TI83 Calculator Programs for Numerical Analysis Problems - Part 4
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)
-------------------------------------------------------
Program #53 DEBASH3 (Adams Bashforth 3 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 3-step method requires three "starting values", wo , w1 , and w2 . The value w1 is an approximation to y ( xo + h ) , and the value w2 is an approximation to y ( xo + 2 h ) . The values for w1 and w2 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:
x2 -> X . ( Note that it is x2 = xo + 2 h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) , w2 -> L1(3) . Thus, the L1 list looks like { wo , w1 , w2 } 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 x3 = xo + 3 h .
Column 1 , row 2 contains the value of w3 = w2 + ( h / 12 ) ( 23 f ( x2 , w2 )
- 16 f ( x1 , w1 ) + 5 f ( xo , wo ) ) .
Column 1 , row 3 contains the value of y ( x3 ) .
Column 1 , row 4 contains the value of y ( x3 ) - w3 = exact error at x3.
Column 2 , row 1 contains the value of x2 .
Column 2 , row 2 contains the value of w2 .
Column 2 , row 3 contains the value f ( x2 , w2 ) .
Column 2 , row 4 contains the value of B ( x3 ) = error bound function at x3 .
Column 3 , row 1 contains the value of x1 .
Column 3 , row 2 contains the value of w1 .
Column 3 , row 3 contains the value f ( x1 , w1 ) .
Column 3 , row 4 contains the value f ( x3 , w3 ) .
Column 4 , row 1 contains the value of xo .
Column 4 , row 2 contains the value of wo .
Column 4 , row 3 contains the value f ( xo , wo ) .
Column 4 , row 4 contains the value of ( h / 12 ) ( 23 f ( x2 , w2 )
- 1 6 f ( x1 , w1 ) + 5 f ( xo , wo ) ) .
In addition, the value of X is updated to contain the value of x3 . The values in the list L1 are unchanged, but are not used when rerunning the program for a continuation of the given problem. The values for w1 , w2 , w3 needed for a continuation rerun are found by the program in the output matrix [A] . Also, the value of H is stored into the variable K on the first run of the program. The number in H is then set to zero, but the program knows to find the correct value of h in the variable K during a continuation rerun of the program 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 x4 , w4 , etc ... . Keep pressing [ENTER] to continue the computation for the given problem.
(The purpose of the procedure described in the preceding paragraph is to save on computation time during the continuation reruns of the program.)
Special Note : The value in the variable H must be reset to its original value under any of the following circumstances:
a) If the value of X is reset to its original value x2 in the middle of a computation for a given problem.
b) If the program is to be used for the numerical solution of a new problem.
c) If a different program (using a different numerical method) is to be used for the same problem or a different problem.
Click here for a DEBASH3 graphic display of the output matrix.
Instruction#1_____If H != 0..........................(See note below.)
Instruction#2_____Then
Instruction#3_____H -> K............................(Note: We use the symbol "->" to
Instruction#4_____0 -> H.............................represent the operation "Store".)
Instruction#5_____{4,4} -> dim( [A] )
Instruction#6_____X - 2K -> X
Instruction#7_____For ( J , 2 , 4 )
Instruction#8_____X -> [A]( 1 , 6 - J )
Instruction#9_____L1(J-1) -> Y
Instruction#10____Y -> [A]( 3 , 6 - J )
Instruction#11____Y1 -> [A]( 3 , 6 - J )
Instruction#12____X + K -> X
Instruction#13____End
Instruction#14____Else
Instruction#15____X + K -> X
Instruction#16____[A](4,3) -> [A](3,1)
Instruction#17____{4,3} -> dim( [A] )
Instruction#18____augment( [ [0,0,0,0] ]T , [A] ) -> [A]
Instruction#19____End
Instruction#20____( K / 12 ) ( 23 [A](3,2) - 16 [A](3,3) + 5 [A](3,4) ) -> [A](4,4)
Instruction#21____[A](2,2) + [A](4,4) -> Y
Instruction#22____Y -> [A](2,1)
Instruction#23____X -> [A](1,1)
Instruction#24____Y2 -> [A](3,1)
Instruction#25____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#26____Y3 -> [A](4,2)
Instruction#27____Y1 -> [A](4,3)
Instruction#28____[A]
Note that in instruction#1, 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 DEBASH3 download in binhex format.
PC Windows users: Click here for DEBASH3 download in zip format.
Program DEBASH3S. This is a short version for DEBASH3. 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. (Description to appear.)
Instruction#1_____1 -> dim ( L2 ).........................(Note: We use the symbol "->" to
Instruction#2_____X - 2 H -> X.............................represent the operation "Store".)
Instruction#3_____For ( J , 1 , 3 )
Instruction#4_____L1 ( J ) -> Y
Instruction#5_____Y1 -> L2 ( J )
Instruction#6_____X + H -> X
Instruction#7_____End
Instruction#8_____L1(3) + ( H / 12 ) ( 23 L2(3) - 16 L2(2) + 5 L2(1) ) -> Y
Instruction#9_____L1(2) -> L1(1)
Instruction#10____L1(3) -> L1(2)
Instruction#11____Y -> L1(3)
Instruction#12____[ [ X , Y , Y2 , Y2 - Y , Y3 ] ]T
Back to Menu
-------------------------------------------------------
Program #54 DEMOULT3 (Adams Moulton 3 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 3-step method requires three "starting values", wo , w1 , and w2 . The value w1 is an approximation to y ( xo + h ) , and the value w2 is an approximation to y ( xo + 2 h ) . The values for w1 and w2 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:
x2 -> X . ( Note that it is x2 = xo + 2 h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) , w2 -> L1(3) . Thus, the L1 list looks like { wo , w1 , w2 } 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 x3 = xo + 3 h .
Column 1 , row 2 contains the value of w3 . w3 solves the equation:
w3 = w2 + ( h / 24 ) ( 9 f( x3 , w3 ) + 19 f ( x2 , w2 ) - 5 f ( x1 , w1 ) + f ( xo , wo ) ) .
Column 1 , row 3 contains the value of y ( x3 ) .
Column 1 , row 4 contains the value of y ( x3 ) - w3 = exact error at x3.
Column 2 , row 1 contains the value of x2 .
Column 2 , row 2 contains the value of w2 .
Column 2 , row 3 contains the value f ( x2 , w2 ) .
Column 2 , row 4 contains the value of B ( x3 ) = error bound function at x3 .
Column 3 , row 1 contains the value of x1 .
Column 3 , row 2 contains the value of w1 .
Column 3 , row 3 contains the value f ( x1 , w1 ) .
Column 3 , row 4 contains the value f ( x3 , w3 ) .
Column 4 , row 1 contains the value of xo .
Column 4 , row 2 contains the value of wo .
Column 4 , row 3 contains the value f ( xo , wo ) .
Column 4 , row 4 contains the value of w2 + ( h / 24 ) ( 19 f ( x2 , w2 )
- 5 f ( x1 , w1 ) + f ( xo , wo ) ) .
Also, list L2 contains Steffensen iterates in the solution (by iteration)
for w3 in the equation:
w3 = w2 + ( h / 24 ) ( 9 f( x3 , w3 ) + 19 f ( x2 , w2 ) - 5 f ( x1 , w1 ) + f ( xo , wo ) ) .
In addition, the value of X is updated to contain the value of x3 . The values in the list L1 are unchanged, but are not used when rerunning the program for a continuation of the given problem. The values for w1 , w2 , w3 needed for a continuation rerun are found by the program in the output matrix [A] . Also, the value of H is stored into the variable K on the first run of the program. The number in H is then set to zero, but the program knows to find the correct value of h in the variable K during a continuation rerun of the program 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 x4 , w4 , etc ... . Keep pressing [ENTER] to continue the computation for the given problem.
(The purpose of the procedure described in the preceding paragraph is to save on computation time during the continuation reruns of the program.)
Special Note : The value in the variable H must be reset to its original value under any of the following circumstances:
a) If the value of X is reset to its original value x2 in the middle of a computation for a given problem.
b) If the program is to be used for the numerical solution of a new problem.
c) If a different program (using a different numerical method) is to be used for the same problem or a different problem.
Click here for a DEMOULT3 graphic display of the output matrix.
Instruction#1_____If H != 0..........................(See note below.)
Instruction#2_____Then
Instruction#3_____H -> K
Instruction#4_____0 -> H
Instruction#5_____1 -> dim( L2 )
Instruction#6_____{4,4} -> dim( [A] )..................(Note: We use the symbol "->" to
Instruction#7_____X-2K -> X.............................represent the operation "Store".)
Instruction#8_____For ( J , 2 , 4 )
Instruction#9_____X -> [A]( 1 , 6-J )
Instruction#10____L1(J-1) -> Y
Instruction#11____Y -> [A]( 2 , 6-J )
Instruction#12____Y1 -> [A]( 3 , 6-J )
Instruction#13____X + K -> X
Instruction#14____End
Instruction#15____Else
Instruction#16____X + K -> X
Instruction#17____[A](4,3) -> [A[(3,1)
Instruction#18____{4,3} -> dim( [A] )
Instruction#19____augment( [ [ 0 , 0 , 0 , 0 ] ]T , [A] ) -> [A]
Instruction#20____End
Instruction#21____Y + ( K / 24 ) ( 19 [A](3,2) - 5 [A](3,3) + [A](3,4) ) -> [A](4,4)
Instruction#22____Y + ( K / 12 ) ( 23 [A](3,2) - 16 [A](3,3) + 5 [A](3,4) ) -> A
Instruction#23____A -> L2(1)
Instruction#24____For( I , 1 , 4 )
Instruction#25____A -> Y
Instruction#26____( K / 24 ) ( 9 Y1 ) + [A](4,4) -> B
Instruction#27____B -> Y
Instruction#28____( K / 24 ) ( 9 Y1 ) + [A](4,4) -> C
Instruction#29____If C - 2 B + A != 0..........................(See note below.)
Instruction#30____A - ( ( B - A )2 ) / ( C - 2 B + A ) -> A
Instruction#31____A -> L2( I + 1 )
Instruction#32____End
Instruction#33____A -> [A](2,1)
Instruction#34____A -> Y
Instruction#35____Y1 -> [A](4,3)
Instruction#36____X -> [A](1,1)
Instruction#37____Y2 -> [A](3,1)
Instruction#38____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#39____Y3 -> [A](4,2)
Instruction#40____[A]
Note that in instruction#1, 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 DEMOULT3 download in binhex format.
PC Windows users: Click here for DEMOULT3 download in zip format.
Program DEMOLT3S. This is a short version for DEMOULT3. 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. (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 - 2 H -> X
Instruction#4_____For ( J , 1 , 3 )
Instruction#5_____L1 ( J ) -> Y
Instruction#6_____Y1 -> L3 ( J )
Instruction#7_____X + H -> X
Instruction#8_____End
Instruction#9_____Y + ( H / 24 ) ( 19 L3(3) - 5 L3(2) + L3(1) ) -> T
Instruction#10____Y + ( H / 12 ) ( 23 L3(3) - 16 L3(2) + 5 L3(1) ) -> Y
Instruction#11____Y -> L2 (1)
Instruction#12____For ( I , 1 , 4 )
Instruction#13____Y -> A
Instruction#14____( H / 24 ) ( 9 Y1 ) + T -> B
Instruction#15____B -> Y
Instruction#16____( H / 24 ) ( 9 Y1 ) + T -> C
Instruction#17____If C - 2 B + A != 0...............................(See note below.)
Instruction#18____A - ( B - A )2 / ( C - 2 B + A ) -> Y
Instruction#19____Y -> L2 ( I + 1 )
Instruction#20____End
Instruction#21____L1(2) -> L1(1)
Instruction#22____L1(3) -> L1(2)
Instruction#23____Y -> L1(3)
Instruction#24____[ [ X , Y , Y2 , Y2 - Y , Y3 ] ]T
Note that in instruction#17, 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.
Back to Menu
-------------------------------------------------------
Program #55 DEPCB4M3 (Bashforth 4 Step Predictor, Moulton 3 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 four "starting values", wo , w1 , w2 , and w3 . The value w1 is an approximation to y ( xo + h ) , the value w2 is an approximation to y ( xo + 2 h ) , and the value of w3 is an approximation to y ( xo + 3 h ) . The values for w1 , w2 and w3 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:
x3 -> X . ( Note that it is x3 = xo + 3 h , and not xo , which is initially stored into X . )
wo -> L1( 1 ) , w1 -> L1( 2 ) , w2 -> L1(3) , w3 -> L1(4) . Thus, the L1 list looks like { wo , w1 , w2 , w3 } 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 5 and contains the following:
Column 1 , row 1 contains the value of x4 = xo + 4 h .
Column 1 , row 2 contains the value of w4 . w4 solves the equation:
w4 = w3 + ( h / 24 ) ( 9 f( x4 , w4(0) ) + 19 f ( x3 , w3 ) - 5 f ( x2 , w2 ) + f ( x1 , w1 ) ) ,
where w4(0) = w3 + ( h / 24 ) ( 55 f ( x3 , w3 )- 59 f ( x2 , w2 )
+ 37 f ( x1 , w1 ) - 9 f ( xo , wo ) )
Column 1 , row 3 contains the value of y ( x4 ) .
Column 1 , row 4 contains the value of y ( x4 ) - w4 = exact error at x4.
Column 2 , row 1 contains the value of x3 .
Column 2 , row 2 contains the value of w3 .
Column 2 , row 3 contains the value f ( x3 , w3 ) .
Column 2 , row 4 contains the value of B ( x4 ) = error bound function at x4 .
Column 3 , row 1 contains the value of x2 .
Column 3 , row 2 contains the value of w2 .
Column 3 , row 3 contains the value f ( x2 , w2 ) .
Column 3 , row 4 contains the value f ( x4 , w4 ) .
Column 4 , row 1 contains the value of x1 .
Column 4 , row 2 contains the value of w1 .
Column 4 , row 3 contains the value f ( x1 , w1 ) .
Column 4 , row 4 contains the value of w4(0) .
Column 5 , row 1 contains the value of xo .
Column 5 , row 2 contains the value of wo .
Column 5 , row 3 contains the value f ( xo , wo ) .
Column 5 , row 4 contains the value f ( x4 , w4(0) ) .
In addition, the value of X is updated to contain the value of x4 . The values in the list L1 are unchanged, but are not used when rerunning the program for a continuation of the given problem. The values for w1 , w2 , w3 , w4 needed for a continuation rerun are found by the program in the output matrix [A] . Also, the value of H is stored into the variable K on the first run of the program. The number in H is then set to zero, but the program knows to find the correct value of h in the variable K during a continuation rerun of the program 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 x5 , w5 , etc ... . Keep pressing [ENTER] to continue the computation for the given problem.
(The purpose of the procedure described in the preceding paragraph is to save on computation time during the continuation reruns of the program.)
Special Note : The value in the variable H must be reset to its original value under any of the following circumstances:
a) If the value of X is reset to its original value x3 in the middle of a computation for a given problem.
b) If the program is to be used for the numerical solution of a new problem.
c) If a different program (using a different numerical method) is to be used for the same problem or a different problem.
Click here for a DEPCB4M3 graphic display of the output matrix.
Instruction#1_____If H != 0..........................(See note below.)
Instruction#2_____Then
Instruction#3_____H -> K............................(Note: We use the symbol "->" to
Instruction#4_____0 -> H.............................represent the operation "Store".)
Instruction#5_____{4,5} -> dim( [A] )
Instruction#6_____X - 3 K -> X
Instruction#7_____For ( J , 2 , 5 )
Instruction#8_____X -> [A]( 1 , 7 - J )
Instruction#9_____L1(J-1) -> Y
Instruction#10____Y -> [A]( 2 , 7 - J )
Instruction#11____Y1 -> [A]( 3 , 7 - J )
Instruction#12____X + K -> X
Instruction#13____End
Instruction#14____Else
Instruction#15____X + K -> X
Instruction#16____[A](4,3) -> [A](3,1)
Instruction#17____{4,4} -> dim( [A] )
Instruction#18____augment( [ [ 0 , 0 , 0 , 0 ] ]T , [A] ) -> [A]
Instruction#19____End
Instruction#20____Y + ( K / 24 ) ( 55 [A](3,2) - 59 [A](3,3) +
___________________37 [A](3,4) - 9 [A](3,5) ) -> Y
Instruction#21____Y -> [A](4,4)
Instruction#22____Y1 -> [A](4,5)
Instruction#23____[A](2,2) + ( K / 24 ) ( 9 [A](4,5) +
___________________19 [A](3,2) - 5 [A](3,3) + [A](3,4) ) -> Y
Instruction#24____Y -> [A](2,1)
Instruction#25____X -> [A](1,1)
Instruction#26____Y1 -> [A](4,3)
Instruction#27____Y2 -> [A](3,1)
Instruction#28____[A](3,1) - [A](2,1) -> [A](4,1)
Instruction#29____Y3 -> [A](4,2)
Instruction#30____[A]
Note that in instruction#1, 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 DEPCB4M3 download in binhex format.
PC Windows users: Click here for DEPCB4M3 download in zip format.
Back to Menu
-------------------------------------------------------
Program #56 DERKV6 (DE Runge Kutta Verner 6th 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 + 3 k1 / 40 + 875 k3 / 2244 +
23 k4 / 72 + 264 k5 / 1955) + 125 k7 / 11592 + 43 k8 / 616 .
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 / 6 , wo + k1 / 6 ) .
Column 3 , row 3 contains the value of k3 =
h f ( xo + 4 h / 15 , wo + 4 k1 / 75 + 16 k2 / 75 ).
Column 3 , row 4 contains the value of k4 =
h f ( xo + 2 h / 3 , wo + 5 k1 / 6 - 8 k2 / 3 + 5 k3 / 2 ) .
Column 4 , row 1 contains the value of k5 =
h f ( xo + 5 h / 6 , wo - 165 k1 / 64 - 55 k2 / 6 + 425 k3 / 64 + 85 k4 / 96 ) .
Column 4 , row 2 contains the value of k6 = h f ( xo + h , wo + 12 k1 / 5
- 8 k2 + 4015 k3 / 612 - 11 k4 / 36 + 88 k5 / 255 ) .
Column 4 , row 3 contains the value of k7 = h f ( xo + h / 15 , wo - 8263 k1 / 15000
+ 124 k2 / 75 - 643 k3 / 680 - 81 k4 / 250 + 2484 k5 / 10625 ) .
Column 4 , row 4 contains the value of k8 = h f ( xo + h , wo + 3501 k1 / 1720
- 300 k2 / 43 + 297275 k3 / 52632 - 319 k4 / 2322
+ 24068 k5 / 84065 + 3850 k7 / 26703 ) .
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 DERKV6 graphic display of the output matrix.
Instruction#1_____X -> V............................(Note: We use the symbol "->" to
Instruction#2_____Y -> W.............................represent the operation "Store".)
Instruction#3_____HY1 -> A
Instruction#4_____V + ( 1 / 6 ) H -> X
Instruction#5_____W + ( 1 / 6 ) A -> Y
Instruction#6_____HY1 -> B
Instruction#7_____V + ( 4 / 15 ) H -> X
Instruction#8_____W + ( 4 / 75 ) A + ( 16 / 75 ) B -> Y
Instruction#9_____HY1 -> C
Instruction#10____V + ( 2 / 3 ) H -> X
Instruction#11____W + ( 5 / 6 ) A - ( 8 / 3 ) B + ( 5 / 2 ) C -> Y
Instruction#12____HY1 -> D
Instruction#13____V + ( 5 / 6 ) H -> X
Instruction#14____W - ( 165 / 64 ) A + ( 55 / 6 ) B - ( 425 / 64 ) C +
__________________( 85 / 96 ) D -> Y
Instruction#15____HY1 -> E
Instruction#16____V + H -> X
Instruction#17____W + ( 12 / 5 ) A - ( 8 ) B + ( 4015 / 612 ) C - ( 11 / 36 ) D +
__________________( 88 / 255 ) E -> Y
Instruction#18____HY1 -> F
Instruction#19____V + ( 1 / 15 ) H -> X
Instruction#20____W - ( 8263 / 15000 ) A + ( 124 / 75 ) B - ( 643 / 680 ) C -
__________________( 81 / 250 ) D + ( 2484 / 10625 ) E -> Y
Instruction#21____HY1 -> G
Instruction#22____V + H -> X
Instruction#23____W + ( 3501 / 1720 ) A - ( 300 / 43 ) B + ( 297275 / 52632 ) C -
__________________( 319 / 2322 ) D + ( 24068 / 84065 ) E +
__________________( 3850 / 26703 ) G -> Y
Instruction#24____HY1 -> L
Instruction#25____W + ( 3 / 40 ) A + ( 875 / 2244 ) C + ( 23 / 72 ) D +
__________________( 264 / 1955 ) E + ( 125 / 11592 ) G + ( 43 / 616 ) L -> Y
Instruction#26____{ X , Y , Y2 , Y2 - Y } -> L1
Instruction#27____{ V , W , H , Y3 } -> L2
Instruction#28____{ A , B , C , D } -> L3
Instruction#29____{ E , F , G , L } -> L4
Instruction#30____List>matr ( L1, L2, L3, L4, [A] )
Instruction#31____[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 DERKV6 download in binhex format.
PC Windows users: Click here for DERKV6 download in zip format.
Back to Menu
-------------------------------------------------------
Program #57 LSPOLY1D (Least Squares Linear Polynomial (Discrete Data))
(Description to appear.)
Inputs:
Place { x1 , x2 , ... , xn } into the L1 List variable.
Place { y1 , y2 , ... , yn } into the L2 List variable.
Outputs:
The output matrix [A] is 2 by 5 and contains the following:
Column 1 contains ao , a1 , the coefficients for the least squares linear polynomial
P(x) = ao + a1 x .
Column 2 , row 1 contains the value of the error = sum[ ( yi - P( xi ) )2 ].
Column 2 , row 2 contains the value 0 .
Columns 3 and 4 contain the matrix of coefficients for the left hand side of the normal equations.
Column 5 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY1D graphic display of the output matrix.
Instruction#1_____dim(L1) -> dim(L3)...........(Note: We use the symbol "->" to
Instruction#2_____Fill ( 1 , L3 )........................represent the operation "Store".)
Instruction#3_____List>matr ( L3, L1, [A] )
Instruction#4_____List>matr ( L2, [B] )
Instruction#5_____[A]T[A] -> [C]
Instruction#6_____[A]T[B] -> [D]
Instruction#7_____[C]-1[D] -> [E]
Instruction#8_____Matr>list ( [B] - [A][E] , L4 )
Instruction#9_____[ [ sum (L4^2 ) ] [0] ] -> [F]
Instruction#10____augment ( augment ( augment ( [E] , [F] ) , [C] ) , [D] ) -> [G]
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 LSPOLY1D download in binhex format.
PC Windows users: Click here for LSPOLY1D download in zip format.
Back to Menu
-------------------------------------------------------
Program #58 LSPOLY2D (Least Squares Quadratic Polynomial (Discrete Data))
(Description to appear.)
Inputs:
Place { x1 , x2 , ... , xn } into the L1 List variable.
Place { y1 , y2 , ... , yn } into the L2 List variable.
Outputs:
The output matrix [A] is 3 by 6 and contains the following:
Column 1 contains ao , a1 , a2, the coefficients for the least squares quadratic
polynomial P(x) = ao + a1 x + a2 x2 .
Column 2 , row 1 contains the value of the error = sum[ ( yi - P( xi ) )2 ].
Column 2 , rows 2 and 3 contain the value 0 .
Columns 3 ,4 , and 5 contain the matrix of coefficients for the left hand side of the normal equations.
Column 6 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY2D graphic display of the output matrix.
Instruction#1_____dim(L1) -> dim(L3)...........(Note: We use the symbol "->" to
Instruction#2_____Fill ( 1 , L3 )........................represent the operation "Store".)
Instruction#3_____List>matr ( L3, L1, L1^2, [A] )
Instruction#4_____List>matr ( L2, [B] )
Instruction#5_____[A]T[A] -> [C]
Instruction#6_____[A]T[B] -> [D]
Instruction#7_____[C]-1[D] -> [E]
Instruction#8_____Matr>list ( [B] - [A][E] , L4 )
Instruction#9_____[ [ sum (L4^2 ) ] [0] [0] ] -> [F]
Instruction#10____augment ( augment ( augment ( [E] , [F] ) , [C] ) , [D] ) -> [G]
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 LSPOLY2D download in binhex format.
PC Windows users: Click here for LSPOLY2D download in zip format.
Back to Menu
-------------------------------------------------------
Program #59 LSPOLY3D (Least Squares Cubic Polynomial (Discrete Data))
(Description to appear.)
Inputs:
Place { x1 , x2 , ... , xn } into the L1 List variable.
Place { y1 , y2 , ... , yn } into the L2 List variable.
Outputs:
The output matrix [A] is 4 by 7 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , the coefficients for the least squares quadratic
polynomial P(x) = ao + a1 x + a2 x2 + a3 x3 .
Column 2 , row 1 contains the value of the error = sum[ ( yi - P( xi ) )2 ].
Column 2 , rows 2 , 3 , and 4 contain the value 0 .
Columns 3 ,4 , 5 , and 6 contain the matrix of coefficients for the left hand side of the normal equations.
Column 7 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY3D graphic display of the output matrix.
(You may need iterative improvement to help reduce the roundoff errors in this method.)
Instruction#1_____dim(L1) -> dim(L3)...........(Note: We use the symbol "->" to
Instruction#2_____Fill ( 1 , L3 )........................represent the operation "Store".)
Instruction#3_____List>matr ( L3, L1, L1^2, L1^3, [A] )
Instruction#4_____List>matr ( L2, [B] )
Instruction#5_____[A]T[A] -> [C]
Instruction#6_____[A]T[B] -> [D]
Instruction#7_____[C]-1[D] -> [E]
Instruction#8_____Matr>list ( [B] - [A][E] , L4 )
Instruction#9_____[ [ sum (L4^2 ) ] [0] [0] [0] ] -> [F]
Instruction#10____augment ( augment ( augment ( [E] , [F] ) , [C] ) , [D] ) -> [G]
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 LSPOLY3D download in binhex format.
PC Windows users: Click here for LSPOLY3D download in zip format.
Back to Menu
-------------------------------------------------------
Program #60 LSPOLY4D (Least Squares Quartic Polynomial (Discrete Data))
(Description to appear.)
Inputs:
Place { x1 , x2 , ... , xn } into the L1 List variable.
Place { y1 , y2 , ... , yn } into the L2 List variable.
Outputs:
The output matrix [A] is 5 by 8 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , a4 , the coefficients for the least squares quadratic
polynomial P(x) = ao + a1 x + a2 x2 + a3 x3 + a4 x4 .
Column 2 , row 1 contains the value of the error = sum[ ( yi - P( xi ) )2 ].
Column 2 , rows 2 , 3 , 4 , and 5 contain the value 0 .
Columns 3 ,4 , 5 , 6 , and 7 contain the matrix of coefficients for the left hand side of the normal equations.
Column 8 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY4D graphic display of the output matrix.
(You may need iterative improvement to help reduce the roundoff errors in this method.)
Instruction#1_____dim(L1) -> dim(L3)...........(Note: We use the symbol "->" to
Instruction#2_____Fill ( 1 , L3 )........................represent the operation "Store".)
Instruction#3_____List>matr ( L3, L1, L1^2, L1^3, L1^4, [A] )
Instruction#4_____List>matr ( L2, [B] )
Instruction#5_____[A]T[A] -> [C]
Instruction#6_____[A]T[B] -> [D]
Instruction#7_____[C]-1[D] -> [E]
Instruction#8_____Matr>list ( [B] - [A][E] , L4 )
Instruction#9_____[ [ sum (L4^2 ) ] [0] [0] [0] [0] ] -> [F]
Instruction#10____augment ( augment ( augment ( [E] , [F] ) , [C] ) , [D] ) -> [G]
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 LSPOLY4D download in binhex format.
PC Windows users: Click here for LSPOLY4D download in zip format.
Back to Menu
-------------------------------------------------------
Program #61 LSEXPD (Least Squares Exponential Function (Discrete Data))
(Description to appear.)
Inputs:
Place { x1 , x2 , ... , xn } into the L1 List variable.
Place { y1 , y2 , ... , yn } into the L2 List variable.
Outputs:
The output matrix [A] is 2 by 5 and contains the following:
Column 1 contains "b" and "a" , the parameters for the least squares exponential
function b [ exp ( a x ) ] .
Column 2 , row 1 contains the value of the error = sum[ ( yi - b [exp ( a xi ) ] )2 ].
Column 2 , row 2 contains the value 0 .
Columns 3 and 4 contain the matrix of coefficients for the left hand side of the "normal equations".
Column 5 contains the vector for the right hand side of the "normal equations".
Click here for a LSEXPD graphic display of the output matrix.
Instruction#1_____L2 -> L6............................(Note: We use the symbol "->" to
Instruction#2_____ln( L2 ) -> L2......................represent the operation "Store".)
Instruction#3_____prgm LSPOLY1D
Instruction#4_____e^( [G](1,1) ) -> [G](1, 1)
Instruction#5_____L6 -> L2
Instruction#6_____L2 - [G](1,1) e^( [G](2, 1) L1 ) -> L5
Instruction#7_____sum ( L5^2 ) -> [G](1, 2)
Instruction#8_____[G]
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 LSEXPD download in binhex format.
PC Windows users: Click here for LSEXPD download in zip format.
Back to Menu
-------------------------------------------------------
Program #62 LSPOLY1C (Least Squares Linear Polynomial (Continuous Data))
(Description to appear.)
Inputs:
Place the value of "a" into the variable A .
Place the value of "b" into the variable B .
Place f ( x ) into the function variable Y1 .
Outputs:
The output matrix [A] is 2 by 5 and contains the following:
Column 1 contains ao , a1 , the coefficients for the least squares linear polynomial
P(x) = ao + a1 x .
Column 2 , row 1 contains the value of the error = integral from "a" to "b" of
[ ( f ( x ) - P( x ) )2 ].
Column 2 , row 2 contains the value 0 .
Columns 3 and 4 contain the matrix of coefficients for the left hand side of the normal equations.
Column 5 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY1C graphic display of the output matrix.
Instruction#1_____[ [ B - A , ( B2 - A2 ) / 2 ] [ ( B2 - A2 ) / 2 , ( B^3 - A^3 ) / 3 ] ] -> [A]
Instruction#2_____[ [ fnInt ( Y1 , X , A , B ) ] [ fnInt ( XY1 , X , A , B ) ] ] -> [B]
Instruction#3_____[A]-1[B] -> [C]............................(Note: We use the symbol "->" to
Instruction#4_____[C]T[B] -> [D].............................represent the operation "Store".)
Instruction#5_____[ [ fnInt ( Y1^2, X, A, B ) - [D](1,1) ] [0] ] -> [E]
Instruction#6_____augment ( augment ( augment ( [C] , [E] ) , [A] ) , [B] ) -> [F]
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 LSPOLY1C download in binhex format.
PC Windows users: Click here for LSPOLY1C download in zip format.
Back to Menu
-------------------------------------------------------
Program #63 LSPOLY2C (Least Squares Quadratic Polynomial (Continuous Data))
(Description to appear.)
Inputs:
Place the value of "a" into the variable A .
Place the value of "b" into the variable B .
Place f ( x ) into the function variable Y1 .
Outputs:
The output matrix [A] is 3 by 6 and contains the following:
Column 1 contains ao , a1 , a2, the coefficients for the least squares quadratic
polynomial P(x) = ao + a1 x + a2 x2 .
Column 2 , row 1 contains the value of the error = integral from "a" to "b" of
[ ( f ( x ) - P( x ) )2 ].
Column 2 , rows 2 and 3 contain the value 0 .
Columns 3 ,4 , and 5 contain the matrix of coefficients for the left hand side of the normal equations.
Column 6 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY2C graphic display of the output matrix.
Instruction#1_____[ [ A, A2 / 2 , A^3 / 3 ] [ A2 / 2 , A^3 / 3 , A^4 / 4 ]
_______________[ A^3 / 3 , A^4 / 4 , A^5 / 5 ] ] -> [A]
Instruction#2_____[ [ B, B2 / 2 , B^3 / 3 ] [ B2 / 2 , B^3 / 3 , B^4 / 4 ]
_______________[ B^3 / 3 , B^4 / 4 , B^5 / 5 ] ] - [A] -> [A]
Instruction#3_____[ [ fnInt ( Y1 , X , A , B ) ] [ fnInt ( XY1 , X , A , B ) ]
_______________[ fnInt ( X2Y1 , X , A , B ) ] ]-> [B]
Instruction#4_____[A]-1[B] -> [C]............................(Note: We use the symbol "->" to
Instruction#5_____[C]T[B] -> [D].............................represent the operation "Store".)
Instruction#6_____[ [ fnInt ( Y1^2, X, A, B ) - [D](1,1) ] [0] [0] ] -> [E]
Instruction#7_____augment ( augment ( augment ( [C] , [E] ) , [A] ) , [B] ) -> [F]
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 LSPOLY2C download in binhex format.
PC Windows users: Click here for LSPOLY2C download in zip format.
Back to Menu
-------------------------------------------------------
Program #64 LSPOLY3C (Least Squares Cubic Polynomial (Continuous Data))
(Description to appear.)
Inputs:
Place the value of "a" into the variable A .
Place the value of "b" into the variable B .
Place f ( x ) into the function variable Y1 .
Outputs:
The output matrix [A] is 4 by 7 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , the coefficients for the least squares cubic
polynomial P(x) = ao + a1 x + a2 x2 + a3 x3 .
Column 2 , row 1 contains the value of the error = integral from "a" to "b" of
[ ( f ( x ) - P( x ) )2 ].
Column 2 , rows 2 , 3 , and 4 contain the value 0 .
Columns 3 ,4 , 5 , and 6 contain the matrix of coefficients for the left hand side of the normal equations.
Column 7 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY3C graphic display of the output matrix.
(You may need iterative improvement to help reduce the roundoff errors in this method.)
Instruction#1_____[ [ A / 2, A2 / 2 , A^3 / 3 , A^4 / 4 ] [ 0 , A^3 / 6 , A^4 / 4 , A^5 / 5 ]
_______________[ 0 , 0 , A^5 / 10 , A^6 / 6 ] [ 0 , 0 , 0 , A^7 / 14 ] ] -> [A]
Instruction#2_____[ [ B / 2, B2 / 2 , B^3 / 3 , B^4 / 4 ] [ 0 , B^3 / 6 , B^4 / 4 , B^5 / 5 ]
_______________[ 0 , 0 , B^5 / 10 , B^6 / 6 ] [ 0 , 0 , 0 , B^7 / 14 ] ] - [A] -> [A]
Instruction#3_____[A]T + [A] -> [A]
Instruction#4_____[ [ fnInt ( Y1 , X , A , B ) ] [ fnInt ( XY1 , X , A , B ) ]
_______________[ fnInt ( X2Y1 , X , A , B ) ] [ fnInt ( ( X^3 ) Y1 , X , A , B ) ] ]-> [B]
Instruction#5_____Matr>list ( rref ( augment ( [A] , [B] ) ) , 5 , L1 )
Instruction#6_____List>matr ( L1 , [C] ]................(Note: We use the symbol "->" to
Instruction#7_____[C]T[B] -> [D]............................represent the operation "Store".)
Instruction#8_____[ [ fnInt ( Y1^2, X, A, B ) - [D](1,1) ] [0] [0] [0] ] -> [E]
Instruction#9_____augment ( augment ( augment ( [C] , [E] ) , [A] ) , [B] ) -> [F]
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 LSPOLY3C download in binhex format.
PC Windows users: Click here for LSPOLY3C download in zip format.
Back to Menu
-------------------------------------------------------
Program #65 LSPOLY4C (Least Squares Quartic Polynomial (Continuous Data))
(Description to appear.)
Inputs:
Place the value of "a" into the variable A .
Place the value of "b" into the variable B .
Place f ( x ) into the function variable Y1 .
Outputs:
The output matrix [A] is 5 by 8 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , a4 , the coefficients for the least squares quartic
polynomial P(x) = ao + a1 x + a2 x2 + a3 x3 + a4 x4 .
Column 2 , row 1 contains the value of the error = integral from "a" to "b" of
[ ( f ( x ) - P( x ) )2 ].
Column 2 , rows 2 , 3 , 4 , and 5 contain the value 0 .
Columns 3 ,4 , 5 , 6 , and 7 contain the matrix of coefficients for the left hand side of the normal equations.
Column 8 contains the vector for the right hand side of the normal equations.
Click here for a LSPOLY4C graphic display of the output matrix.
(You may need iterative improvement to help reduce the roundoff errors in this method.)
Instruction#1_____[ [ A / 2, A2 / 2 , A^3 / 3 , A^4 / 4 , A^5 / 5 ]
_______________[ 0 , A^3 / 6 , A^4 / 4 , A^5 / 5 , A^6 / 6 ]
_______________[ 0 , 0 , A^5 / 10 , A^6 / 6 , A^7 / 7 ]
_______________[ 0 , 0 , 0 , A^7 / 14 , A^8 / 8 ]
_______________[ 0 , 0 , 0 , 0 , A^9 / 18 ] ] -> [A]
Instruction#2_____[ [ B / 2, B2 / 2 , B^3 / 3 , B^4 / 4 , B^5 / 5 ]
_______________[ 0 , B^3 / 6 , B^4 / 4 , B^5 / 5 , B^6 / 6 ]
_______________[ 0 , 0 , B^5 / 10 , B^6 / 6 , B^7 / 7 ]
_______________[ 0 , 0 , 0 , B^7 / 14 , B^8 / 8 ]
_______________[ 0 , 0 , 0 , 0 , B^9 / 18 ] ] - [A] -> [A]
Instruction#3_____[A]T + [A] -> [A]......................(Note: We use the symbol "->" to
___________________________.....................represent the operation "Store".)
Instruction#4_____[ [ fnInt ( Y1 , X , A , B ) ] [ fnInt ( XY1 , X , A , B ) ]
_______________[ fnInt ( X2Y1 , X , A , B ) ] [ fnInt ( ( X^3 ) Y1 , X , A , B ) ]
_______________[ fnInt ( ( X^4 )Y1 , X , A , B ) ] ]-> [B]
Instruction#5_____Matr>list ( rref ( augment ( [A] , [B] ) ) , 6 , L1 )
Instruction#6_____List>matr ( L1 , [C] ]
Instruction#7_____[C]T[B] -> [D]
Instruction#8_____[ [ fnInt ( Y1^2, X, A, B ) - [D](1,1) ] [0] [0] [0] [0] ] -> [E]
Instruction#9_____augment ( augment ( augment ( [C] , [E] ) , [A] ) , [B] ) -> [F]
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 LSPOLY4C download in binhex format.
PC Windows users: Click here for LSPOLY4C download in zip format.
Back to Menu
-------------------------------------------------------
Program #66 LSLEGNDR ( Least Squares Legendre Polynomials ( Degrees 0 - 5 , [ -1 , 1 ] ) )
(Description to appear.)
Inputs:
Let y = f ( x ) on [ - 1 , 1 ] .
Let P ( x ) = ao Po ( x ) + a1 P1 ( x ) + . . . + a5 P5 ( x ) on [ - 1 , 1 ] , where
Pi(x) = the i th monic Legendre polynomial.
Place f ( x ) into the function variable Y1 .
Outputs:
The output matrix [A] is 6 by 10 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , a4 , a5 , the coefficients for the least squares
polynomial P(x) = ao Po ( x ) + a1 P1 ( x ) + . . . + a5 P5 ( x ) .
Column 2 , row k + 1 , contains the value of the k th error = integral from " a= -1" to " b = 1"
of [ ( f ( x ) - P(k)( x ) )2 ] , where P(k)( x ) =
ao Po ( x ) + a1 P1 ( x ) + . . . + ak Pk ( x ) .
Column 3 contains the vector for the right hand side of the normal equtions.
Column 4 , row k + 1 , contains the integral from " a= -1" to " b = 1" of
[ ( Pk( x ) )2 ]
Columns 5 , 6 , 7 , 8 , 9 , and 10 contain the coefficients of the regular polynomial
forms of the least squares polynomials, degrees 0 to 5 .
Click here for a LSLEGNDR graphic display of the output matrix.
Instruction#1_____{ 1, 0, 0, 0, 0, 0 } -> L1..............(Note: We use the symbol "->" to
Instruction#2_____{ 0, 1, 0, 0, 0, 0 } -> L2..............represent the operation "Store".)
Instruction#3_____{ -1/3, 0, 1, 0, 0, 0 } -> L3
Instruction#4_____{ 0, -3/5, 0, 1, 0, 0 } -> L4
Instruction#5_____{ 3/35, 0, -6/7, 0, 1, 0 } -> L5
Instruction#6_____{ 0, 5/21, 0, -10/9, 0, 1 } -> L6
Instruction#7_____List>matr (L1, L2, L3, L4, L5, L6, [A] )
Instruction#8_____{ 2 , 2/3 , 2/5 - 2/9 , 2/7 - 6/25 , 2/9 - 12/49 + 6/175 ,
___________________2/11 - 20/81 + 10/147 } -> L1
Instruction#9_____augment ( { fnInt ( Y1, X, -1, 1 ) } ,
____________________seq ( fnInt ( X^I Y1 , X , -1 , 1 ) , I , 1 , 5 ) ) -> L2
Instruction#10____List>matr ( L2, [B] )
Instruction#11____Matr>list ( [A]T[B] , L2 )
Instruction#12____L2 / L1 -> L3
Instruction#13____L3(1) -> [A](1,1)
Instruction#14____fnInt ( Y1^2 , X , -1 , 1 ) - L1(1) L3(1)^2 -> L4(1)
Instruction#15____For ( J, 2, 6 )
Instruction#16____L4(J-1) - L1(J)L3(J)^2 -> L4(J)
Instruction#17____For ( I, 1, J )
Instruction#18____[A](I,J-1) + [A](I,J) L3(J) -> [A](I,J)
Instruction#19____End
Instruction#20____End
Instruction#21____List>matr ( L3, L4, L2, L1, [C] )
Instruction#22____augment ( [C] , [A] ) -> [D]
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 LSLEGNDR download in binhex format.
PC Windows users: Click here for LSLEGNDR download in zip format.
Back to Menu
-------------------------------------------------------
Program #67 LSLGDRAB ( Least Squares Legendre Polynomials ( Degrees 0 - 5 , [ a , b ] ) )
(Description to appear.)
Inputs:
Let y = f ( x ) on [ a , b ] .
Let Q ( x ) = ao Qo ( x ) + a1 Q1 ( x ) + . . . + a5 Q5 ( x ) on [ a , b ] , where
Qi(x) = the i th monic Legendre polynomial on [ a , b ].
Place f ( x ) into the function variable Y1 .
Also, place "a" into the variable A and "b" into the variable B.
Outputs:
The output matrix [A] is 6 by 10 and contains the following:
Column 1 contains ao , a1 , a2 , a3 , a4 , a5 , the coefficients for the least squares
polynomial Q(x) = ao Qo ( x ) + a1 Q1 ( x ) + . . . + a5 Q5 ( x ) .
Column 2 , row k + 1 , contains the value of the k th error = integral from " a " to " b "
of [ ( f ( x ) - Q(k)( x ) )2 ] , where Q(k)( x ) =
ao Qo ( x ) + a1 Q1 ( x ) + . . . + ak Qk ( x ) .
Column 3 contains the vector for the right hand side of the normal equtions.
Column 4 , row k + 1 , contains the integral from " a " to " b " of
[ ( Qk( x ) )2 ]
Columns 5 , 6 , 7 , 8 , 9 , and 10 contain the coefficients of the regular polynomial
forms of the least squares polynomials, degrees 0 to 5 .
Click here for a LSLGDRAB graphic display of the output matrix.
Instruction#1_____2 / ( B - A ) -> C..................(Note: We use the symbol "->" to
Instruction#2_____( A + B ) / ( A - B ) -> D.......represent the operation "Store".)
Instruction#3_____{ 1, 0, 0, 0, 0, 0 } -> L1
Instruction#4_____{ 0, 1, 0, 0, 0, 0 } -> L2
Instruction#5_____{ -1/3, 0, 1, 0, 0, 0 } -> L3
Instruction#6_____{ 0, -3/5, 0, 1, 0, 0 } -> L4
Instruction#7_____{ 3/35, 0, -6/7, 0, 1, 0 } -> L5
Instruction#8_____{ 0, 5/21, 0, -10/9, 0, 1 } -> L6
Instruction#9_____List>matr (L1 , L2 / C , L3 / C^2 , L4 / C^3 ,
_____________________L5 / C^4 , L6 / C^5 , [A] )
Instruction#10____{ D , C , 0 , 0 , 0 , 0 } -> L2
Instruction#11____{ D^2 , 2CD , C^2 , 0 , 0 , 0 } -> L3
Instruction#12____{ D^3 , 3 (D^2) C , 3 D (C^2) , C^3 } -> L4
Instruction#13____{ D^4 , 4 (D^3) C , 6 (D^2) (C^2) , 4 D (C^3) , C^4 , 0 } -> L5
Instruction#14____{ D^5 , 5 (D^4) C , 10 (D^3) (C^2) , 10 (D^2) (C^3) ,
_____________________5 D (C^4) , C^5 } -> L6
Instruction#15____List>matr (L1, L2, L3, L4, L5, L6, [B] )
Instruction#16____[B] [A] -> [C]
Instruction#17____{ 2 / C , (2/3) / (C^3) , (2/5 - 2/9) / (C^5) ,
___________________(2/7 - 6/25) / (C^7) , (2/9 - 12/49 + 6/175) / (C^9) ,
___________________(2/11 - 20/81 + 10/147) / (C^11) } -> L1
Instruction#18____augment ( { fnInt ( Y1, X, A, B ) } ,
___________________seq ( fnInt ( X^I Y1 , X , A , B ) , I , 1 , 5 ) ) -> L2
Instruction#19____List>matr ( L2, [D] )
Instruction#20____Matr>list ( [C]T[D] , L3 )
Instruction#21____L3 / L1 -> L4
Instruction#22____[C] -> [J]
Instruction#23____L4(1) -> [C](1,1)
Instruction#24____fnInt ( Y1^2 , X , A , B ) - L1(1) L4(1)^2 -> L5(1)
Instruction#25____For ( J, 2, 6 )
Instruction#26____L5(J-1) - L1(J)L4(J)^2 -> L5(J)
Instruction#27____For ( I, 1, J )
Instruction#28____[C](I,J-1) + [C](I,J) L4(J) -> [C](I,J)
Instruction#29____End
Instruction#30____End
Instruction#31____List>matr ( L4, L5, L3, L1, [D] )
Instruction#32____augment ( [D] , [C] ) -> [E]
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 LSLGDRAB download in binhex format.
PC Windows users: Click here for LSLGDRAB download in zip format.
Back to Menu
-------------------------------------------------------
Program #68 SYSNEWT3 (Newton Iteration for 3 x 3 Systems)
(Description to appear.)
Inputs:
The equations are F 1 ( x , w , z ) = 0 , F 2 ( x , w , z ) = 0 , F 3 ( x , w , z ) = 0 .
The initial Newton iterate vector is xo = ( xo , wo , zo )
Place F 1 ( x , w , z ) into the Y1 function variable
Place F 2 ( x , w , z ) into the Y2 function variable
Place F 3 ( x , w , z ) into the Y3 function variable
Place xo into the variable X .
Place wo into the variable Y .
Place zo into the variable Z .
Outputs:
The output matrix [A] is 3 by 9 and contains the following:
Column 1 contains the Newton iterate vector x1 = ( x1 , w1 , z1 ) ,
(where x1 = xo - J -1 ( xo ) F ( xo ) ).
Columns 2 , 3 , and 4 contain the Jacobian matrix J ( xo ) .
Columns 5 , 6 and 7 contain the matrix J -1 ( xo ) .
Column 8 contains the vector :
F ( xo ) = ( F 1 ( xo , wo , zo ) , F 2 ( xo , wo , zo ) , F 3 ( xo , wo , zo ) ) .
Column 9 contains the vector J -1 ( xo ) F ( xo ) .
Click here for a SYSNEWT3 graphic display of the output matrix.
Instruction#1_____{3,3} -> dim( [A] )..................(Note: We use the symbol "->" to
Instruction#2_____{3,1} -> dim( [B] )..................represent the operation "Store".)
Instruction#3_____Y1 -> [B](1,1)
Instruction#4_____Y2 -> [B](2,1)
Instruction#5_____Y3 -> [B](3,1)
Instruction#6_____nDeriv( Y1, X, X, .00001 ) -> [A](1,1)
Instruction#7_____nDeriv( Y1, W, W, .00001 ) -> [A](1,2)
Instruction#8_____nDeriv( Y1, Z, Z, .00001 ) -> [A](1,3)
Instruction#9_____nDeriv( Y1, X, X, .00001 ) -> [A](2,1)
Instruction#10____nDeriv( Y1, W, W, .00001 ) -> [A](2,2)
Instruction#11____nDeriv( Y1, Z, Z, .00001 ) -> [A](2,3)
Instruction#12____nDeriv( Y1, X, X, .00001 ) -> [A](3,1)
Instruction#13____nDeriv( Y1, W, W, .00001 ) -> [A](3,2)
Instruction#14____nDeriv( Y1, Z, Z, .00001 ) -> [A](3,3)
Instruction#15____[A]-1 -> [C]
Instruction#16____[C] [B] -> [D]
Instruction#17____X - [D](1,1) -> X
Instruction#18____W - [D](2,1) -> W
Instruction#19____Z - [D](3,1) -> Z
Instruction#20____augment ( augment ( [ [X] [W] [Z] ] , [A] ) , [C] ) -> [E]
Instruction#21____augment ( augment ( [ E ] , [B] ) , [D] ) -> [F]
Note: In instruction#20, X, W, and Z are real variables and should be typed as such. [A], [C], and [E] are matrix variables and should be entered as matrix names. In instruction#21, [E], [B], [D], and [F] are all matrix names.
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 SYSNEWT3 download in binhex format.
PC Windows users: Click here for SYSNEWT3 download in zip format.
Back to Menu
-------------------------------------------------------
Program #69 ECONOMIZ (Chebyshev Economization, (Degrees 0-8, [ -1 , 1 ] ) )
(Description to appear.)
Inputs:
Let P ( x ) = ao + a1 x + ... + an xn , where n is between 0 and 8 ,
and where x is between -1 and +1 .
Then, place { ao , a1 , ... , an } into the L1 list variable.
Outputs:
Let P ( x ) = bo + b1 T1(x) + b2 T2(x) + ... + bn Tn(x) .
The output matrix [ F ] is 9 by 11 and contains the following:
Column 1 , row " i " , contains | bn | + | bn-1 | + ... + | b i-1 | , where
" i " is between 1 and n+1 . ( | bn | is the absolute value of bn . )
Column 2 , row " i " , contains b i-1 , where " i " is between 1 and n+1 .
Let Pk ( x ) = bo + b1 T1(x) + b2 T2(x) + ... + bk Tk(x) =
the economized polynomial of degree " k " =
co + c1 x + ... + c k xk , where " k " is between 0 and n .
Then, column ( k + 3 ) , row " i " contains c i-1 , where " i " is between 1 and k + 1 ,
and where " k " is between 0 and n .
All other entries of the output matrix are zero .
Click here for a ECONOMIZ graphic display of the output matrix.
Instruction#1_____{ 9 , 1 } -> dim ( [ E ] )............(See note below.)
Instruction#2_____[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ] [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
__________________[ -1, 0, 2, 0, 0, 0, 0, 0, 0 ] [0, -3, 0, 4, 0, 0, 0, 0, 0 ]
__________________[ 1, 0, -8, 0, 8, 0, 0, 0, 0 ] [0, 5, 0, -20, 0, 16, 0, 0, 0 ]
__________________[ -1, 0, 18, 0, -48, 0, 32, 0, 0 ] [ 0, -7, 0, 56, 0, -112, 0, 64, 0 ]
__________________[ 1, 0, -32, 0, 160, 0, -256, 0, 128 ] ] T -> [ A ]
Instruction#3_____List>matr ( L1, [ B ] )
Instruction#4_____{ 9 , 1 } -> dim ( [ B ] )
Instruction#5_____Matr>list ( rref ( augment ( [ A ] , [ B ] ) ) , 10 , L2 )
Instruction#6_____List>matr ( L2 , [ C ] )
Instruction#7_____L2(1) -> E : L2(2) -> F : L2(3) -> G : L2(4) -> H
Instruction#8_____L2(5) -> K : L2(6) -> L : L2(7) -> M : L2(8) -> N
Instruction#9_____L2(9) -> P
Instruction#10____abs ( L2(9) ) -> [ E ] ( 9 , 1 )
Instruction#11____For ( J , 2 , 9 )
Instruction#12____abs ( L2( 10 - J ) ) + [ E ] ( 11 - J , 1 ) -> [ E ] ( 10 - J , 1 )
Instruction#13____End
Instruction#14____[ [ E, E, E, E, E, E, E, E, E ] [ 0, F, F, F, F, F, F, F, F ]
_________________[ 0, 0, G, G, G, G, G, G, G ] [ 0, 0, 0, H, H, H, H, H, H ]
_________________[ 0, 0, 0, 0, K, K, K, K, K ] [0, 0, 0, 0, 0, L, L, L, L ]
_________________[ 0, 0, 0, 0, 0, 0, M, M, M ] [ 0, 0, 0, 0, 0, 0, 0, N, N ]
_________________[ 0, 0, 0, 0, 0, 0, 0, 0, P ] ] -> [ D ]
Instruction#15____[ A ] [ D ] -> [ D ]
Instruction#16____augment ( augment ( [ E ] , [ C ] ) , [ D ] ) -> [ F ]
Note: We use the symbol "->" to represent the operation "Store".
Note: Use the MATRIX menu to enter the matrix name [E] onto a program line. Do not type the characters "[" , "E" , "]" 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 ECONOMIZ download in binhex format.
PC Windows users: Click here for ECONOMIZ download in zip format.
Back to Menu
-------------------------------------------------------
Program #70 ECONOMAB (Chebyshev Economization, (Degrees 0-8, [ a , b ] ) )
(Description to appear.)
Inputs:
Let P ( x ) = ao + a1 x + ... + an xn , where n is between 0 and 8 ,
and where x is between " a " and " b " .
Then, place { ao , a1 , ... , an } into the L1 list variable.
Also, place " a " into the variable A and " b " into the variable B .
Outputs:
Let u = ( a + b - 2 x ) / ( a - b ) . Then, x = " a " corresponds to u = - 1 ,
and x = " b " corresponds to u = +1 .
Let P ( x ) = bo + b1 T1(u) + b2 T2(u) + ... + bn Tn(u) .
The output matrix [ F ] is 9 by 11 and contains the following:
Column 1 , row " i " , contains | bn | + | bn-1 | + ... + | b i-1 | , where
" i " is between 1 and n+1 . ( | bn | is the absolute value of bn . )
Column 2 , row " i " , contains b i-1 , where " i " is between 1 and n+1 .
Let Pk ( x ) = bo + b1 T1(u) + b2 T2(u) + ... + bk Tk(u) =
the economized polynomial of degree " k " , which is
co + c1 x + ... + c k xk , where " k " is between 0 and n .
Then, column ( k + 3 ) , row " i " contains c i-1 , where " i " is between 1 and k + 1 ,
and where " k " is between 0 and n .
All other entries of the output matrix are zero .
Click here for a ECONOMAB graphic display of the output matrix.
Instruction#1_____.5 ( B + A ) -> C..................(Note: We use the symbol "->" to
Instruction#2_____.5 ( B - A ) -> D....................represent the operation "Store".)
Instruction#3_____prgm ECNSETUP
Instruction#4_____List>matr ( L1 , [ B ] )...........(See note below.)
Instruction#5_____{ 9 , 1 } -> dim ( [ B ] )
Instruction#6_____Matr>list ( [ G ] [ B ] , L1 )
Instruction#7_____prgm ECONOMIZ
Instruction#8_____(-) C / D -> C
Instruction#9_____1 / D -> D
Instruction#10____prgm ECNSETUP
Instruction#11____augment ( augment ( [ E ] , [ C ] ) , [ G ] [ D ] ) -> [ F ]
Subroutine ECNSETUP (Matrix Setup for Shift of Coordinates)
(Description to appear.)
Instruction#1_____CC -> E : CD -> F............(Note: We use the symbol "->" to
Instruction#2_____DD -> G : EE -> H............represent the operation "Store".)
Instruction#3_____FF -> I : GG -> J : EF -> K : FG -> L
Instruction#4_____HE -> M : GJ -> N : E I -> P : G I -> Q
Instruction#5_____[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ] [ C, D, 0, 0, 0, 0, 0, 0, 0 ]
__________________[ E, 2F , G, 0, 0, 0, 0, 0, 0 ] [ CE, 3DE, 3CG, DG, 0, 0, 0, 0, 0 ]
__________________[ H, 4K, 6 I, 4L, J, 0, 0, 0, 0 ]
__________________[ CH, 5DH, 10 C I, 10 D I, 5CJ, DJ, 0, 0, 0 ]
__________________[ M, 6FH, 15P, 20 F I, 15Q, 6FJ, N, 0, 0]
__________________[ CM, 7DM, 21 PC , 35 PD , 35 QC , 21 QD , 7CN, DN, 0 ]
__________________[ HH, 8KH, 28 H I , 56 K I , 70 I2 , 56 L I , 28 I J , 8LJ, JJ ] ]T
_____________________________ -> [ G ]
Note: Use the MATRIX menu to enter the matrix name [B] onto a program line. Do not type the characters "[" , "B" , "]" separately.
These programs may be downloaded to your computer for use with the TI-GRAPH LINK application or the equivalent.
Macintosh OS users: Click here for ECONOMAB download in binhex format.
Macintosh OS users: Click here for ECNSETUP download in binhex format.
PC Windows users: Click here for ECONOMAB download in zip format.
PC Windows users: Click here for ECNSETUP download in zip format.
Back to Menu
-------------------------------------------------------