TI83 Calculator Programs for Numerical Analysis Problems - Part 5



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 #71 OPTINTP2 (Optimal Interpolation, 2 Points)


(Description to appear.)


Inputs:

Let y = f ( x ) on [ a , b ] .

Place f ( x ) into the function variable Y1 .

Place "a" into the variable A .
Place "b" into the variable B .
Place "x" into the variable X .


Outputs:

The output matrix [ B ] is 2 by 3 and contains the following:

Column 1 contains x1 and x2 , the Chebyshev zeroes on the interval [ a , b ] .

Column 2 contains f ( x1 ) and f ( x2 ) .

Column 3 contains the values f ( x ) and P ( x ) for the given value of "x".
Here, P ( x ) is the interpolating polynomial for the data in columns 1 and 2 of
the output matrix [ B ] .

In addition:

The list variable L1 contains the list { x1 , x2 } .

The list variable L2 contains the list { f ( x1 ) , f ( x2 ) } .

The list variable L3 contains the list { t 1 , t 2 } . Here, t 1 and t 2 are the
Chebyshev zeroes on the interval [ - 1 , 1 ] .
( t k = cos ( ( 2 k - 1 ) ( Pi ) / 4 ) )

The matrix variable [ A ] contains the divided difference table for the
data in L1 and L2 , as created by the program NEWTDIV.

The function variable Y2 contains the polynomial P ( x ) in divided
difference form.

After running the program OPTINTP2, one can then execute the instruction
" LinReg L1 , L2 , Y3 " on the home screen in order to place P ( x ) into
the Y3 variable in standard polynomial form.

Click here for a OPTINTP2 graphic display of the output matrix.



Instruction#1_____2 -> N............................(Note: We use the symbol "->" to
Instruction#2_____( B - A ) / 2 -> C...............represent the operation "Store".)
Instruction#3_____( B + A ) / 2 -> D
Instruction#4_____seq ( cos ( (Pi) ( 2K - 1 ) / ( 2N ) ) , K, 1, N ) -> L3.....(See note below.)
Instruction#5_____C L3 + D -> L1
Instruction#6_____Y1 ( L1 ) -> L2
Instruction#7_____prgm NEWTDIV
Instruction#8_____" [A]( 1 , 1 ) + ( X - L1(1) ) [A]( 2 , 2 ) " -> Y2
Instruction#9_____List > matr ( L1, L2, { Y1, Y2 } ,[B] )........(See note below.)
Instruction#10____[B]

Note: In Instruction#4, "Pi" represents the Pi button on the calculator keyboard.

Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" 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 OPTINTP2 download in binhex format.

PC Windows users: Click here for OPTINTP2 download in zip format.


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Program #72 OPTINTP3 (Optimal Interpolation, 3 Points)


(Description to appear.)


Inputs:

Let y = f ( x ) on [ a , b ] .

Place f ( x ) into the function variable Y1 .

Place "a" into the variable A .
Place "b" into the variable B .
Place "x" into the variable X .

Outputs:

The output matrix [ B ] is 3 by 3 and contains the following:

Column 1 contains x1 , x2 and x3 , the Chebyshev zeroes on the interval [ a , b ] .

Column 2 contains f ( x1 ) , f ( x2 ) and f ( x3 ) .

Column 3 contains the values f ( x ) and P ( x ) for the given value of "x".
Here, P ( x ) is the interpolating polynomial for the data in columns 1 and 2 of
the output matrix [ B ] .

In addition:

The list variable L1 contains the list { x1 , x2 , x3 } .

The list variable L2 contains the list { f ( x1 ) , f ( x2 ) , f ( x3 ) } .

The list variable L3 contains the list { t 1 , t 2 , t 3 } . Here, t 1 , t 2 and t 3 are the
Chebyshev zeroes on the interval [ - 1 , 1 ] .
( t k = cos ( ( 2 k - 1 ) ( Pi ) / 6 ) )

The matrix variable [ A ] contains the divided difference table for the
data in L1 and L2 , as created by the program NEWTDIV.

The function variable Y2 contains the polynomial P ( x ) in divided
difference form.

After running the program OPTINTP3, one can then execute the instruction
" QuadReg L1 , L2 , Y3 " on the home screen in order to place P ( x ) into
the Y3 variable in standard polynomial form.

Click here for a OPTINTP3 graphic display of the output matrix.



Instruction#1_____3 -> N............................(Note: We use the symbol "->" to
Instruction#2_____( B - A ) / 2 -> C...............represent the operation "Store".)
Instruction#3_____( B + A ) / 2 -> D
Instruction#4_____seq ( cos ( (Pi) ( 2K - 1 ) / ( 2N ) ) , K, 1, N ) -> L3.....(See note below.)
Instruction#5_____C L3 + D -> L1
Instruction#6_____Y1 ( L1 ) -> L2
Instruction#7_____prgm NEWTDIV
Instruction#8_____" [A]( 1 , 1 ) + ( X - L1(1) )
______________( [A]( 2 , 2 ) + ( X - L1(2) ) [A]( 3 , 3 ) ) " -> Y2
Instruction#9_____List > matr ( L1, L2, { Y1, Y2 } ,[B] )........(See note below.)
Instruction#10____[B]

Note: In Instruction#4, "Pi" represents the Pi button on the calculator keyboard.

Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" 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 OPTINTP3 download in binhex format.

PC Windows users: Click here for OPTINTP3 download in zip format.


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Program #73 OPTINTP4 (Optimal Interpolation, 4 Points)


(Description to appear.)


Inputs:

Let y = f ( x ) on [ a , b ] .

Place f ( x ) into the function variable Y1 .

Place "a" into the variable A .
Place "b" into the variable B .
Place "x" into the variable X .

Outputs:

The output matrix [ B ] is 4 by 3 and contains the following:

Column 1 contains x1 , x2 , x3 and x4 , the Chebyshev zeroes on the interval [ a , b ] .

Column 2 contains f ( x1 ) , f ( x2 ) , f ( x3 ) and f ( x4 ) .

Column 3 contains the values f ( x ) and P ( x ) for the given value of "x".
Here, P ( x ) is the interpolating polynomial for the data in columns 1 and 2 of
the output matrix [ B ] .

In addition:

The list variable L1 contains the list { x1 , x2 , x3 , x4 } .

The list variable L2 contains the list { f ( x1 ) , f ( x2 ) , f ( x3 ) , f ( x4 ) } .

The list variable L3 contains the list { t 1 , t 2 , t 3 , t 4 } . Here, t 1 , t 2 , t 3 and t 4 are the
Chebyshev zeroes on the interval [ - 1 , 1 ] .
( t k = cos ( ( 2 k - 1 ) ( Pi ) / 8 ) )

The matrix variable [ A ] contains the divided difference table for the
data in L1 and L2 , as created by the program NEWTDIV.

The function variable Y2 contains the polynomial P ( x ) in divided
difference form.

After running the program OPTINTP4, one can then execute the instruction
" CubicReg L1 , L2 , Y3 " on the home screen in order to place P ( x ) into
the Y3 variable in standard polynomial form.

Click here for a OPTINTP4 graphic display of the output matrix.



Instruction#1_____4 -> N............................(Note: We use the symbol "->" to
Instruction#2_____( B - A ) / 2 -> C...............represent the operation "Store".)
Instruction#3_____( B + A ) / 2 -> D
Instruction#4_____seq ( cos ( (Pi) ( 2K - 1 ) / ( 2N ) ) , K, 1, N ) -> L3.....(See note below.)
Instruction#5_____C L3 + D -> L1
Instruction#6_____Y1 ( L1 ) -> L2
Instruction#7_____prgm NEWTDIV
Instruction#8_____" [A]( 1 , 1 ) + ( X - L1(1) )
______________( [A]( 2 , 2 ) + ( X - L1(2) )
______________( [A]( 3 , 3 ) + ( X - L1(3) ) [A]( 4 , 4 ) ) ) " -> Y2
Instruction#9_____List > matr ( L1, L2, { Y1, Y2 } ,[B] )........(See note below.)
Instruction#10____[B]

Note: In Instruction#4, "Pi" represents the Pi button on the calculator keyboard.

Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" 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 OPTINTP4 download in binhex format.

PC Windows users: Click here for OPTINTP4 download in zip format.


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Program #74 OPTINTP5 (Optimal Interpolation, 5 Points)


(Description to appear.)


Inputs:

Let y = f ( x ) on [ a , b ] .

Place f ( x ) into the function variable Y1 .

Place "a" into the variable A .
Place "b" into the variable B .
Place "x" into the variable X .

Outputs:

The output matrix [ B ] is 5 by 3 and contains the following:

Column 1 contains x1 , x2 , x3 , x4 and x5 , the Chebyshev
zeroes on the interval [ a , b ] .

Column 2 contains f ( x1 ) , f ( x2 ) , f ( x3 ) , f ( x4 ) and f ( x5 ) .

Column 3 contains the values f ( x ) and P ( x ) for the given value of "x".
Here, P ( x ) is the interpolating polynomial for the data in columns 1 and 2 of
the output matrix [ B ] .

In addition:

The list variable L1 contains the list { x1 , x2 , x3 , x4 , x5 } .

The list variable L2 contains the list { f ( x1 ) , f ( x2 ) , f ( x3 ) , f ( x4 ) , f ( x5 ) } .

The list variable L3 contains the list { t 1 , t 2 , t 3 , t 4 , t 5 } . Here, t 1 , t 2 , t 3 , t 4
and t 5 are the Chebyshev zeroes on the interval [ - 1 , 1 ] .
( t k = cos ( ( 2 k - 1 ) ( Pi ) / 8 ) )

The matrix variable [ A ] contains the divided difference table for the
data in L1 and L2 , as created by the program NEWTDIV.

The function variable Y2 contains the polynomial P ( x ) in divided
difference form.

After running the program OPTINTP5, one can then execute the instruction
" QuartReg L1 , L2 , Y3 " on the home screen in order to place P ( x ) into
the Y3 variable in standard polynomial form.

Click here for a OPTINTP5 graphic display of the output matrix.



Instruction#1_____5 -> N............................(Note: We use the symbol "->" to
Instruction#2_____( B - A ) / 2 -> C...............represent the operation "Store".)
Instruction#3_____( B + A ) / 2 -> D
Instruction#4_____seq ( cos ( (Pi) ( 2K - 1 ) / ( 2N ) ) , K, 1, N ) -> L3.....(See note below.)
Instruction#5_____C L3 + D -> L1
Instruction#6_____Y1 ( L1 ) -> L2
Instruction#7_____prgm NEWTDIV
Instruction#8_____" [A]( 1 , 1 ) + ( X - L1(1) )
______________( [A]( 2 , 2 ) + ( X - L1(2) )
______________( [A]( 3 , 3 ) + ( X - L1(3) )
______________( [A]( 4 , 4 ) + ( X - L1(4) ) [A]( 5 , 5 ) ) ) ) " -> Y2
Instruction#9_____List > matr ( L1, L2, { Y1, Y2 } ,[B] )........(See note below.)
Instruction#10____[B]

Note: In Instruction#4, "Pi" represents the Pi button on the calculator keyboard.

Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" 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 OPTINTP5 download in binhex format.

PC Windows users: Click here for OPTINTP5 download in zip format.


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Program #75 LSCHEBYS (Least Squares Chebyshev Polynomials, (Degrees 0-3, [ -1 , 1 ] ) )


(Description to appear.)



(This program may take about 20 seconds to run.)


Instruction#1_____[ [ 1, 0, -1, 0 ] [ 0, 1, 0, -3 ] [ 0, 0, 2, 0 ] [ 0, 0, 0, 4 ] ] -> [A]
Instruction#2_____"XY1" -> Y2 :
______________"X2Y1" -> Y3 :.....................(Note: We use the symbol "->" to
______________"X3Y1" -> Y4...........................represent the operation "Store".)
Instruction#3_____[ [ fnInt( Y1( cos( X ) ) , X , 0 , Pi ]
_______________[ fnInt( Y2( cos( X ) ) , X , 0 , Pi ]
_______________[ fnInt( Y3( cos( X ) ) , X , 0 , Pi ]
_______________[ fnInt( Y4( cos( X ) ) , X , 0 , Pi ] ] -> [B]......(See notes below.)
Instruction#4_____Matr>list ( [A]T[B] , L1 )
Instruction#5_____( 2 / Pi ) L1 -> L2 : .5L2(1) -> L2(1)
Instruction#6_____fnInt( Y1( cos(X) )2 , X , 0 , Pi ) -> S
Instruction#7_____S - cumSum ( L1 L2 ) -> L3
Instruction#8_____L2(1) -> G : L2(2) -> H : L2(3) -> I : L2(4) -> J
Instruction#9_____[ [ G, G, G, G ] [ 0, H, H, H ] [ 0, 0, I, I ] [ 0, 0, 0, J ] ] -> [C]
Instruction#10____[A] [C] -> [D]
Instruction#11____List>matr ( L2, L3, L1, [E] )
Instruction#12____augment ( [E] , [D] ) -> [F]

Note: In Instruction#3, "Pi" represents the Pi button on the calculator keyboard.

Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" 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 LSCHEBYS download in binhex format.

PC Windows users: Click here for LSCHEBYS download in zip format.



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Program #76 LSCHEBAB (Least Squares Chebyshev Polynomials, (Degrees 0-3, [ a , b ] ) )


(Description to appear.)



(This program may take about 30 seconds to run.)


Instruction#1_____(B+A) / 2 -> C :
______________(B-A) / 2 -> D..................(Note: We use the symbol "->" to
Instruction#2_____(-)C / D -> E : D-1 -> F..........represent the operation "Store".)
Instruction#3_____[ [ 1, E, E2, E3 ] [ 0, F, 2EF, 3E2F ]
_______________[ 0, 0, F2, 3EF2 ] [ 0, 0, 0, F3 ] ] -> [G]...........(See note below.)
Instruction#4_____Equ>String ( Y1 , Str 1 )
Instruction#5_____" " -> Y5
Instruction#6_____String>Equ ( Str 1 , Y5 )
Instruction#7_____" Y5 ( C + DX ) " -> Y1
Instruction#8_____prgm LSCHEBYS
Instruction#9_____" " -> Y1 : String>Equ ( Str 1 , Y1 )
Instruction#10____[G] [D] -> [H]
Instruction#11____List>matr ( L2, DL3, L1, [ I ] )
Instruction#12____augment ( [ I ] , [ H ] ) -> [ J ]

Note: Use the MATRIX menu to enter the matrix name [G] onto a program line. Do not type the characters "[" , "G" , "]" 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 LSCHEBAB download in binhex format.

PC Windows users: Click here for LSCHEBAB download in zip format.


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Program #77 NEWTON (Newton Iteration with Test)


(Description to appear.)





Instruction#1_____"X - Y1 / nDeriv( Y1, X, X, .00001)" -> Y3
Instruction#2_____X -> A............................(Note: We use the symbol "->" to
Instruction#3_____Y3 -> B.............................represent the operation "Store".)
Instruction#4_____Y3(B) -> C
Instruction#5_____If B-A != 0..............................(See note 1 below.)
Instruction#6_____( C - B ) / ( B - A ) -> D
Instruction#7_____B -> X
Instruction#8_____[ [ A ] [ B ] [ C ] [ D ] ].............(See note 2 below.)

Note 1: In instruction#5, 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 2: In instruction#8, do not use the Matrix menu to type any part of the instruction. Instead, type the characters one at a time: First type " [ " , then " [ " , then " A " , then " ] " , then " [ " , etc... . This can be done on the TI83 keyboard itself.

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 NEWTON download in binhex format.

PC Windows users: Click here for NEWTON download in zip format.


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Program #78 MODNEWT (Modified Newton Method)


(Description to appear.)





Instruction#1_____Y1 -> A.....................(See note below.)
Instruction#2_____nDeriv(Y1, X, X, .00001) -> B
Instruction#3_____nDeriv( nDeriv(Y1, X, X, .00001) , X, X, .0001 ) -> C
Instruction#4_____X - A B / ( B2 - A C ) -> X

Note: We use the symbol "->" to represent the operation "Store".


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Program #79 UNSCLPWR (Unscaled Power Method)

Let A be an n by n real matrix. We assume there is a unique dominant eigenvalue k1, so that abs(k1) > abs(k2) > ... > abs(kn) . In the unscaled power method, we start with a vector x(0) , which we assume to be an approximation to the eigenvector v(1) corresponding to the eigenvalue k1. (In practice, x(0) is usually chosen arbitrarily. However, if an approximation to the eigenvector v(1) is available, this approximation should be used for x(0).)

We form the product Ax(0) and define this vector to be x(1). If x(0) approximates the eigenvector v(1), then x(1) approximates the vector k1v(1). Let xi(0) , i=1, ... , n be the components of x(0) . Then the set of numbers abs( xi(0) ) has a maximum value. Let p be the smallest index such that abs( xp(0) ) is this maximum value. Then the ratio xp(1) / xp(0) is approximately the dominant eigenvalue k1 . At this point, we replace the vector x(0) with x(1), and the process is repeated.

Before executing the program, we store the matrix A into the matrix variable [A]. The vector x(0) is stored in [B]. The dimension of A is determined in instructions#1,2. Instructions#3,4 are used to set up an ( n by 1 ) matrix [D] which will be used to store all the ratios xi(1) / xi(0) , i = 1, ... , n . The values of the components of x(1) are stored in [C] (see instruction#5). Instructions#6,7,8,9 are used to compute the ratios xi(1) / xi(0) , i = 1, ... , n , and store the results in the column matrix [D]. If any zero denominator is encountered in these computations, the corresponding entry of [D] is left with a zero entry .

An output matrix [E] is set up in instruction#10. In this matrix, the first column is the vector x(0), the second column is the vector x(1), and the third column contains the ratios xi(1) / xi(0) , i = 1, ... , n . In particular, the ratio xp(1) / xp(0) is in the third column. It is easily identified, and it is used as the approximation to the eigenvalue k1. The output matrix [E] is displayed in instruction#12. Instruction#11 is used to replace x(0) with x(1) in preparation for rerunning the program to continue the iterations.

After executing the program, keep pressing [ENTER]. The iterates for x(0), x(1), and approximations to k1 appear on the home screen. Convergence to k1 is, in general, linear.

Instruction#1_____dim([A]) -> L1..........(See note at end of program.)
Instruction#2_____L1(1) -> N..................(Note: We use the symbol "->" to
Instruction#3_____{ N , 1 } -> dim([D]).........represent the operation "Store".)
Instruction#4_____Fill( 0 , [D] )
Instruction#5_____[A][B] -> [C]
Instruction#6_____For ( I , 1 , N )
Instruction#7_____If [B] != 0..................(See note at end of program.)
Instruction#8_____[C]( I , 1 ) / [B]( I , 1 ) -> [D]( I , 1 )
Instruction#9_____End
Instruction#10____augment ( augment ( [B] , [C] ) , [D] ) -> [E]
Instruction#11____[C] -> [B]
Instruction#12____[E]

Note: Use the MATRIX menu to enter the matrix names [A] , [B] , ... onto a program line. Do not type the characters "[" , "A" , "]" , "B" , ... separately.

Note that in instruction#7, 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.

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 UNSCLPWR download in binhex format.

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Program #80 BEZIER (Bezier Curve, 2 Points)


(Description to appear.)


Inputs:

The initial point for the curve is ( x1 , y1 ) . The endpoint point for the curve is ( x4 , y4 )
The "control" point for the initial point is ( x2 , y2 ).
The "control" point for the terminal point is ( x3 , y3 ).

Place { x1 , x2 , x3 , x4 } into the L1 List variable.
Place { y1 , y2 , y3 , y4 } into the L2 List variable.

Outputs:

The parametric equations for the Bezier curve are placed into the variables
X1T and Y1T . These variables are "turned on" .

The window variables are set up to guarantee that a plot of the Bezier curve will appear in the graph window after the GRAPH button is pressed. After the plot, you may want to use a zoom box around the graph to see more detail. The plot will contain 21 points.

The calculator is set to parametric mode by the program.



Instruction#1_____L1(1) -> A.....................(See note below.)
Instruction#2_____3 ( L1(2) - L1(1) ) ) -> B
Instruction#3_____3 ( L1(3) - 2 L1(2) + L1(1) ) -> C
Instruction#4_____L1(4) - 3 ( L1(3) - L1(2) ) - L1(1) -> D
Instruction#5_____" ( ( D T + C ) T + B ) T + A " -> X1T
Instruction#6_____L2(1) -> E
Instruction#7_____3 ( L2(2) - L2(1) ) ) -> F
Instruction#8_____3 ( L2(3) - 2 L2(2) + L2(1) ) -> G
Instruction#9_____L2(4) - 3 ( L2(3) - L2(2) ) - L2(1) -> H
Instruction#10____" ( ( H T + G ) T + F ) T + E " -> Y1T
Instruction#11____2 min ( L1 ) - max ( L1 ) -> Xmin
Instruction#12____2 max ( L1 ) - min ( L1 ) -> Xmax
Instruction#13____2 min ( L2 ) - max ( L2 ) -> Ymin
Instruction#14____2 max ( L2 ) - min ( L2 ) -> Ymax
Instruction#15____Param
Instruction#16____0 -> Tmin
Instruction#17____1 -> Tmax
Instruction#18____.05 -> Tstep
Instruction#19____Plot1 ( Scatter , L1 , L2 , + )...........(See note below.)


Note: We use the symbol "->" to represent the operation "Store".

Note: For instruction#19, use the [2nd] , [STAT PLOT ] menu repeatedly.

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 BEZIER download in binhex format.

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Program #81 LSTRCN2 (LS Trig Curve, Continuous Data, Case N=2)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Outputs:

The function S2(x) = ao / 2 + a1 cos(x) + a2 cos(2x) + b1 sin(x) is stored in the Y2 function variable.

The 4 by 2 matrix [A] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 , b1 } .
Column 2 contains the four values { E , 2 , 0 , 0 } .
The value of E is the "error" E = Integ [ ( f(x) - S2(x) )2 ] on the interval [ - pi , pi ] .



Instruction#1_____{ 4 , 2 } -> dim ( [ A ] ).....................(See notes below.)
Instruction#2_____Fill ( 0 , [ A ] )
Instruction#3_____( 1 / pi ) fnInt ( Y1 , X , - pi , pi ) -> [ A ]( 1 , 1 )
Instruction#4_____( 1 / pi ) fnInt ( Y1 cos(X) , X , - pi , pi ) -> [ A ]( 2 , 1 )
Instruction#5_____( 1 / pi ) fnInt ( Y1 cos(2X) , X , - pi , pi ) -> [ A ]( 3 , 1 )
Instruction#6_____( 1 / pi ) fnInt ( Y1 sin(X) , X , - pi , pi ) -> [ A ]( 4 , 1 )
Instruction#7_____.5 [A](1,1) -> [A](1,1)
Instruction#8_____fnInt ( Y1 ^ 2 , X , - pi, pi ) - (pi) ( 2 [A](1,1) ^ 2 + [A](2,1) ^ 2 +
______________[A](3,1) ^ 2 + [A](4,1) ^ 2 ) -> [ A ]( 1 , 2 )
Instruction#9_____" [A](1,1) + [A](2,1) cos(X) + [A](3,1) cos(2X) +
______________[A](4,1) sin(X) " -> Y2
Instruction#10____[A]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.

Note that in instructions#3,4,5,6,8, we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRCN2 download in binhex format.

PC Windows users: Click here for LSTRCN2 download in zip format.


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Program #82 LSTRCN3 (LS Trig Curve, Continuous Data, Case N=3)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Outputs:

The function S3(x) = ao / 2 + a1 cos(x) + a2 cos(2x) + a3 cos(3x) + b1 sin(x) + b2 sin(2x) is stored in the Y2 function variable.

The 6 by 2 matrix [A] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 , a3 , b1 , b2 } .
Column 2 contains the four values { E , 3 , 0 , 0 , 0 , 0 } .
The value of E is the "error" E = Integ [ ( f(x) - S3(x) )2 ] on the interval [ - pi , pi ] .



Instruction#1_____{ 4 , 2 } -> dim ( [ A ] ).....................(See notes below.)
Instruction#2_____Fill ( 0 , [ A ] )
Instruction#3_____( 1 / pi ) fnInt ( Y1 , X , - pi , pi ) -> [ A ]( 1 , 1 )
Instruction#4_____( 1 / pi ) fnInt ( Y1 cos(X) , X , - pi , pi ) -> [ A ]( 2 , 1 )
Instruction#5_____( 1 / pi ) fnInt ( Y1 cos(2X) , X , - pi , pi ) -> [ A ]( 3 , 1 )
Instruction#6_____( 1 / pi ) fnInt ( Y1 cos(3X) , X , - pi , pi ) -> [ A ]( 4 , 1 )
Instruction#7_____( 1 / pi ) fnInt ( Y1 sin(X) , X , - pi , pi ) -> [ A ]( 5 , 1 )
Instruction#8_____( 1 / pi ) fnInt ( Y1 sin(2X) , X , - pi , pi ) -> [ A ]( 6 , 1 )
Instruction#9_____.5 [A](1,1) -> [A](1,1)
Instruction#10____fnInt ( Y1 ^ 2 , X , - pi, pi ) - (pi) ( 2 [A](1,1) ^ 2 + [A](2,1) ^ 2 +
______________[A](3,1) ^ 2 + [A](4,1) ^ 2 + [A](5,1) ^ 2 +
______________[A](6,1) ^ 2 ) -> [ A ]( 1 , 2 )
Instruction#11____" [A](1,1) + [A](2,1) cos(X) + [A](3,1) cos(2X) +
______________[A](4,1) cos(3X) + [A](5,1) sin(X) + [A](6,1) sin(2X) " -> Y2
Instruction#12____[A]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix name [A] onto a program line. Do not type the characters "[" , "A" , "]" separately.

Note that in instructions#3,4,5,6,7,8,10 we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRCN3 download in binhex format.

PC Windows users: Click here for LSTRCN3 download in zip format.


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Program #83 LSTRDN2 (LS Trig Curve, Discrete Data, Case N=2, M>2)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Place the value of m into the variable M . The value of m is defined by the condition that 2 m is the number of equal subintervals in the partition of the interval [ - pi , pi ] . Also, m must be greater than 2 .

Outputs:

The function S2(x) = ao / 2 + a1 cos(x) + a2 cos(2x) + b1 sin(x) is stored in the Y2 function variable.

The 4 by 2 matrix [E] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 , b1 } .
Column 2 contains the four values { E , 2 , m , 0 } .

The value of the real variable E is the "error" E = sum [ ( f(xi) - S2(xi) )2 ]
on the interval [ - pi , pi ] . Here, xi = - (pi) + i h , where h = pi / m ,
and where i = 0 , 1 , 2 , ... , 2 m - 1 .



Instruction#1_____pi / M -> H.....................(See notes below.)
Instruction#2_____seq ( - pi + I H , I , 0 , 2 M - 1 ) -> L1
Instruction#3_____dim(L1) -> dim(L2)
Instruction#4_____Fill( 1 , L2 )
Instruction#5_____List>matr ( L2 , cos(L1) , cos (2L1) , sin(L1) , [A] )
Instruction#6_____List>matr ( Y1(L1) , [B] )
Instruction#7_____( 1 / M ) [A]T [B] -> [ C ]
Instruction#8_____.5 [C](1,1) -> [C](1,1)
Instruction#9_____Matr>list ( [B] - [A] [C] , L3 )
Instruction#10____[ [ sum ( L3 ^ 2 ) ] [ 2 ] [ M ] [ 0 ] ] -> [ D ]
Instruction#11____" [C](1,1) + [C](2,1) cos(X) + [C](3,1) cos(2X) +
______________[C](4,1) sin(X) " -> Y2
Instruction#12____augment ( [C] , [D] ) -> [E]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix names [A] , [B] , ... onto a program line. Do not type the characters "[" , "A" , "]" , etc ... separately.

Note that in instructions#1 and 2 we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRDN2 download in binhex format.

PC Windows users: Click here for LSTRDN2 download in zip format.


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Program #84 LSTRDN3 (LS Trig Curve, Discrete Data, Case N=3, M>3)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Place the value of m into the variable M . The value of m is defined by the condition that 2 m is the number of equal subintervals in the partition of the interval [ - pi , pi ] . Also, m must be greater than 3 .

Outputs:

The function S3(x) = ao / 2 + a1 cos(x) + a2 cos(2x) + a3 cos(3x) + b1 sin(x) + b2 sin(2x) + is stored in the Y2 function variable.

The 6 by 2 matrix [E] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 , a3 , b1 , b2 } .
Column 2 contains the four values { E , 2 , m , 0 , 0 , 0 } .

The value of the real variable E is the "error" E = sum [ ( f(xi) - S3(xi) )2 ]
on the interval [ - pi , pi ] . Here, xi = - (pi) + i h , where h = pi / m ,
and where i = 0 , 1 , 2 , ... , 2 m - 1 .



Instruction#1_____pi / M -> H.....................(See notes below.)
Instruction#2_____seq ( - pi + I H , I , 0 , 2 M - 1 ) -> L1
Instruction#3_____dim(L1) -> dim(L2)
Instruction#4_____Fill( 1 , L2 )
Instruction#5_____List>matr ( L2 , cos(L1) , cos (2L1) , cos (3L1) ,
______________sin(L1) , sin(2L1) , [A] )
Instruction#6_____List>matr ( Y1(L1) , [B] )
Instruction#7_____( 1 / M ) [A]T [B] -> [ C ]
Instruction#8_____.5 [C](1,1) -> [C](1,1)
Instruction#9_____Matr>list ( [B] - [A] [C] , L3 )
Instruction#10____[ [ sum ( L3 ^ 2 ) ] [ 3 ] [ M ] [ 0 ] [ 0 ] [ 0 ] ] -> [ D ]
Instruction#11____" [C](1,1) + [C](2,1) cos(X) + [C](3,1) cos(2X) +
______________[C](4,1) cos(3X) + [C](5,1) sin(X) + [C](6,1) sin(2X) " -> Y2
Instruction#12____augment ( [C] , [D] ) -> [E]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix names [A] , [B] , ... onto a program line. Do not type the characters "[" , "A" , "]" , etc ... separately.

Note that in instructions#1 and 2 we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRDN3 download in binhex format.

PC Windows users: Click here for LSTRDN3 download in zip format.


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Program #85 LSTRIN2 (LS Trig Interpolation Curve, Case N=2)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Outputs:

The function S2(x) = ao / 2 + a1 cos(x) + ( a2 / 2 ) cos(2x) + b1 sin(x) is stored in the Y2 function variable.

The 4 by 1 matrix [C] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 / 2 , b1 } .



Instruction#1_____2 -> M.............................(See notes below.)
Instruction#2_____pi / M -> H
Instruction#3_____seq ( - pi + I H , I , 0 , 2 M - 1 ) -> L1
Instruction#4_____dim(L1) -> dim(L2)
Instruction#5_____Fill( 1 , L2 )
Instruction#6_____List>matr ( L2 , cos(L1) , cos (2L1) , sin(L1) , [A] )
Instruction#7_____List>matr ( Y1(L1) , [B] )
Instruction#8_____( 1 / M ) [A]T [B] -> [ C ]
Instruction#9_____.5 [C](1,1) -> [C](1,1)
Instruction#10____.5 [C](3,1) -> [C](3,1)
Instruction#11____" [C](1,1) + [C](2,1) cos(X) + [C](3,1) cos(2X) +
______________[C](4,1) sin(X) " -> Y2
Instruction#12____[C]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix names [A] , [B] , ... onto a program line. Do not type the characters "[" , "A" , "]" , etc ... separately.

Note that in instructions#2 and 3 we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRIN2 download in binhex format.

PC Windows users: Click here for LSTRIN2 download in zip format.


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Program #86 LSTRIN3 (LS Trig Interpolation Curve, Case N=3)


(Description to appear.)



Inputs:

Place the data function f(x) ( defined on the interval [ - pi , pi ] ) into the Y1 function
variable. The function f(x) may be piecewise continuous. ( "pi" is defined in the
usual way as 3.1415926535898 . )

Outputs:

The function S3(x) = ao / 2 + a1 cos(x) + ( a2 ) cos(2x) +
( a3 / 2 ) cos(3x) + b1 sin(x)+ b2 sin(2x) is stored in the Y2 function variable.

The 6 by 1 matrix [C] is displayed and contains the following:
Column 1 contains the four values { ao / 2 , a1 , a2 , a3 / 2 , b1 , b2 } .



Instruction#1_____3 -> M.............................(See notes below.)
Instruction#2_____pi / M -> H
Instruction#3_____seq ( - pi + I H , I , 0 , 2 M - 1 ) -> L1
Instruction#4_____dim(L1) -> dim(L2)
Instruction#5_____Fill( 1 , L2 )
Instruction#6_____List>matr ( L2 , cos(L1) , cos (2L1) , sin(L1) , [A] )
Instruction#7_____List>matr ( Y1(L1) , [B] )
Instruction#8_____( 1 / M ) [A]T [B] -> [ C ]
Instruction#9_____.5 [C](1,1) -> [C](1,1)
Instruction#10____.5 [C](4,1) -> [C](4,1)
Instruction#11____" [C](1,1) + [C](2,1) cos(X) + [C](3,1) cos(2X) +
______________[C](4,1) cos(3X) + [C](5,1) sin(X)+
______________[C](6,1) sin(2X) " -> Y2
Instruction#12____[C]


Note: We use the symbol "->" to represent the operation "Store".

Note: Use the MATRIX menu to enter the matrix names [A] , [B] , ... onto a program line. Do not type the characters "[" , "A" , "]" , etc ... separately.

Note that in instructions#2 and 3 we are using the symbol " pi " to represent the number 3.1415926535898 . There is an equivalent key on the right side of the TI83 keyboard for this number.



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 LSTRIN3 download in binhex format.

PC Windows users: Click here for LSTRIN3 download in zip format.


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Program #87 SYSITER3 (Iteration for 3 x 3 Systems)


This program solves the system X = f1( X , W , Z ) , W = f2( X , W , Z ) , Z = f3( X , W , Z ) by iteration. If ( X0 , W0 , Z0 ) is an approximate solution, we compute ( X1 , W1 , Z1 ) from the equations X1 = f1( X0 , W0 , Z0 ) , W1 = f2( X0 , W0 , Z0 ) , Z1 = f3( X0 , W0 , Z0 ) . The values for X1 , W1 , Z1 are used to update X0 , W0 , Z0 , and the process is repeated.

Before executing the program, the function f1( X , W , Z ) is stored in the function variable Y1 (as an expression in X , W and Z), the function f2( X , W , Z ) is stored in the function variable Y2 (as an expression in X , W and Z) , and the function f3( X , W , Z ) is stored in the function variable Y3 (as an expression in X , W and Z) .The value for X0 is stored in the variable X, value for W0 is stored in the variable W , and the value for Z0 is stored in the variable Z. Instruction#1 computes X1 and stores the value temporarily in the variable U. Instruction#2 computes W1 and stores the value temporarily in the variable V. Instruction#3 computes Z1 and stores the value in the variable Z (thus,Z is "updated"). Instruction#4 updates X with the value in U (which is X1), and Instruction#5 updates W with the value in V (which is W1) . Instruction#6 displays the updated values for ( X , W , Z ).

After executing the program, keep pressing ENTER. The iterates will appear on the home screen.




Instruction#1_____Y1 -> U.....................(See notes below.)
Instruction#2_____Y2 -> V
Instruction#3_____Y3 -> Z
Instruction#4_____U -> X
Instruction#5_____V -> W
Instruction#6_____Disp X , W , Z


Note: We use the symbol "->" to represent the operation "Store".



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Program #88 SYSGS2 (Gauss-Seidel for 2 x 2 Nonlinear Systems)


(Description to appear.)



Inputs:

The equations to solve are X = f1( X , Z ) and Z = f2( X , Z ) .

Place f1( X , Z ) into the function variable Y1 (as an expression in X and Z), and place f2( X , Z ) into the function variable Y2 (as an expression in X and Z).

Place the value for X0 into the variable X, and place the value for Z0 into the variable Z.

Outputs:

The values for X1 and Z1 are displayed on the homescreen, where X1 = f1( X0 , Z0 ) and Z1 = f2( X1 , Z0 ) .



Instruction#1_____Y1 -> X.....................(See notes below.)
Instruction#2_____Y2 -> Z
Instruction#3_____Disp X , Z



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Program #89 SYSGS3 (Gauss-Seidel for 3 x 3 Nonlinear Systems)


(Description to appear.)



Inputs:

The equations to solve are X = f1( X , W , Z ) , W = f2( X , W , Z ) , and Z = f3( X , W , Z ).

Place f1( X , W , Z ) into the function variable Y1 (as an expression in X , W , and Z), place f2( X , W , Z ) into the function variable Y2 (as an expression in X , W , and Z), and place f3( X , W , Z ) into the function variable Y3 (as an expression in X , W , and Z).

Place the value for X0 into the variable X, place the value for W0 into the variable W , and place the value for Z0 into the variable Z.

Outputs:

The values for X1 , W1 and Z1 are displayed on the homescreen, where X1 = f1( X0 , W0 , Z0 ) , W1 = f2( X1 , W0 , Z0 ) , and Z1 = f3( X1 , W1 , Z0 ) .



Instruction#1_____Y1 -> X.....................(See notes below.)
Instruction#2_____Y2 -> W
Instruction#3_____Y3 -> Z
Instruction#4_____Disp X , W , Z


Note: We use the symbol "->" to represent the operation "Store".



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