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**CS-110 Computational Engineering Part 3**

A. Avudainayagam Department of Mathematics

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**Root Finding: f(x)=0 Method 1: The Bisection method**

Th: If f(x) is continuous in [a,b] and if f(a)f(b)<0, then there is atleast one root of f(x)=0 in (a,b). y y f(b) > 0 x2 a a x4 b x x3 x5 x x1 f (a) < 0 b x1 Multiple roots in (a,b) Single root in (a,b)

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The Method Find an interval [x0 ,x1] so that f(x0)f(x1) < 0 (This may not be easy. Use your knowledge of the physical phenomenon which the equation represents). Cut the interval length into half with each iteration,by examining the sign of the function at the mid point. If f(x2) = 0, x2 is the root. If f(x2) 0 and f(x0)f(x2) < 0, root lies in [x0,x2]. Otherwise root lies in [x2,x1]. Repeat the process until the interval shrinks to a desired level.

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**Number of Iterations and Error Tolerance**

Length of the interval (where the root lies) after n iterations We can fix the number of iterations so that the root lies within an interval of chosen length (error tolerance). If n satisfies this, root lies within a distance of / 2 of the actual root.

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**Though the root lies in a small interval, |f(x)| may not be small if f(x) has a large slope.**

Conversely if |f(x)| small, x may not be close to the root if f(x) has a small slope. So, we use both these facts for the termination criterion. We first choose an error tolerance on f(x) : |f(x)| < and m the maximum number of iterations.

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**Pseudo code (Bisection Method)**

1. Input > 0 , m > 0, x1 > x0 so that f(x0) f(x1) < 0. Compute f0 = f(x0) . k = 1 (iteration count) 2. Do { (a) Compute f2 = f(x2) = f (b) If f2f0 < 0 , set x1= x2 otherwise set x0 = f2 and f0 = f2. (c) Set k = k+1. } 3. While |f2| > and k m set x = x2 , the root.

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**False Position Method (Regula Falsi)**

Instead of bisecting the interval [x0,x1], we choose the point where the straight line through the end points meet the x-axis as x2 and bracket the root with [x0,x2] or [x2,x1] depending on the sign of f(x2).

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**False Position Method y x Straight line through (x0,f0) ,(x1,f1) :**

y=f(x) (x1,f1) (x2,f2) x0 x2 x1 x (x0,f0) Straight line through (x0,f0) ,(x1,f1) : New end point x2 :

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**False Position Method (Pseudo Code)**

1. Choose > 0 (tolerance on |f(x)| ) m > 0 (maximum number of iterations ) k = 1 (iteration count) x0 ,x1 (so that f0 ,f1 < 0) 2. { a. Compute f2 = f(x2) b. If f0f2 < 0 set x1 = x2 , f0 = f2 c. k = k+1 } 3. While (|f2| ) and (k m) 4. x = x2 , the root.

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**Newton-Raphson Method / Newton’s Method**

At an approximate xk to the root ,the curve is approximated by the tangent to the curve at xk and the next approximation xk+1 is the point where the tangent meets the x-axis. y root x x2 x1 x0 y = f(x)

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Tangent at (xk, fk) : y = f(xk) + f ´(xk)(x-xk) This tangent cuts the x-axis at xk+1 Warning : If f´(xk) is very small , method fails. Two function Evaluations per iteration

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**Newton’s Method - Pseudo code**

1. Choose > 0 (function tolerance |f(x)| < ) m > 0 (Maximum number of iterations) x0 - initial approximation k - iteration count Compute f(x0) 2. Do { q = f (x0) (evaluate derivative at x0) x1= x0 - f0/q x0 = x1 f0 = f(x0) k = k+1 } 3. While (|f0| ) and (k m) 4. x = x1 the root.

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**Getting caught in a cycle of Newton’s Method**

xk+1 xk x Alternate iterations fall at the same point . No Convergence.

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**Newton’s Method for finding the square root of a number x = a**

f(x) = x2 - a2 = 0 Example : a = 5 , initial approximation x0 = 2. x1 = 2.25 x2 = x3 = x4 =

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The secant Method The Newton’s Method requires 2 function evaluations (f,f ). The Secant Method requires only 1 function evaluation and converges as fast as Newton’s Method at a simple root. Start with two points x0,x1 near the root (no need for bracketing the root as in Bisection Method or Regula Falsi Method) . xk-1 is dropped once xk+1 is obtained.

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**The Secant Method (Geometrical Construction)**

y y = f(x1) x2 x0 x1 x (x0,f0) (x1,f1) Two initial points x0, x1 are chosen The next approximation x2 is the point where the straight line joining (x0,f0) and (x1,f1) meet the x-axis Take (x1,x2) and repeat.

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**The secant Method (Pseudo Code)**

1. Choose > 0 (function tolerance |f(x)| ) m > 0 (Maximum number of iterations) x0 , x1 (Two initial points near the root ) f0 = f(x0) f1 = f(x1) k = 1 (iteration count) 2. Do { x0 = x1 f0 = f1 x1= x2 f1 = f(x2) k = k } 3. While (|f1| ) and (m k)

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**General remarks on Convergence**

The false position method in general converges faster than the bisection method. (But not always ) counter example follows. The bisection method and the false position method are guaranteed for convergence. The secant method and the Newton-Raphson method are not guaranteed for convergence.

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**Order of Convergence A measure of how fast an algorithm converges .**

Let be the actual root : f () = 0 Let xk be the approximate root at the kth iteration . Error at the kth iteration = ek = |xk- | The algorithm converges with order p if there exist such that

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**Order of Convergence of**

Bisection method p = 1 ( linear convergence ) False position - generally Super linear ( 1 < p < 2 ) Secant method (super linear) Newton Raphson method p = 2 quadratic

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**Machine Precision The smallest positive float M that can be added to**

one and produce a sum that is greater than one.

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**Pseudo code to find ‘Machine Epsilon’**

1. Set M = 1 2. Do { M = M / 2 x = 1 + M } 3. While ( x < 1 ) 4. M = 2 M

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**Solutions Of Linear Algebraic Systems**

AX = B A n n matrix B n 1 vector X n 1 vector (unknown) If |A| 0, the system has a unique solution, provided B 0. If |A| = 0, the matrix A is called a singular matrix. Direct Methods Iterative Methods

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The Augmented Matrix [A | B]

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**Elementary row operations**

1. Interchange rows j and k. 2. Multiply row k by . 3. Add times row j to row k. Use elementary row operations and transform the equation so that the solution becomes apparent

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Diagonal Matrix (D) = Upper Triangular Matrix (U) = Lower Triangular Matrix (L) =

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**Gauss Elimination Method**

Using row operations, the matrix is reduced to an upper triangular matrix and the solutions obtained by back substitution

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Partial Pivot Method Step 1 We look for the largest element in absolute value (called pivot) in the first column and interchange this row with the first row Step 2 Multiply the I row by an appropriate number and subtract this from the II row so that the first element of the second row becomes zero. Do this for the remaining rows so that all the elements in the I column except the top most element are zero. Now the matrix looks like:

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Now look for the pivot in the second column below the first row and interchange this with the second row. Multiply the II row by an appropriate numbers and subtract this from remaining rows (below the second row) so that the matrix looks like :

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Now look for the pivot in the III column below the second row and interchange this row with the third row. Proceed till the matrix looks like : Now the solution is obtained by back substitution.

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**Gauss Elimination (An example)**

Augmented Matrix :

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**Solution by back substitution**

x3 = / = 5 x2 = (2 - 5 )/1.5 = -2 x1 = (2-8(5)+2)/4 = -9 x1 = -9 , x2 = -2 , x3 = 5

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**Gauss Elimination (Pseudo Code)**

1. Input matrix A of order n n and r h s vector B and form the augmented matrix [A|B]. 2. For j = 1 to n do { (a) Compute the pivot index j p n such that (b) If Apj = 0 , print “singular matrix” and exit. (c) If p > j , interchange rows p and j (d) For each i > j , subtract Aij/Ajj times row j from row i. } 3. Now the augmented matrix is [C|D] where C is upper triangular. 4. Solve by back substitution For j = 1= n down to 1 compute :

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**Determinant of a Matrix**

Some properties of the determinant. 1. det (AT) = det (A) 2. det (AB) = det (A) det (B) 3. det (U) = U11U22U33...Unn 4. det [E1(k,j)] = -1 . det A 5. det [E2(k,)] = det A 6. det [E3(k,j,)] = det A

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Matrix Determinant Gauss Elimination procedure uses two types of row operations. 1. Interchange of rows for pivoting 2. Add times row j to row k. The first operation changes the sign of the determinant while the second has no effect on the determinant. To obtain the matrix determinant, follow Gauss elimination and obtain the upper triangular form U, keeping count of (r) the interchanges of rows Then det A = (-1)r U11U Unn

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**Matrix Determinant (Pseudo Code)**

1. Set r = 0 2. For j = 1 to n do { (a) Compute the pivot index j p n such that (b) If Apj = 0 , print “Singular Matrix” and exit. (c) If p > j interchange rows p and j , set r = r +1. (d) For each i > j , subtract Aij/Ajj times row j and row i } 3. det = (-1)r A11A Ann

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Matrix Determinant A = det = (-1)1 (1)(4)(5.5) = -22

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LU Decomposition When Gauss elimination is used to solve a linear system, the row operations are applied simultaneously to the RHS vector. If the system is solved again for a different RHS vector, the row operations must be repeated. The LU decomposition eliminates the need to repeat the row operators each time a new RHS vector is used.

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The Matrix A (AX = B) is written as a product of an upper triangular matrix U and a lower triangular matrix L. The diagonal elements of the upper triangular matrix are chosen to be 1 to get a consistent system of equations n = 3 = 9 Equations for 9 unknowns L11, L21, L22, L31, L32, L33, U12, U13, U23.

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= First column gives L11 L21 L31 = (A11, A21, A31) First row gives U11 U12 U13 = (1, A12/L11, A13/L11) Second column gives L12 L22 L32 = (0, A22 - L21U12, A32 - L31U22) Second row gives U21 U22 U23 = (0, 1, (A23 - L21U13)/L22 ) Third column gives L13 L23 L33 = (0, 0, A33 - L31U13 - L32 U23)

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**LU Decomposition (Pseudo Code)**

Li1 = Ai , i = 1, 2, ... , n U1j = A1j/L , j = 2, 3, ... , n for j = 2, 3, ... , n-1 , i = j, j+1, ... , n , k = j, j+1, ... , n

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**LU Decomposition (Example)**

=

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**The value of U are stored in the zero space of L .**

Q = [L \ U] = After each element of A is employed , it is not needed again. So Q can be stored in place of A.

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After LU decomposition of the matrix A, the system AX = B is solved in 2 steps : (1) Forward substitution, (2) Backward Substitution AX = LUX = B Put UX = Y LY = = Forward Substitution : Y1 = B1/ L11 Y2 = (B2 - L21Y1) / L22 Y3 = (B3 - L31B1 - L32B2) / L33

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UX = = Back Substitution : x3 = Y3 x2 = Y2 - x3 U23 x1 = Y1 - x2U12 - x3 U23

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As in Gauss elimination, LU decomposition must employ pivoting to avoid division by zero and to minimize round off errors. The pivoting is done immediately after computing each column.

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**Matrix Inverse Using the LU Decomposition**

LU decomposition can be used to obtain the inverse of the original coefficient matrix. Each column j of the inverse is determined by using a unit vector (with 1 in the jth row ) as the RHS vector

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**Matrix inverse using LU decomposition-an example**

= L U A First column of the inverse of A is given by = The second and third columns are obtained by taking the RHS vector as (0,1,0) and (0,0,1).

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**Iterative Method for solution of Linear systems**

Jacobi iteration Gauss - Seidel iteration

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**Jacobi Iteration AX = B n = 3 a11x1 + a12x2 + a13x3 = b1**

Preprocessing : 1. Arrange the equations so that the diagonal terms of the coefficient matrix are not zero. 2. Make row transformation if necessary to make the diagonal elements as large as possible.

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**Jacobi Pseudo Code Rewrite the equation as**

x1 = 1/a11 (b1 - a12x2 - a13x3) x2 = 1/a22 (b2 - a21x1 - a23x3) x3 = 1/a33 (b3 - a31x1 - a32x2) Jacobi Pseudo Code Choose an initial approximation vector (x0, x1, x2). If no prior information is available then (x0, x1, x2) = (0, 0, 0) will do. Iterate : , 1 i n Stop when

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**Diagonally Dominant Matrices**

ie., the diagonal element of each row is larger than the sum of the absolute values of the other elements in the row. Diagonal dominance is a sufficient but not necessary condition for the convergence of the Jacobi or Gauss-Seidel iteration.

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**The Gauss-Seidel iteration**

In the Jacobi iteration all the components of the new estimate of the vector are computed from the current estimate However when is computed the updated estimates are already available. The Gauss - Seidel iteration takes advantage of the latest information available where updating an estimate.

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Gauss - Seidel Jacobi First Iteration x1 = (b1 - a12x2 - a13x3) / a11 x2 = (b2 - a21x1 - a23x3) / a22 x3 = (b3 - a31x1 - a32x2) / a33 x1 = (b1 - a12x2 - a13x3) / a11 x2 = (b2 - a21x1 - a23x3) / a22 x3 = (b3 - a31x1 - a23x3) / a33 Second Iteration x1 = (b1 - a12x2 - a13x3) / a11 x2 = (b2 - a21x1 - a23x3) / a22 x3 = (b3 - a31x1 - a32x2) / a33 x1 = (b1 - a12x2 - a13x3) / a11 x2 = (b2 - a21x1 - a23x3) / a22 x3 = (b3 - a31x1 - a32 x2) / a33

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**Matrix Condition Number**

Vector and Matrix norms Norm is a real valued function that gives a measure of size or length of the vector or the matrix. Example : Length of the vector A = (a, b, c) is a norm. For the n dimensional vector A = (x1, x2, x3, ... , xn) For the matrix Aij , These are called Euclidean norms.

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**Other Useful Norms For Vectors - (the p - norm)**

- (sum of the absolute values of the element ) - (Maximum magnitude norm) For Matrices - (Column sum norm) - (row sum norm)

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**Properties of norms 1. || x || > 0 for x 0**

3. || x || = || || . || x || 4. || x + y || || x || +|| y ||

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**Ill Conditioning Consider the system :**

The exact solution : x1 = 201, x2 = 100 Consider the slightly perturbed system : A11 is increased by 1 percent A22 is decreased by 0.25 percent (approx.) Solution of the perturbed system : x1 = 50, x2 = 24.75 Near 75% change in the solution.

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When the solution of system, is highly sensitive to the values of the coefficient matrix or the RHS vector, we say that the system is ill - conditioned or the matrix is ill - conditioned Ill conditioned systems will result in large accumulated round off errors.

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**Condition number of a Matrix**

k(A) = || A || || A-1 || This is a measure of the relative sensitivity of the solution to small change in the RHS vector. If k is large, the system is regarded as being ill - conditioned The row - sum norm is usually employed to find k(A)

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**Condition Number of the Matrix**

Row - sum norm of A = || A || = Max (300, 601) = 601. Row - sum norm of A-1 = || A-1 || = Max (6.01, 3) = 6.01. Condition Number k(A) = 601 (6.01) = 3612 (large). A is ill - conditioned.

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**Equations that are not properly scaled may give rise to large condition numbers.**

Consider k (A) = || A || || A-1 || = 2000 × = 1001(large)

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**Equilibration : Scale each row of A by its largest element.**

K(A) = || A || || A-1 || = 2(1) = 2s Great Improvement in the condition number.

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**Checking the answer is not enough when the condition number is high**

Consider x = and y = given 34 (-0.11) + 55 (0.45) - 21 = 0.01 0 55 (-0.11) + 89 (0.45) - 34 = 0.00 0

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**Interpolation P4 y P3 × P2 × × P1 × Given the data :**

(xk , yk) , k = 1, 2, 3, ... , n , find a function f passing through each of the data samples. f(xk) = yk , k n Once f is determined, we can use it to predict the value of y at points other than the samples. x

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**Piecewise Linear Interpolation**

A straight line segment is used between each adjacent pair of the data points. × × × × × x1 x2 x3 x4 x5 1 k n

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**Polynomial Interpolation**

For the data set (xk , yk), k = 1, ... , n , we find a polynomial of degree (n - 1) that has n constants : The n interpolation constraints f(xk) = yk gives Not feasible for large data sets, since the condition number increases rapidly with increasing n.

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**Lagrange Interpolating Polynomial**

The Lagrange interpolating polynomial of degree k, Qk(x) is constructed as follows : . . .

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**Each Qk(x) is a polynomial of degree (n - 1)**

Qk (xj) = 1 , j = k = 0 , j k The polynomial curve that passes through the data set (xk, yk), k = 1, 2, ..., n is f(x) = y1Q1(x) + y2Q2(x) yn Qn(x) Polynomial is written directly without having to solve a system of equations.

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**Lagrange interpolations (Pseudo Code)**

Choose x, the point where the function value is required y = 0 Do for i = 1 to n product = f1 Do for j = 1 to n if ( i j ) product = product (x - xj)/ (xi - xj) end if end Do y = y + product End Do

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Least Square Fit If the number of sample is large or if the dependent variable contains measurement noise, it is often letter to find a function which minimizes an error criterion such as A function that minimizes E is called the Least Square Fit

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Straight Line Fit To fit a straight line through the n points (x1, y1), (x2, y2), ... ,(xn, yn) Assume f(x) = a1 + a2x Error Find a1, a2 which minimizes E.

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**Straight Line Fit (An example )**

Fit a straight line through the five points (0, 2.10), (1, 2.83), (2, 1.10), (3, 3.20), (4, 3.90) a11 = n = 5 a21 = a12 a1 = 1.84 , a2 = , f(x) = x

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**Data Representation Integers - Fixed Point Numbers**

Decimal System - base uses 0,1,2,…,9 (396)10 = (6 100) + (9 101) + (3 102) = (396)10 Binary System - base uses 0,1 (11001)2 = (1 20) + (0 21) + (0 22) + (1 23) + (1 24) = (25)10

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**Decimal to Binary Conversion**

Convert (39)10 to binary form base = 2 2 39 Remainder 1 Remainder 1 Remainder 1 Remainder 0 Remainder 0 Remainder 1 Put the remainder in reverse order (100111)2 = (1 20 ) + (1 21) + (1 22) + (0 23) + (0 24) + (1 25) = (39)10

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**Largest number that can be stored in m-digits**

base - 10 : (99999…9) = 10m - 1 base - 2 : (11111…1) = 2m - 1 m = (999) = (111) = Limitation : Memory cells consist of 8 bits(1byte) multiples each position containing 1 binary digit.

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**Sign - Magnitude Notation**

Common cell lengths for integers : k = 16 bits k = 32 bits Sign - Magnitude Notation First bit is used for a sign 0 - non negative number negative number The remaining bits are used to store the binary magnitude of the number. Limit of 16 bit cell : (32767)10 Limit of 32 bit cell : ( )10

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**Two’s Compliment notation**

Definition : The two’s complement of a negative integer I in a k - bit cell : Two’s Compliment of I = 2k + I (Eg) : Two’s Complement of (-3)10 in a 3 - bit cell = ( )10 = (5)10 = (101)2 (-3)10 will be stored as 101

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**Storage Scheme for storing an integer I in a k - bit cell in Two’ Compliment notation**

I , I 0 , first bit = 0 Stored Value C = 2k + I , I < 0

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**The Two’s Compliment notation admits one more negative number than the sign - magnitude notation.**

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**(Eg) Take a 2 bit cell (k = 2)**

Range in Sign - magnitude notation : = 1 -1 = 11 1 = 01 Range in Two’s Compliment notation Two’s Compliment of -1 = 22-1 = (3)10 = (11)2 Two’s Compliment of -2 = = (2)10 = (10)2 Two’s Compliment of -3 = = 0 - Not possible

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**Floating Point Numbers**

Integer Part + Fractional Part Decimal System - base Binary System - base Fractional Part (0.7846)10 Fractional Part ( )2

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**Binary Fraction Decimal Fraction**

(10.11) Integer Part (10)2 = = 2 Fractional Part (11)2 = Decimal Fraction = ( 2.75 )10

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**Decimal Fraction Binary Fraction**

Convert (0.9)10 to binary fraction 0.9 2 integer part 2 integer part 2 integer part 1 2 integer part 0 2 Repetition integer part 0 (0.9)10 = ( )2

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**Scientific Notation (Decimal)**

= 7.47 10-5 = 10 9,700,000,000 = 9.7 109 Binary (10.01)2 = (1.001)2 21 (0.110)2 = (1.10)2 2-1

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**Computer stores a binary approximation to x**

x q 2n q - mantissa , n exponent (-39.9)10 = ( )2 = ( )2 (25)10

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**Decimal Value of stored number (-39.9)10 **

= 23 bit 32 bits : First bit for sign Next 8 bits for exponent 23 bits for mantissa =

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**Round of Errors can be reduced by Efficient Programming Practice**

The number of operations (multiplications and additions ) must be kept minimum. (Complexity theory)

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**An Example of Efficient Programming**

Problem : Evaluate the value of the Polynomial. P(x) = a4x4 + a3x3 + a2x2 + a1x + a0 at a given x. Requires 13 mulitiplications and 4 additions. Bad Programme !

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**An Efficient method (Horner’s method)**

P(x) = a4x4 + a3x3 + a2x2 + a1x + a0 = a0 + x(a1 + x(a2 + x(a3 + xa4) )) Requires 4 multiplications and 4 additions. Psuedo code for an nth degree polynomial Input a’s, n,x p = 0 For i = n, n-1, … , 0 p = xP + a(i) End

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**Numerical Integration**

When do we need numerical integration ? Antiderivate of f(x) difficult to find. f(x) is available only in a tabular form

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Trapezoidal Rule

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Simpson’s rule

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**EULER’S METHOD-Pseudocode**

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**Euler’s Method-Pseudo code**

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**A Second order method: Taylor Series method**

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Taylor’s method

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**Taylor Series Method: An example**

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**Taylor Series solution at x = 0.1**

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**Finite Difference Method for Boundary Value Problems**

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Chapter: 3c System of Linear Equations

Chapter: 3c System of Linear Equations

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