e S t y l e, none ) h o l d on ; % rub out a r e a above t h e c u r v e y = s u b s ( f2, r a n g e ) ; %sample v a l u e s on c u r v e f 2 a r e a ( range, y, FaceColor, [ 1, 1, 1 ], L i n e S t y l e, none ) % P l o t f u n t i o n s o v e r p l o t r a n g e e z p l o t ( f1, p l o t r a n g e ) ; h f 2 = e z p l o t ( f2, p l o t r a n g e ) ; s e t ( hf2, Color, b l a c k ) ; MATLAB code for this at integral area between.m 22 / 24
23 MATLAB Example: Area Under x-axis (1) MATLAB Code: integral area under.m % d e c l a r e f u n c t i o n syms x p l o t r a n g e = [ 5, 5 ] ; i n t l i m i t s = [ 3, 3 ] ; f = x ˆ 3 ; % D e f i n i t e I n t e g r a l v i a I n d e f i n i t e r e s u l t i n t a r e a = i n t ( f, i n t l i m i t s ( 1 ), i n t l i m i t s ( 2 ) ) % WRONG ( f o r Area )! Answer i s ZERO % For x ˆ3 need to s p l i t a t x = 0 % E x e r c i s e : Write a g e n e r a l f u n c t i o n to work t h i s p o i n t out. % Do Area P r o p e r l y i n t a r e a = abs ( i n t ( f, i n t l i m i t s ( 1 ), 0 ) ) + abs ( i n t ( f, 0, i n t l i m i t s ( 2 ) 23 / 24
24 MATLAB code for this at integral area under.m 24 / 24 MATLAB Example: Area Under x-axis (2) MATLAB Code: integral area under.m % Set up p l o t r a n g e = i n t l i m i t s ( 1 ) : 0. 1 : 0 ; y = s u b s ( f, r a n g e ) ; %sample v a l u e s on c u r v e % Shade a r e a below t h e c u r v e a r e a ( range, y, FaceColor, [ 1, 0, 0 ], L i n e S t y l e, none ) ; h o l d on ; r a n g e = 0 : 0. 1 : i n t l i m i t s ( 2 ) ; y = s u b s ( f, r a n g e ) ; %sample v a l u e s on c u r v e % Shade a r e a below t h e c u r v e a r e a ( range, y, FaceColor, [ 0, 0, 1 ], L i n e S t y l e, none ) ; %p l o t f u n c t i o n h f =e z p l o t ( f, p l o t r a n g e ) ; s e t ( hf, Color, b l a c k ) ;
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