TI83 Calculator Programs for Numerical Analysis Problems - Part 3
These programs are copyrighted (1997-2007), but you may copy them for instructional purposes as long as no profit is made from their use. The author is not responsible for any data loss which may be caused to any calculator or its memory by the use of these programs.
The following are TI-83 calculator programs for the solution of numerical analysis problems of a type which are usually found in college courses or textbooks in numerical analysis. These programs are also suitable, with minor modifications, for use in other TI calculators, such as the TI82, TI85, TI86, TI89, TI92, etc... . (The easier programs could be used, with minor modifications, in a TI81 calculator.) These TI-83 programs, together with their descriptions, will probably be read more easily by students who are either taking a course in numerical analysis, or who have already taken a course in numerical analysis. (Click here for comments concerning calculators other than the TI83.)
Comments and questions concerning these programs may be addressed to Gerald Roskes, Department of Mathematics, Queens College, Flushing, New York 11367, or send email to gerald_roskes@qc.edu.
This web page will be continually expanding as we add more programs to our list. If you have an interest in calculator programs for numerical analysis, you should view our page every few months. (This web page was last updated in December, 2007.)
The following are links to the various TI83 programs:
Part 1 (Programs 1 - 23)
Prog1 SIMPITER (Simple Iteration)
Prog2 BISECT (Bisection Method)
Prog3 SECANT (Secant Method)
Prog4 STEFFEN (Steffensen's Method)
Prog5 AITKEN (Aitken's Delta2 Method)
Prog6 HORNER (Horner's Method)
Prog7 FALSEPOS (False Position Method)
Prog8 MULLER (Muller's Method)
Prog9 SYSITER2 (Iteration for 2 x 2 Systems)
Prog10 SYSNEWT2 (Newton Iteration for 2 x 2 Systems)
Prog11 JACOBI (Jacobi Iteration)
Prog12 GSEIDEL (Gauss-Seidel Iteration)
Prog13 SOR (Successive Over-Relaxation)
Prog14 SCLPWR (Scaled Power Method)
Prog15 WDEFLATE (Wielandt Deflation)
Prog16 INTERPN (N-Point Interpolation)
Prog17 INTERP2 (2-Point Interpolation)
Prog18 INTERP3 (3-Point Interpolation)
Prog19 INTERP4 (4-Point Interpolation)
Prog20 INTERP5 (5-Point Interpolation)
Prog21 NEVILLE (Neville's Method)
Prog22 NEWTDIV (Newton's Divided Difference Method)
Prog23 LAGRANGE (Lagrange Interpolation Polynomial)
Part 2 (Programs 24 - 42)
Prog24 HERMITEN (N Point Hermite Interpolation)
Prog25 HERMITDD (Hermite Divided Difference Method)
Prog26 HERMITLA (Hermite-Lagrange Interpolating Polynomial)
Prog27 DERIV2PT (Derivative 2-Point Approximation)
Prog28 DERIV3PT (Derivative 3-Point Approximation)
Prog29 DERIV5PT (Derivative 5-Point Approximation)
Prog30 DDER3PT (Double Derivative 3-Point Approximation)
Prog31 NEWFORDF (Newton's Forward Difference Method)
Prog32 INEWCOTE (Newton-Cotes Integration Method
Prog33 ICOMTRAP (Composite Trapezoid Rule)
Prog34 ICOMMIDP (Composite Midpoint Rule)
Prog35 ICOMSIMP (Composite Simpson Rule)
Prog36 IROMBERG (Romberg Integration)
Prog37 IGAUSS (Gauss Quadrature Method)
Prog38 IGAUSSAB (Gaussian Quadrature Method on [a,b])
Prog39 IGAUSAB2 (Gauss 2 Point Quadrature on [a,b])
Prog40 IGAUSAB3 (Gauss 3 Point Quadrature on [a,b])
Prog41 IGAUSAB4 (Gauss 4 Point Quadrature on [a,b])
Prog42 IGAUSAB5 (Gauss 5 Point Quadrature on [a,b])
Part 3 (Programs 43 - 52)
Prog43 DEEULER (Euler's Method)
Prog44 DETAYLR2 (Taylor Method, Order 2)
Prog45 DETAYLR4 (Taylor Method, Order 4)
Prog46 DEMIDPT (DE Midpoint Method)
Prog47 DEMODEUL (DE Modified Euler Method)
Prog48 DEHEUN (DE Heun's Method)
Prog49 DERK4 (DE Runge Kutta 4th Order Method)
Prog50 DEBASH2 (Adams Bashforth 2 Step)
Prog51 DEMOULT2 (Adams Moulton 2 Step)
Prog52 DEPCB2M2 (Bashforth 2 Step Predictor, Moulton 2 Step Corrector)
Part 4 (Programs 53 - 70)
Prog53 DEBASH3 (Adams Bashforth 3 Step)
Prog54 DEMOULT3 (Adams Moulton 3 Step)
Prog55 DEPCB4M3 (Bashforth 4 Step Predictor, Moulton 3 Step Corrector)
Prog56 DERKV6 (DE Runge Kutta Verner 6th Order Method)
Prog57 LSPOLY1D (Least Squares Linear Polynomial (Discrete Data))
Prog58 LSPOLY2D (Least Squares Quadratic Polynomial (Discrete Data))
Prog59 LSPOLY3D (Least Squares Cubic Polynomial (Discrete Data))
Prog60 LSPOLY4D (Least Squares Quartic Polynomial (Discrete Data))
Prog61 LSEXPD (Least Squares Exponential Function (Discrete Data))
Prog62 LSPOLY1C (Least Squares Linear Polynomial (Continuous Data))
Prog63 LSPOLY2C (Least Squares Quadratic Polynomial (Continuous Data))
Prog64 LSPOLY3C (Least Squares Cubic Polynomial (Continuous Data))
Prog65 LSPOLY4C (Least Squares Quartic Polynomial (Continuous Data))
Prog66 LSLEGNDR (Least Squares Legendre Polynomials ( Degrees 0-5, [ -1 , 1 ] ) )
Prog67 LSLGDRAB (Least Squares Legendre Polynomials ( Degrees 0-5, [ a , b ] ) )
Prog68 SYSNEWT3 (Newton Iteration for 3 x 3 Systems)
Prog69 ECONOMIZ (Chebyshev Economization, (Degrees 0-8, [ -1 , 1 ] ) )
Prog70 ECONOMAB (Chebyshev Economization, (Degrees 0-8, [ a , b ] ) )
Part 5 (Programs 71 - 86)
Prog71 OPTINTP2 (Optimal Interolation, 2 Points)
Prog72 OPTINTP3 (Optimal Interolation, 3 Points)
Prog73 OPTINTP4 (Optimal Interolation, 4 Points)
Prog74 OPTINTP5 (Optimal Interolation, 5 Points)
Prog75 LSCHEBYS (Least Squares Chebyshev Polynomials, (Degrees 0-3, [ -1 , 1 ] ) )
Prog76 LSCHEBAB (Least Squares Chebyshev Polynomials, (Degrees 0-3, [ a , b ] ) )
Prog77 NEWTON (Newton Iteration with Test)
Prog78 MODNEWT (Modified Newton Method)
Prog79 UNSCLPWR (Unscaled Power Method)
Prog80 BEZIER (Bezier Curve, 2 Points)
Prog81 LSTRCN2 (LS Trig Curve , Continuous Data , Case N=2)
Prog82 LSTRCN3 (LS Trig Curve , Continuous Data , Case N=3)
Prog83 LSTRDN2 (LS Trig Curve , Discrete Data , Case N=2 , M > 2)
Prog84 LSTRDN3 (LS Trig Curve , Discrete Data , Case N=3 , M > 3)
Prog85 LSTRIN2 (LS Trig Interpolation Curve , Case N=2)
Prog86 LSTRIN2 (LS Trig Interpolation Curve , Case N=3)
Prog87 SYSITER3 (Iteration for 3 x 3 Systems)
Prog88 SYSGS2 (Gauss-Seidel for 2 x 2 Nonlinear Systems)
Prog89 SYSGS3 (Gauss-Seidel for 3 x 3 Nonlinear Systems)
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Program #43 DEEULER (Euler's Method)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 5 by 2 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + h wo' = wo + h fo ,
where fo = f ( xo , wo ) .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 1 , row 5 contains the value of B ( x1 ) = error bound function at x1 .
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of f ( xo , wo ) .
Column 2 , row 5 contains the value of ( h ) * f ( xo , wo ) .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DEEULER graphic display of the output matrix.
Instruction#1_____{5,2} -> dim([A])......................(Note: We use the symbol "->" to
Instruction#2_____X -> [A](1,2).............................represent the operation "Store".)
Instruction#3_____Y -> [A](2,2)
Instruction#4_____H -> [A](3,2)
Instruction#5_____Y1 -> [A](4,2)
Instruction#6_____HY1 -> [A](5,2)
Instruction#7_____Y + H Y1 -> Y
Instruction#8_____X + H -> X
Instruction#9_____X -> [A](1,1)
Instruction#10____Y -> [A](2,1)
Instruction#11____Y2 -> [A](3,1)
Instruction#12____Y2 - Y -> [A](4,1)
Instruction#13____Y3 -> [A](5,1)
Instruction#14____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEEULER download in binhex format.
PC Windows users: Click here for DEEULER download in zip format.
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Program #44 DETAYLR2 (Taylor Method, Order 2)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ' ' = ( d / dx ) ( y ' ) = ( d / dx ) f ( x , y ) = fx ( x , y ) + fy ( x , y ) f ( x , y ) is placed into the Y2 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y3 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y3 variable.
B ( x ) (if available) is placed into the Y4 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y4 variable.
Outputs:
The output matrix [A] is 4 by 3 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + h wo' + ( h2 / 2 ) wo' ' ,
where wo' = fo = f ( xo , wo ) , and where wo' ' = ( d / dx ) f ( xo , wo ) =
fx ( xo , wo , ) + fy ( xo , wo ) f ( xo , wo ) .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of wo' = fo =f ( xo , wo ) .
Column 3 , row 2 contains the value of wo' ' = ( d / dx ) f ( xo , wo )
Column 3 , row 3 contains the value of ( h ) wo' = ( h ) * f ( xo , wo ) .
Column 3 , row 4 contains the value of ( h2 / 2 ) wo' ' .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DETAYLR2 graphic display of the output matrix.
Instruction#1_____{4,3} -> dim([A])......................(Note: We use the symbol "->" to
Instruction#2_____X -> [A](1,2).............................represent the operation "Store".)
Instruction#3_____Y -> [A](2,2)
Instruction#4_____H -> [A](3,2)
Instruction#5_____Y1 -> [A](1,3)
Instruction#6_____Y2 -> [A](2,3)
Instruction#7_____H [A](1,3) -> [A](3,3)
Instruction#8_____( H2 / 2 ) [A](2,3) -> [A](4,3)
Instruction#9_____[A](2,2) + [A](3,3) + [A](4,3) -> Y
Instruction#10____X + H -> X
Instruction#11____X -> [A](1,1)
Instruction#12____Y -> [A](2,1)
Instruction#13____Y3 -> [A](3,1)
Instruction#14____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#15____Y4 -> [A](4,2)
Instruction#16____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DETAYLR2 download in binhex format.
PC Windows users: Click here for DETAYLR2 download in zip format.
Back to Menu
-------------------------------------------------------
Program #45 DETAYLR4 (Taylor Method, Order 4)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ' ' = ( d / dx ) ( y ' ) = ( d / dx ) f ( x , y ) = fx ( x , y ) + fy ( x , y ) f ( x , y ) is placed into the Y2 variable as an expression in X and Y .
y ' ' ' = ( d / dx ) ( y ' ' ) is placed into the Y3 variable as an expression in X and Y .
y ' ' ' ' = ( d / dx ) ( y ' ' ' ) is placed into the Y4 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y5 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y5 variable.
B ( x ) (if available) is placed into the Y6 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y6 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + h wo' + ( h2 / 2 ) wo' ' +
( h3 / 6 ) wo' ' ' + (h4 / 24 ) wo' ' ' ' ,
where wo' = fo = f ( xo , wo ) ,
wo' ' = ( d / dx ) f ( xo , wo ) = fx ( xo , wo , ) + fy ( xo , wo ) f ( xo , wo ) ,
wo' ' ' = ( d./.dx )2 f ( xo , wo ) ,
wo' ' ' ' = ( d./.dx )3 f ( xo , wo ) .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of wo' = fo =f ( xo , wo ) .
Column 3 , row 2 contains the value of wo' ' = ( d / dx ) f ( xo , wo )
Column 3 , row 3 contains the value of wo' ' ' = ( d / dx )2 f ( xo , wo )
Column 3 , row 4 contains the value of wo' ' ' ' = ( d / dx )3 f ( xo , wo )
Column 4 , row 1 contains the value of ( h ) wo' = ( h ) * f ( xo , wo ) .
Column 4 , row 2 contains the value of ( h2 / 2 ) wo' ' .
Column 4 , row 3 contains the value of ( h3 / 6 ) wo' ' ' .
Column 4 , row 4 contains the value of ( h4 / 24 ) wo' ' ' ' .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DETAYLR4 graphic display of the output matrix.
Instruction#1_____{4,4} -> dim([A])......................(Note: We use the symbol "->" to
Instruction#2_____X -> [A](1,2).............................represent the operation "Store".)
Instruction#3_____Y -> [A](2,2)
Instruction#4_____H -> [A](3,2)
Instruction#5_____Y1 -> [A](1,3)
Instruction#6_____Y2 -> [A](2,3)
Instruction#7_____Y3 -> [A](3,3)
Instruction#8_____Y4 -> [A](4,3)
Instruction#9_____H [A](1,3) -> [A](1,4)
Instruction#10____( H2 /2 ) [A](2,3) -> [A](2,4)
Instruction#11____( ( H^3) /6 ) [A](3,3) -> [A](3,4)
Instruction#12____( ( H^4) /24 ) [A](4,3) -> [A](4,4)
Instruction#13____[A](2,2) + [A](1,4) + [A](2,4) + [A](3,4) + [A](4,4) -> Y
Instruction#14____X + H -> X
Instruction#15____X -> [A](1,1)
Instruction#16____Y -> [A](2,1)
Instruction#17____Y5 -> [A](3,1)
Instruction#18____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#19____Y6 -> [A](4,2)
Instruction#20____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DETAYLR4 download in binhex format.
PC Windows users: Click here for DETAYLR4 download in zip format.
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-------------------------------------------------------
Program #46 DEMIDPT (DE Midpoint Method)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + k2
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of f ( xo , wo ) .
Column 3 , row 2 contains the value of k1 = h f ( xo , wo ) .
Column 3 , row 3 contains the value of xo + h / 2 .
Column 3 , row 4 contains the value of wo + k1 / 2 .
Column 4 , row 1 contains the value of f ( xo + h / 2 , wo + k1 / 2 ) .
Column 4 , row 2 contains the value of k2 = h f ( xo + h / 2 , wo + k1 / 2 ) .
Column 4 , row 3 contains the value 0 .
Column 4 , row 4 contains the value 0 .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DEMIDPT graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...............(Note: We use the symbol "->" to
Instruction#2_____Fill( 0 , [A] ).............................represent the operation "Store".)
Instruction#3_____X -> [A](1,2)
Instruction#4_____Y -> [A](2,2)
Instruction#5_____H -> [A](3,2)
Instruction#6_____Y1 -> [A](1,3)
Instruction#7_____H [A](1,3) -> [A](2,3)
Instruction#8_____X + (.5)H -> X
Instruction#9_____X -> [A](3,3)
Instruction#10____Y + (.5)[A](2,3) -> Y
Instruction#11____Y -> [A](4,3)
Instruction#12____Y1 -> [A](1,4)
Instruction#13____H [A](1,4) -> [A](2,4)
Instruction#14____X + (.5) H -> X
Instruction#15____X -> [A](1,1)
Instruction#16____[A](2,2) + [A](2,4) -> Y
Instruction#17____Y -> [A](2,1)
Instruction#18____Y2 -> [A](3,1)
Instruction#19____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#20____Y3 -> [A](4,2)
Instruction#21____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEMIDPT download in binhex format.
PC Windows users: Click here for DEMIDPT download in zip format.
Back to Menu
-------------------------------------------------------
Program #47 DEMODEUL (DE Modified Euler Method)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + ( k1 + k2 ) / 2 .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of f ( xo , wo ) .
Column 3 , row 2 contains the value of k1 = h f ( xo , wo ) .
Column 3 , row 3 contains the value of xo + h .
Column 3 , row 4 contains the value of wo + k1 .
Column 4 , row 1 contains the value of f ( xo + h , wo + k1 ) .
Column 4 , row 2 contains the value of k2 = h f ( xo + , wo + k1 ) .
Column 4 , row 3 contains the value ( k1 + k2 )/2 .
Column 4 , row 4 contains the value 0 .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DEMODEUL graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...............(Note: We use the symbol "->" to
Instruction#2_____Fill( 0 , [A] ).............................represent the operation "Store".)
Instruction#3_____X -> [A](1,2)
Instruction#4_____Y -> [A](2,2)
Instruction#5_____H -> [A](3,2)
Instruction#6_____Y1 -> [A](1,3)
Instruction#7_____H [A](1,3) -> [A](2,3)
Instruction#8_____X + H -> X
Instruction#9_____X -> [A](3,3)
Instruction#10____Y + [A](2,3) -> Y
Instruction#11____Y -> [A](4,3)
Instruction#12____Y1 -> [A](1,4)
Instruction#13____H [A](1,4) -> [A](2,4)
Instruction#14____X -> [A](1,1)
Instruction#15____[A](2,2) + ( [A](2,3) + [A](2,4) ) / 2 -> Y
Instruction#16____Y -> [A](2,1)
Instruction#17____Y2 -> [A](3,1)
Instruction#18____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#19____Y3 -> [A](4,2)
Instruction#20____( [A](2,3) + [A](2,4) ) / 2 -> [A](3,4)
Instruction#21____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
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Program #48 DEHEUN (DE Heun's Method)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + ( k1 + 3 k2 ) / 4 .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of f ( xo , wo ) .
Column 3 , row 2 contains the value of k1 = h f ( xo , wo ) .
Column 3 , row 3 contains the value of xo + 2 h / 3 .
Column 3 , row 4 contains the value of wo + 2 k1 / 3 .
Column 4 , row 1 contains the value of f ( xo + 2 h / 3 , wo + 2 k1 / 3 ) .
Column 4 , row 2 contains the value of k2 = h f ( xo + 2 h / 3 , wo + 2 k1 / 3 ) .
Column 4 , row 3 contains the value ( k1 + 3 k2 ) / 4 .
Column 4 , row 4 contains the value 0 .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DEHEUN graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...............(Note: We use the symbol "->" to
Instruction#2_____Fill( 0 , [A] ).............................represent the operation "Store".)
Instruction#3_____X -> [A](1,2)
Instruction#4_____Y -> [A](2,2)
Instruction#5_____H -> [A](3,2)
Instruction#6_____Y1 -> [A](1,3)
Instruction#7_____H [A](1,3) -> [A](2,3)
Instruction#8_____X + 2H / 3 -> X
Instruction#9_____X -> [A](3,3)
Instruction#10____Y + ( 2 / 3 )[A](2,3) -> Y
Instruction#11____Y -> [A](4,3)
Instruction#12____Y1 -> [A](1,4)
Instruction#13____H [A](1,4) -> [A](2,4)
Instruction#14____X + H / 3 -> X
Instruction#15____X -> [A](1,1)
Instruction#16____[A](2,2) + ( 1 / 4 ) ( [A](2,3) + 3 [A](2,4) ) -> Y
Instruction#17____Y -> [A](2,1)
Instruction#18____Y2 -> [A](3,1)
Instruction#19____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#20____Y3 -> [A](4,2)
Instruction#21____( 1 / 4 ) ( [A](2,3) + 3 [A](2,4) ) -> [A](3,4)
Instruction#22____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEHEUN download in binhex format.
PC Windows users: Click here for DEHEUN download in zip format.
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Program #49 DERK4 (DE Runge Kutta 4th Order Method)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
Then in the TI83 calculator, we first store the inputs as follows:
xo -> X .
wo -> Y .
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x1 = xo + h .
Column 1 , row 2 contains the value of w1 = wo + ( k1 + 2 k2 + 2 k3 + k4 ) / 6 .
Column 1 , row 3 contains the value of y ( x1 ) .
Column 1 , row 4 contains the value of y ( x1 ) - w1 = exact error at x1.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of wo .
Column 2 , row 3 contains the value of h .
Column 2 , row 4 contains the value of B ( x1 ) = error bound function at x1 .
Column 3 , row 1 contains the value of k1 = h f ( xo , wo ) .
Column 3 , row 2 contains the value of k2 = h f ( xo + h / 2 , wo + k1 / 2 ) .
Column 3 , row 3 contains the value of k3 = h f ( xo + h / 2 , wo + k2 / 2 ).
Column 3 , row 4 contains the value of k4 = h f ( xo + h , wo + k3 ) .
Column 4 , row 1 contains the value of wo + k1 / 2 .
Column 4 , row 2 contains the value of wo + k2 / 2 ).
Column 4 , row 3 contains the value wo + k3 .
Column 4 , row 4 contains the value ( k1 + 2 k2 + 2 k3 + k4 ) / 6 .
In addition, the value of X is updated to contain the value of x1 . The value of Y is updated to contain the value of w1 . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x2 , w2 , etc ... .
Click here for a DERK4 graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...............(Note: We use the symbol "->" to
Instruction#2_____X -> [A](1,2).............................represent the operation "Store".)
Instruction#3_____Y -> [A](2,2)
Instruction#4_____H -> [A](3,2)
Instruction#5_____H Y1 -> [A](1,3)
Instruction#6_____X + H / 2 -> X
Instruction#7_____[A](2,2) + (.5) [A](1,3) -> Y
Instruction#8_____Y -> [A](1,4)
Instruction#9_____H Y1 -> [A](2,3)
Instruction#10____[A](2,2) + (.5) [A](2,3) -> Y
Instruction#11____Y -> [A](2,4)
Instruction#12____H Y1 -> [A](3,3)
Instruction#13____X + H / 2 -> X
Instruction#14____[A](2,2) + [A](3,3) -> Y
Instruction#15____Y -> [A](3,4)
Instruction#16____H Y1 -> [A](4,3)
Instruction#17____( [A](1,3) + 2 ( [A](2,3) + [A](3,3) ) + [A](4,3) ) / 6 -> [A](4,4)
Instruction#18____[A](2,2) + [A](4,4) -> Y
Instruction#19____X -> [A](1,1)
Instruction#20____Y -> [A](2,1)
Instruction#21____Y2 -> [A](3,1)
Instruction#22____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#23____Y3 -> [A](4,2)
Instruction#24____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DERK4 download in binhex format.
PC Windows users: Click here for DERK4 download in zip format.
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Program #50 DEBASH2 (Adams Bashforth 2 Step)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
The Bashforth 2-step method requires two "starting values", wo , and w1 . The value w1 is an approximation to y ( xo + h ) . The value for w1 will generally come from using some one step method, such as a Runge Kutta method.
Then in the TI83 calculator, we first store the inputs as follows:
x1 -> X . ( Note that it is x1 = xo + h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) . Thus, the L1 list looks like { wo , w1 } initially.
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x2 = x1 + h .
Column 1 , row 2 contains the value of w2 = w1 + ( h / 2 ) ( 3 f ( x1 , w1 ) - f ( xo , wo ) ) .
Column 1 , row 3 contains the value of y ( x2 ) .
Column 1 , row 4 contains the value of y ( x2 ) - w2 = exact error at x2.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of x1 .
Column 2 , row 3 contains the value 0 .
Column 2 , row 4 contains the value of B ( x2 ) = error bound function at x2 .
Column 3 , row 1 contains the value of wo .
Column 3 , row 2 contains the value of w1 .
Column 3 , row 3 contains the value 0 .
Column 3 , row 4 contains the value 0 .
Column 4 , row 1 contains the value of f ( xo , wo ) .
Column 4 , row 2 contains the value of f ( x1 , w1 ) .
Column 4 , row 3 contains the value 0 .
Column 4 , row 4 contains the value of ( h / 2 ) ( 3 f ( x1 , w1 ) - f ( xo , wo ) ) .
In addition, the value of X is updated to contain the value of x2 . The values in the list L1 are updated so that L1 will contain { w1 , w2 } . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x3 , w3 , etc ... .
Click here for a DEBASH2 graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...... ...........(Note: We use the symbol "->" to
Instruction#2_____Fill( 0, [A] ).............................represent the operation "Store".)
Instruction#3_____X-H -> X
Instruction#4_____For ( I , 1 , 2 )
Instruction#5_____X -> [A]( I , 2 )
Instruction#6_____L1(I) -> Y
Instruction#7_____Y -> [A]( I , 3 )
Instruction#8_____Y1 -> [A]( I , 4 )
Instruction#9_____X + H -> X
Instruction#10____End
Instruction#11____( H / 2 ) ( 3 [A](2,4) - [A](1,4) ) -> [A](4,4)
Instruction#12____[A](2,3) + [A](4,4) -> [A](2,1)
Instruction#13____X -> [A](1,1)
Instruction#14____Y2 -> [A](3,1)
Instruction#15____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#16____Y3 -> [A](4,2)
Instruction#17____L1(2) -> L1(1)
Instruction#18____[A](2,1) -> L1(2)
Instruction#19____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEBASH2 download in binhex format.
PC Windows users: Click here for DEBASH2 download in zip format.
Program DEBASH2S. This is a short version for DEBASH2. The short version has a smaller output matrix, has fewer instructions, but has some redundant computations within the program. These redundancies do not slow the program in any significant manner. (Program description to appear.)
Instruction#1_____1 -> dim ( L2 ).........................(Note: We use the symbol "->" to
Instruction#2_____X - H -> X.............................represent the operation "Store".)
Instruction#3_____For ( J , 1 , 2 )
Instruction#4_____L1 ( J ) -> Y
Instruction#5_____Y1 -> L2 ( J )
Instruction#6_____X + H -> X
Instruction#7_____End
Instruction#8_____L1(2) + ( H / 2 ) ( 3 L2(2) - L2(1) ) -> Y
Instruction#9_____L1(2) -> L1(1)
Instruction#10____Y -> L1(2)
Instruction#11____[ [ X , Y , Y2 , Y2 - Y , Y3 ] ]T
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Program #51 DEMOULT2 (Adams Moulton 2 Step)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
The Moulton 2-step method requires two "starting values", wo , and w1 . The value w1 is an approximation to y ( xo + h ) . The value for w1 will generally come from using some one step method, such as a Runge Kutta method.
Then in the TI83 calculator, we first store the inputs as follows:
x1 -> X . ( Note that it is x1 = xo + h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) . Thus, the L1 list looks like { wo , w1 } initially.
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x2 = x1 + h .
Column 1 , row 2 contains the value of w2 . w2 solves the equation:
w2 = w1 + ( h / 12 ) ( 5 f ( x2 , w2 ) + 8 f ( x1 , w1 ) - f ( xo , wo ) ) .
Column 1 , row 3 contains the value of y ( x2 ) .
Column 1 , row 4 contains the value of y ( x2 ) - w2 = exact error at x2.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of x1 .
Column 2 , row 3 contains the value 0 .
Column 2 , row 4 contains the value of B ( x2 ) = error bound function at x2 .
Column 3 , row 1 contains the value of wo .
Column 3 , row 2 contains the value of w1 .
Column 3 , row 3 contains the value 0 .
Column 3 , row 4 contains the value of ( h / 12 ) ( 5 f ( x2 , w2 ) ) .
Column 4 , row 1 contains the value of f ( xo , wo ) .
Column 4 , row 2 contains the value of f ( x1 , w1 ) .
Column 4 , row 3 contains the value of f ( x2 , w2 ) .
Column 4 , row 4 contains the value of w1 + ( h / 12 ) ( 8 f ( x1 , w1 ) - f ( xo , wo ) ) .
Also, list L2 contains Steffensen iterates in the solution (by iteration)
for w2 in the equation:
w2 = w1 + ( h / 12 ) ( 5 f ( x2 , w2 ) + 8 f ( x1 , w1 ) - f ( xo , wo ) ) .
In addition, the value of X is updated to contain the value of x2 . The values in the list L1 are updated so that L1 will contain { w1 , w2 } . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x3 , w3 , etc ... .
Click here for a DEMOULT2 graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] )...............(Note: We use the symbol "->" to
Instruction#2_____1 -> dim( L2 )......................represent the operation "Store".)
Instruction#3_____Fill( 0, [A] )
Instruction#4_____X - H -> X
Instruction#5_____For ( I , 1 , 2 )
Instruction#6_____X -> [A]( I , 2 )
Instruction#7_____L1(I) -> Y
Instruction#8_____Y -> [A]( I , 3 )
Instruction#9_____Y1 -> [A]( I , 4 )
Instruction#10____X + H -> X
Instruction#11____End
Instruction#12____Y + ( H / 12 ) ( 8 [A](2,4) - [A](1,4) ) -> [A](4,4)
Instruction#13____X -> [A](1,1)
Instruction#14____Y2 -> [A](3,1)
Instruction#15____Y + ( H / 2 ) ( 3 [A](2,4) - [A](1,4) ) -> A
Instruction#16____A -> L2(1)
Instruction#17____For( I , 1 , 4 )
Instruction#18____A -> Y
Instruction#19____( H / 12 ) ( 5 Y1 ) + [A](4,4) -> B
Instruction#20____B -> Y
Instruction#21____( H / 12 ) ( 5 Y1 ) + [A](4,4) -> C
Instruction#22____If C - 2 B + A != 0..........................(See note below.)
Instruction#23____A - ( ( B - A )2 ) / ( C - 2 B + A ) -> A
Instruction#24____A -> L2( I + 1 )
Instruction#25____End
Instruction#26____A -> [A](2,1)
Instruction#27____A -> Y
Instruction#28____Y1 -> [A](3,4)
Instruction#29____( H / 12 ) ( 5 [A](3,4) ) -> [A](4,3)
Instruction#30____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#31____Y3 -> [A](4,2)
Instruction#32____L1(2) -> L1(1)
Instruction#33____[A](2,1) -> L1(2)
Instruction#34____[A]
Note that in instruction#22, we are using the symbol != to represent "not equal". When entering this program into the TI83, one should use the more standard symbol from the TEST menu.
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEMOULT2 download in binhex format.
PC Windows users: Click here for DEMOULT2 download in zip format.
Program DEMOLT2S. This is a short version for DEMOULT2. The short version has a smaller output matrix, has fewer instructions, but has some redundant computations within the program. These redundancies do not slow the program in any significant manner. (Program description to appear.)
Instruction#1_____1 -> dim ( L2 ).........................(Note: We use the symbol "->" to
Instruction#2_____1 -> dim ( L3 )..........................represent the operation "Store".)
Instruction#3_____X - H -> X
Instruction#4_____For ( J , 1 , 2 )
Instruction#5_____L1 ( J ) -> Y
Instruction#6_____Y1 -> L3 ( J )
Instruction#7_____X + H -> X
Instruction#8_____End
Instruction#9_____Y + ( H / 12 ) ( 8 L3(2) - L3(1) ) -> T
Instruction#10____Y + ( H / 2 ) ( 3 L3(2) - L3(1) ) -> Y
Instruction#11____Y -> L2 (1)
Instruction#12____For ( I , 1 , 4 )
Instruction#13____Y -> A
Instruction#14____( H / 12 ) ( 5 Y1 ) + T -> Y
Instruction#15____( H / 12 ) ( 5 Y1 ) + T -> C
Instruction#16____If C - 2 Y + A != 0...............................(See note below.)
Instruction#17____A - ( Y - A )2 / ( C - 2 Y + A ) -> Y
Instruction#18____Y -> L2 ( I + 1 )
Instruction#19____End
Instruction#20____L1(2) -> L1(1)
Instruction#21____Y -> L1(2)
Instruction#22____[ [ X , Y , Y2 , Y2 - Y , Y3 ] ]T
Note that in instruction#16, we are using the symbol != to represent "not equal". When entering this program into the TI83, one should use the more standard symbol from the TEST menu.
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Program #52 DEPCB2M2 (Bashforth 2 Step Predictor, Moulton 2 Step Corrector)
(Description to appear.)
Inputs:
The differential equation is dy / dx = f ( x , y ), with the initial condition y ( a ) = b .We set xo = a and wo = b = y ( a ). In the numerical method, the "spacing" of the partition points along the x-axis is denoted by "h" . The exact solution is denoted by y ( x ), and the error bound function ( if available ) is denoted by B ( x ) .
This predictor-corrector method requires two "starting values", wo , and w1 . The value w1 is an approximation to y ( xo + h ) . The value for w1 will generally come from using some one step method, such as a Runge Kutta method.
Then in the TI83 calculator, we first store the inputs as follows:
x1 -> X . ( Note that it is x1 = xo + h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) . Thus, the L1 list looks like { wo , w1 } initially.
h -> H .
f ( x , y ) is placed into the Y1 variable as an expression in X and Y .
y ( x ) (if available) is placed into the Y2 variable as an expression in X . If y ( x ) is not available, then 0 is placed into the Y2 variable.
B ( x ) (if available) is placed into the Y3 variable as an expression in X . If B ( x ) is not available, then 0 is placed into the Y3 variable.
Outputs:
The output matrix [A] is 4 by 4 and contains the following:
Column 1 , row 1 contains the value of x2 = x1 + h .
Column 1 , row 2 contains the value of w2 . w2 solves the equation:
w2 = w1 + ( h / 12 ) ( 5 f ( x2 , w2(o) ) + 8 f ( x1 , w1 ) - f ( xo , wo ) ) .
Column 1 , row 3 contains the value of y ( x2 ) .
Column 1 , row 4 contains the value of y ( x2 ) - w2 = exact error at x2.
Column 2 , row 1 contains the value of xo .
Column 2 , row 2 contains the value of x1 .
Column 2 , row 3 contains the value 0 .
Column 2 , row 4 contains the value of B ( x2 ) = error bound function at x2 .
Column 3 , row 1 contains the value of wo .
Column 3 , row 2 contains the value of w1 .
Column 3 , row 3 contains the value w2(o) =
w1 + ( h / 2 ) ( 3 f ( x1 , w1 ) - f ( xo , wo ) ) .
Column 3 , row 4 contains the value of ( h / 12 ) ( 5 f ( x2 , w2(o) ) ) .
Column 4 , row 1 contains the value of f ( xo , wo ) .
Column 4 , row 2 contains the value of f ( x1 , w1 ) .
Column 4 , row 3 contains the value of f ( x2 , w2(o) ) .
Column 4 , row 4 contains the value of w1 + ( h / 12 ) ( 8 f ( x1 , w1 ) - f ( xo , wo ) ) .
In addition, the value of X is updated to contain the value of x2 . The values in the list L1 are updated so that L1 will contain { w1 , w2 } . Thus, after the output matrix is displayed on the home screen, one can press [ENTER] to rerun the program to get a second output matrix containing the values of x3 , w3 , etc ... .
Click here for a DEPCB2M2 graphic display of the output matrix.
Instruction#1_____{4,4} -> dim( [A] ).............(Note: We use the symbol "->" to
Instruction#2_____Fill( 0, [A] )........................represent the operation "Store".)
Instruction#3_____X - H -> X
Instruction#4_____For ( I , 1 , 2 )
Instruction#5_____X -> [A]( I , 2 )
Instruction#6_____L1(I) -> Y
Instruction#7_____Y -> [A]( I , 3 )
Instruction#8_____Y1 -> [A]( I , 4 )
Instruction#9_____X + H -> X
Instruction#10____End
Instruction#11____Y + ( H / 12 ) ( 8 [A](2,4) - [A](1,4) ) -> [A](4,4)
Instruction#12____X -> [A](1,1)
Instruction#13____Y2 -> [A](3,1)
Instruction#14____Y + ( H / 2 ) ( 3 [A](2,4) - [A](1,4) ) -> Y
Instruction#15____Y -> [A](3,3)
Instruction#16____Y1 -> [A](3,4)
Instruction#17____( H / 12 ) ( 5 [A](3,4) ) -> [A](4,3)
Instruction#18____[A](4,3) + [A](4,4) -> [A](2,1)
Instruction#19____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#20____Y3 -> [A](4,2)
Instruction#21____L1(2) -> L1(1)
Instruction#22____[A](2,1) -> L1(2)
Instruction#23____[A]
Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.
This program may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for DEPCB2M2 download in binhex format.
PC Windows users: Click here for DEPCB2M2 download in zip format.
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