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			Blitz++ 計算二階導數
		
            
		
		
            
				
				//
				整理?by?RobinKin
				
						
            
						
				            
				
						
            
						#include?            
				<
				blitz
				/
				vector.h
				>
				
						
            
						
						
            
						
				            
				using
				?
				namespace
				?blitz;

            
				int
				?main()
            
				
				
						{
            ????            
						//
						?In?this?example,?the?function?cos(x)^2?and?its?second?derivative
            ????            
						//
						?2?(sin(x)^2?-?cos(x)^2)?are?sampled?over?the?range?[0,1).
            ????            
						//
						?The?second?derivative?is?approximated?numerically?using?a
            ????            
						//
						?[?1?-2??1?]?mask,?and?the?approximation?error?is?computed.
						
								
            
								
						            
						
								
            
								
								
						            
						
						
								/*
								?cos(x)^2??的二階導數?是2?(sin(x)^2?-?cos(x)^2)

            下面在[0,1)?的范圍中，以delta為步長?，用[?1?-2??1?]?的方法計算二階導數

            看看和精確值的誤差
            
								*/
						            
						
								
            
								
								
            
								
								
            
								????            
						const
						?
						int
						?numSamples?
						=
						?
						100
						;??????????????
						//
						?Number?of?samples
						
								
            
								
						            
						????
						double
						?delta?
						=
						?
						1
						.?
						/
						?numSamples;??????????
						//
						?Spacing?of?samples
						
								
            
								
						            
						????Range?R(
						0
						,?numSamples?
						-
						?
						1
						);??????????????
						//
						?Index?set?of?the?vector

            ????            
						//
						?Sample?the?function?y?=?cos(x)^2?over?[0,1)
            ????            
						//
						
								
            
								????            
						//
						?An?object?of?type?Range?can?be?treated?as?a?vector,?and?used
            ????            
						//
						?as?a?term?in?vector?expressions.
            ????            
						//
						
								
            
								????            
						//
						?The?initialization?for?y?(below)?will?be?translated?via?expression
            ????            
						//
						?templates?into?something?of?the?flavour
            ????            
						//
						
								
            
								????            
						//
						?for?(unsigned?i=0;?i?<?99;?++i)
            ????            
						//
						?{
            ????            
						//
						?????double?_t1?=?cos(i?*?delta);
            ????            
						//
						?????y[i]?=?_t1?*?_t1;
            ????            
						//
						?}
						
								
            
								
						            
						????
            ????Vector            
						<
						double
						>
						?y?
						=
						?sqr(cos(R?
						*
						?delta));

            ????            
						//
						?Sample?the?exact?second?derivative
						
								
            
								
						            
						????Vector
						<
						double
						>
						?y2exact?
						=
						?
						2.0
						?
						*
						?(sqr(sin(R?
						*
						?delta))?
						-
						?sqr(cos(R?
						*
						?delta)));

            ????            
						//
						?Approximate?the?2nd?derivative?using?a?[?1?-2??1?]?mask
            ????            
						//
						?We?can?only?apply?this?mask?to?the?elements?1?..?98,?since
            ????            
						//
						?we?need?one?element?on?either?side?to?apply?the?mask.
						
								
            
								
						            
						????Range?I(
						1
						,numSamples
						-
						2
						);
            ????Vector            
						<
						double
						>
						?y2(numSamples);

            ????y2(I)?            
						=
						?(y(I
						-
						1
						)?
						-
						?
						2
						?
						*
						?y(I)?
						+
						?y(I
						+
						1
						))?
						/
						?(delta
						*
						delta);
            ??
            ????            
						//
						?The?above?difference?equation?will?be?transformed?into
            ????            
						//
						?something?along?the?lines?of
            ????            
						//
						
								
            
								????            
						//
						?double?_t2?=?delta*delta;
            ????            
						//
						?for?(int?i=1;?i?<?99;?++i)
            ????            
						//
						?????y2[i]?=?(y[i-1]?-?2?*?y[i]?+?y[i+1])?/?_t2;
            ?
            ????            
						//
						?Now?calculate?the?root?mean?square?approximation?error:
						
								
            
								
						            
						
								
            
								????            
						double
						?error?
						=
						?sqrt(mean(sqr(y2(I)?
						-
						?y2exact(I))));
            ?
            ????            
						//
						?Display?a?few?elements?from?the?vectors.
            ????            
						//
						?This?range?constructor?means?elements?1?to?91?in?increments
            ????            
						//
						?of?15.
						
								
            
								
						            
						????Range?displayRange(
						1
						,?
						91
						,?
						15
						);
            ?
            ????cout?            
						<<
						?
						"
						Exact?derivative:
						"
						?
						<<
						?y2exact(displayRange)?
						<<
						?endl
            ?????????            
						<<
						?
						"
						Approximation:???
						"
						?
						<<
						?y2(Range(displayRange))?
						<<
						?endl
            ?????????            
						<<
						?
						"
						RMS?Error:???????
						"
						?
						<<
						?error?
						<<
						?endl;

            ????            
						return
						?
						0
						;
            }            
				            
				
						
            
						
						
            
						
						
            
						Output:

            Exact?derivative:[????            
				-
				1.9996
				??
				-
				1.89847
				??
				-
				1.62776
				??
				-
				1.21164
				?
				-
				0.687291
				?
				-
				0.1015495
				
						
            
						???]
            Approximation:???[???            
				-
				1.99953
				??
				-
				1.89841
				???
				-
				1.6277
				???
				-
				1.2116
				?
				-
				0.687269
				?
				-
				0.1015468
				
						
            
						???]
            RMS?Error:???????            
				4.24826e-05
				
						
            
						
						
            
						
				            
		
            

		
            
			posted on 2006-07-01 13:23 楊粼波 閱讀(703) 評論(0)  編輯 收藏 引用  
		
            
	
            
	
	





            
	    
    




            






            

				

            



    
    







            
	   
	



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