# 2250-1 7:30am Lecture Record Week 4 S2010

Last Modified: January 31, 2010, 13:58 MST.    Today: September 24, 2018, 01:22 MDT.

## 1 Feb: Numerical Solutions for y'=f(x,y)

Second lecture on numerical methods.
```Euler, Heun, RK4 algorithms
Computer implementation in maple
Geometric and algebraic ideas in the derivations.
Numerical Integration   Numerical Solutions of DE
RECT                    Euler
TRAP                    Heun [modified Euler]
SIMP                    Runge-Kutta 4 [RK4]

Numerical work maple L3.1, L3.2, L3.3, L4.1, L4.2, L4.3 will be
submitted after Spring Break. No numerical problems from ch 2
are assigned.
All discussion of maple programs will be based in the TA session
[Cox and Murphy].
There will be one additional presentation of maple lab details
in the main lecture. The examples used in maple labs 3, 4 are
the same as those in exam review problems ER-1, ER-2. Each has
form dy/dx=f(x,y) and requires a non-quadrature algorithm, e.g.,
Euler, Heun, RK4.
```
```Numerical Solution of y'=f(x,y)
Two problems will be studied, in maple labs 3, 4.
First problem
y' = -2xy, y(0)=2
Symbolic solution y = 2 exp(-x^2)
Second problem
y' = (1/2)(y-1)^2, y(0)=2
Symbolic solution y = (x-4)/(x-2)
The work begins in exam review problems ER-1, ER-2, both due
before the first midterm exam. The maple numerical work is due
much later, after Spring Break.
Exam Review: Problems ER-1, ER-2 (46.9 K, pdf, 01 Jan 2010)```
```Examples
Web references contain two kinds of examples.
The first three are quadrature problems dy/dx=F(x).
The fourth is of the form dy/dx=f(x,y), which requires a
non-quadrature algorithm like Euler, Heun, RK4.

y'=3x^2-1, y(0)=2, solution y=x^3-x+2
y'=exp(x^2), y(0)=2, solution y=2+int(exp(t^2),t=0..x).
y'=2x+1, y(0)=3 with solution y=x^2+x+3.
y'=1-x-y, y(0)=3, solution y=2-x+exp(-x).
```
```Rect, Trap, Simp rules from calculus
Introduction to the Euler, Heun, RK4 rules from this course.
Example: y'=3x^2-1, y(0)=2 with solution y=x^3-x+2.
Example: y'=2x+1, y(0)=3 with solution y=x^2+x+3.
Dot tables,  connect the dots graphic.
How to draw a graphic without knowing the solution equation for y.
Key example y'=sqrt(x)exp(x^2), y(0)=2.
Challenge: Can you integrate sqrt(x) exp(x^2)?
Making the dot table by approximation of the integral of F(x).
Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
```
```Example for your study:
Problem:  y'=x+1, y(0)=1
It has a dot table with x=0, 0.25, 0.5, 0.75, 1 and
y= 1, 1.25, 1.5625, 1.9375, 2.375.
The exact solution y = 0.5(1+(x+1)^2) has values
y=1, 1.28125, 1.625, 2.03125, 2.5000.
Determine how the dot table was constructed and identify
which rule, either Rect, Trap, or Simp, was applied.
```
Introduction: Maple Labs 3 and 4, due after Spring Break.
Maple lab 3 S2010. Numerical DE (82.5 K, pdf, 01 Jan 2010)
Maple lab 4 S2010. Numerical DE (78.9 K, pdf, 01 Jan 2010)

The work for book sections 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4.
The numerical work using Euler, Heun, RK4 appears in L3.1, L3.2, L3.3.
The actual symbolic solution derivation and answer check were submitted as Exam Review ER-1. Confused? Follow the details in the next link, which duplicates what was done in ER-1.
Sample symbolic solution report for 2.4-3 (22.6 K, pdf, 19 Sep 2006)
Confused about what to put in your L3.1 report? Do the same as what appears in the sample report for 2.4-3 (link below). Include the hand answer check. Include the maple code appendix. Then fill in the table in maple Lab 3, by hand. The example shows a hand answer check and the maple code appendix.

Sample Report for 2.4-3 (175.9 K, pdf, 02 Jan 2010)
Download all .mws maple work-sheets to disk, then run the worksheet in xmaple.
In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...". Some browsers require SHIFT and then mouse-click. Open the saved file in xmaple or maple.
Extension .mws [or .mpl] allows interchange between different versions of maple. Mouse copies of the worksheet pasted into email allow easy transfer of code between versions of maple.

## 2 Feb: Linear Algebraic Equations. No matrices. Section 3.1.

```Maple lab 2 problem 1
Discussion: Option 1: Freezing pipes maple lab 2
Problem: u' + ku = kA(t)
Integration methods
Tables
Maple
```
Option 1: Maple Lab 2, Newton cooling freezing pipes (112.1 K, pdf, 01 Jan 2010)
Option 1: maple worksheet text freezing pipes (1.2 K, txt, 02 Jan 2010)
Option 2: Maple Lab 2, Newton cooling swamp cooler (159.0 K, pdf, 01 Jan 2010)
Option 2: maple worksheet text swamp cooler (1.3 K, txt, 02 Jan 2010)
For more on superposition y=y_p + y_h, see Theorem 2 in the link
Linear DE part I (152.7 K, pdf, 07 Aug 2009)
Manuscript: Linear equation applications, brine tanks, home heating (374.2 K, pdf, 28 Jul 2009)
Slides: Brink tanks (62.9 K, pdf, 30 Nov 2009)
Slides: Home heating (73.8 K, pdf, 30 Nov 2009)
```Linear Algebraic Equations sections 3.1, 3.2
Frame sequences
Toolkit: combo, swap, multiply
Plane and space geometry
The three possibilities
Unique solution
No solution
Infinitely many solutions
Method of elimination
Example for a unique solution
x + 2y =  1
x -  y = -2
Example for no solution
x + 2y = 1
x + 2y = 2
Example for infinitely many solutions
x + 2y = 1
0 = 0
Parameters in the general solution
Differential equations example, problem 3.1-26
y'' -121y = 0, y(0)=44, y'(0)=22
General solution given: y=A exp(11 x) + B exp(-11 x)
Substitute y into y(0)=44, y'(0)=22 to obtain a 2x2
system for unknowns A,B that has the unique solution
A=23, B=21.
```
Prepare 3.1 problems for next collection. See problem notes section 3.1:
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)

## 3 Feb: Frame Sequences. Three Possibilities. No matrices. Sections 3.2, 3.3.

```Lecture: 3.1, 3.2, 3.3
Frame sequences
Toolkit: combo, swap, multiply
Plane and space geometry
The three possibilities
Unique solution
No solution
Infinitely many solutions
Free variable
Signal equation
Echelon form
The last frame test
The last frame algorithm
A detailed account of the three possibilities
Unique solution == zero free variables
No solution == signal equation
Infinitely many solutions == one+ free variables
```
```How to solve a linear system using the toolkit
Toolkit: swap, combo, mult
Toolkit operations neither create nor destroy solutions!
Frame sequence examples
Computer algebra systems and error-free frame sequences.
How to program maple to make a frame sequence without errors.
```
```Solved Problems
Example 4 in 3.2
Back-substitution should be presented as combo operations in a
frame sequence, not as isolated, incomplete algebraic jibberish.
Technically, back-substitution is identical to applying the
frame sequence method to variables in reverse order.
The textbook observes that an echelon matrix as frame one is
a special case, when only combo operations are required to
determine the last frame. Then, and only then, does the last
frame algorithm apply to write out the general solution.
Problem 3.2-24
The book's answer is wrong, it should involve k-4.
See references on 3 possibilities with symbol k.
```

Beamer slides: 3 possibilities with symbol k (60.0 K, pdf, 31 Jan 2010)
Slides: 3 possibilities with symbol k (72.8 K, pdf, 31 Jan 2010)
Manuscript: Example 10 in Linear algebraic equations no matrices (292.8 K, pdf, 01 Feb 2010) In all your solved problems, to be submitted for grading, please use frame sequences to display the solution, as in today's lecture examples. Expected is a sequence of augmented matrices. Yes, you may use maple to make the frame sequence. The maple answer check for the last frame is rref(A).

## 3-4 Feb: Murphy and Cox

Exam 1 review, questions and examples on exam problems 1,2,3,4,5.
Murphy's Lecture, time permitting: Maple Lab 2, problem 1,2,3 details.
Exam 1 date is Feb 17, 2-4pm in WEB 104 or Feb 18, 6:50am in JTB 140.
Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.
Answer Key: Exam 1, f2009, 7:30am (62.9 K, pdf, 30 Sep 2009)
Answer Key: Exam 1, f2009, 12:25pm (339.3 K, pdf, 11 Oct 2009)
Answer Key: Exam 1, S2009, 7:30am (395.7 K, pdf, 02 Mar 2009)
Answer Key: Exam 1, S2009, 10:45am (310.2 K, pdf, 02 Mar 2009)

## 5 Feb: Augmented Matrix for System Ax=b. RREF. Last Frame Algorithm. Sections 3.3, 3.4.

```Review
Last frame test. The RREF of a matrix.
Last frame algorithm.
Scalar form of the solution.
Vector form of the solution.
```
```Lecture: 3.3 and 3.4
Translation of equation models
Equality of vectors
Scalar equations to augmented matrix
Augmented matrix to scalar equations
Matrix toolkit: Combo, swap and multiply
Frame sequences for matrix models.
Special matrices
Zero matrix
identity matrix
diagonal matrix
upper and lower triangular matrices
square matrix
THEOREM. Homogeneous system with a unique solution.
THEOREM. Homogeneous system with more variables than equations.
Equation ideas can be used on a matrix A.
View matrix A as the set of coefficients of a homogeneous
linear system Ax=0. The augmented matrix B for this homogeneous
system would be the given matrix with a column of zeros appended:
B=aug(A,0).
```

## Last Frame Algorithm

How to use maple to compute a frame sequence. Example is Exercise 3.2-14 from Edwards-Penney.
Maple Worksheet: Frame Sequence in maple, Exercise 3.2-14 (3.1 K, mws, 23 Sep 2009)
Maple Text: Frame Sequence in maple, Exercise 3.2-14 (2.8 K, txt, 23 Sep 2009)
Answer checks should also use the online FAQ.
html: Problem notes S2010 (4.4 K, html, 31 Jan 2010)

## 8 Feb: Matrix Operations. Frame Sequence Analysis for Matrices. Section 3.4, 3.5.

Review: Answer checks with matlab, maple and mathematica. Pitfalls.
Review of the three possibilities and frame sequence analysis to find the general solution.
```Matrices
Vector.
Matrix multiply
The college algebra definition
Examples.
Matrix rules
Vector space rules.
Matrix multiply rules.
```

Manuscript: Vectors and Matrices (266.8 K, pdf, 09 Aug 2009)
Manuscript: Matrix Equations (162.6 K, pdf, 09 Aug 2009)
Examples: how to multiply matrices on paper.
Slides: Matrix add, scalar multiply and matrix multiply (122.5 K, pdf, 02 Oct 2009)

Maple: Lab 2 problem L2.1 to be discussed today. Solution projected for L2.1.