Numerical Methods Contents - SAM

35 h= d i f f ( t ) ;
36 d e l t a = d i f f ( y ) . / h ; % slopes of data
37 c=zeros ( size ( t ) ) ;
38 for j =1:n−1
39 i f ( d e l t a ( j ) ∗ d e l t a ( j +1) >0)
40 c ( j +1) = 2 / ( 1 / d e l t a ( j ) + 1 / d e l t a ( j +1) ) ;
41 end
42 end
43 c ( 1 ) =2∗ d e l t a ( 1 )−c ( 2 ) ; c ( n+1)=2∗ d e l t a ( n )−c ( n ) ;
44
45 % plot segments indicating the slopes c(i):
46 % use (vector) plot handle ’hplot’ to reduce the linewidth in step 2
47 h p l o t s =zeros ( 1 , n+1) ;
48 for j =2:n
49 h p l o t s l ( j ) =plot ( [ t ( j ) −0.3∗h ( j −1) , t ( j ) +0.3∗h ( j ) ] ,
[ y ( j ) −0.3∗h ( j −1)∗c ( j ) , y ( j ) +0.3∗h ( j ) ∗c ( j ) ] , ’− ’ , ’ l i n e w i d t h ’ ,2) ;
50 end
51 h p l o t s l ( 1 ) =plot ( [ t ( 1 ) , t ( 1 ) +0.3∗h ( 1 ) ] , [ y ( 1 ) , y ( 1 ) +0.3∗h ( 1 ) ∗c ( 1 ) ] ,
’− ’ , ’ l i n e w i d t h ’ ,2) ;
52 h p l o t s l ( n+1)= plot ( [ t ( end ) −0.3∗h ( end ) , t ( end ) ] ,
[ y ( end ) −0.3∗h ( end ) ∗c ( end ) , y ( end ) ] , ’− ’ , ’ l i n e w i d t h ’ ,2) ;
53 legend ( [ h p l o t ( 1 ) , h p l o t s l ( 1 ) ] , leg { 1 : 2 } ) ;
Ôº º
Ôº º
54 pause ; 97 % tangent in t(j) and p(j) to the linear spline of step 3
78 % -t(j)
79 % -the middle points between t(j) and p(j)
80 % -the middle points between p(j) and t(j+1)
81 % -t(j+1)
82 % and with slopes c(j) in t(j), for every j
83
84 disp ( ’STEP 3 ’ )
85 t i t l e ( ’ A u x i l i a r y l i n e a r s p l i n e − Press enter to continue ’ )
86
87 for j =1:n
88 h p l o t ( 3 ) =plot ( [ t ( j ) 0.5∗( p ( j ) + t ( j ) ) 0.5∗( p ( j ) + t ( j +1) ) t ( j +1) ] ,
[ y ( j ) y ( j ) +0.5∗(p ( j )−t ( j ) ) ∗c ( j ) y ( j +1) +0.5∗(p ( j )−t ( j +1) ) ∗c ( j +1)
y ( j +1) ] , ’m−^ ’ ) ;
89 plot ( [ t ( j ) 0.5∗( p ( j ) + t ( j ) ) 0.5∗( p ( j ) + t ( j +1) ) t ( j +1) ] ,
ones ( 1 , 4 ) ∗( newaxis ( 3 ) +0.25) , ’m^ ’ ) ;
90 end
91 legend ( [ h p l o t ( 1 ) , h p l o t s l ( 1 ) , h p l o t ( 2 ) , h p l o t ( 3 ) ] , leg { 1 : 4 } ) ;
92 pause ;
93
94 % ===== Step 4: quadratic spline =====
95 % final quadratic shape preserving spline
96 % quadratic polynomial in the intervals [t(j), p(j)] and [p(j), t(j)]
55
56 % ===== Step 2: choice of middle points =====
57 % fix points p(j) in [t(j), t(j+1)], depending on the slopes c(j), c(j+1)
58
59 disp ( ’STEP 2 ’ )
60 t i t l e ( ’ Middle p o i n t s − Press enter to continue ’ )
61 set ( h p l o t s l , ’ l i n e w i d t h ’ ,1)
62
63 p = ( t ( 1 ) −1)∗ones ( 1 , length ( t ) −1) ;
64 for j =1:n
65 i f ( c ( j ) ~= c ( j +1) )
66 p ( j ) =( y ( j +1)−y ( j ) + t ( j ) ∗c ( j )−t ( j +1)∗c ( j +1) ) / ( c ( j )−c ( j +1) ) ;
67 end
68 % check and repair if p(j) is outside its interval:
69 i f ( ( p ( j ) < t ( j ) ) | | ( p ( j ) > t ( j +1) ) ) ; p ( j ) = 0.5∗( t ( j ) + t ( j +1) ) ; end ;
70 end
71
72 h p l o t ( 2 ) =plot ( p , ones ( 1 , n ) ∗( newaxis ( 3 ) +0.25) , ’ go ’ ) ;
73 legend ( [ h p l o t ( 1 ) , h p l o t s l ( 1 ) , h p l o t ( 2 ) ] , leg { 1 : 3 } ) ;
74 pause ;
Ôº º
Ôº º
77 % build the linear spline with nodes in: 120 w = y ( j +1) +0.5∗(p ( j )−t ( j +1) ) ∗c ( j +1) ;
75
76 % ===== Step 3: auxiliary linear spline =====
98
99 disp ( ’STEP 4 ’ )
100 t i t l e ( ’ Quadratic s p l i n e ’ )
101
102 % for every interval 2 quadratic interpolations
103 % a, b, ya, yb = extremes and values in the subinterval
104 % w = value in middle point that gives the right slope
105 for j =1:n
106 a= t ( j ) ;
107 b=p ( j ) ;
108 ya = y ( j ) ;
109 w = y ( j ) +0.5∗(p ( j )−t ( j ) ) ∗c ( j ) ;
110 yb = ( ( t ( j +1)−p ( j ) ) ∗( y ( j ) +0.5∗(p ( j )−t ( j ) ) ∗c ( j ) ) + . . .
111 ( p ( j )−t ( j ) ) ∗( y ( j +1) +0.5∗(p ( j )−t ( j +1) ) ∗c ( j +1) ) ) / ( t ( j +1)−t ( j ) ) ;
112 x=linspace ( a , b ,100) ;
113 pb = ( ya ∗(b−x ) .^2 + 2∗w∗( x−a ) . ∗ ( b−x ) +yb ∗( x−a ) . ^ 2 ) / ( ( b−a ) ^2) ;
114 h p l o t ( 4 ) =plot ( x , pb , ’ r−’ , ’ l i n e w i d t h ’ ,2) ;
115
116 a = b ;
117 b = t ( j +1) ;
118 ya = yb ;
119 yb = y ( j +1) ;