Lecture 2 Number Representation and accuracy Normalized Floating Point Representation Significant Digits Accuracy and Precision Rounding and Chopping Reading assignment: Chapter 2 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Representing Real Numbers You are familiar with the decimal system Decimal System Base =10 , Digits(0,1,…9) Standard Representations EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Normalized Floating Point Representation No integral part, Advantage Efficient in representing very small or very large numbers EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Binary System Binary System Base=2, Digits{0,1} EE 3561_Unit_1 (c)Al-Dhaifallah 1435

7-Bit Representation (sign: 1 bit, Mantissa 3bits,exponent 3 bits) EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Fact Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system You can never represent 0.1 exactly in any computer EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Representation Hypothetical Machine (real computers use ≥ 23 bit mantissa) Example: If a machine has 5 bits representation distributed as follows Mantissa 2 bits exponent 2 bit sign 1 bit Possible machine numbers (0.25=00001) (0.375= 01111) (1.5=00111) EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Representation Gap near zero EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Remarks Numbers that can be exactly represented are called machine numbers Difference between machine numbers is not uniform. So, sum of machine numbers is not necessarily a machine number 0.25 + .375 =0.625 (not a machine number) EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Significant Digits Significant digits are those digits that can be used with confidence. 0 1 2 3 4 Length of green rectangle = 3.45 significant EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Loss of Significance Mathematical operations may lead to reducing the number of significant digits 0.123466 E+02 6 significant digits ─ 0.123445 E+02 6 significant digits ────────────── 0.000021E+02 2 significant digits 0. 210000E-02 Subtracting nearly equal numbers causes loss of significance EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Accuracy and Precision Accuracy is related to closeness to the true value Precision is related to the closeness to other estimated values EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Accuracy and Precision Better Precision Accuracy is related to closeness to the true value Precision is related to the closeness to other estimated values Better accuracy EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Rounding and Chopping Rounding (1.2) Chopping (1.1) Rounding: Replace the number by the nearest machine number Chopping: Throw all extra digits True 1.1681 0 1 2 Rounding (1.2) Chopping (1.1) EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Error Definitions True Error can be computed if the true value is known EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Error Definitions Estimated error Used when the true value is not known EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Notation We say the estimate is correct to n decimal digits if We say the estimate is correct to n decimal digits rounded if EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Summary Number that have finite expansion in one numbering system may Number Representation Number that have finite expansion in one numbering system may have an infinite expansion in another numbering system. Normalized Floating Point Representation Efficient in representing very small or very large numbers Difference between machine numbers is not uniform Representation error depends on the number of bits used in the mantissa. EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Summary Rounding Chopping Error Definitions: Absolute true error True Percent relative error Estimated absolute error Estimated percent relative error EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Lecture 3 Taylor Theorem Motivation Taylor Theorem Examples Reading assignment: Chapter 4 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Motivation We can easily compute expressions like b a 0.6 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 1 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 1 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 2 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Convergence of Taylor Series (Observations, Example 1) The Taylor series converges fast (few terms are needed) when x is near the point of expansion. If |x-c| is large then more terms are needed to get good approximation. EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 3 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Example 3 remarks Can we apply Taylor series for x>1?? How many terms are needed to get good approximation??? These questions will be answered using Taylor Theorem EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Theorem (n+1) terms Truncated Taylor Series Reminder EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Theorem EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Error Term EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Example 4 (The Approximation Error) EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Example 5 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Example 5 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Example 5 Error term EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Alternative form of Taylor Theorem EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Theorem Alternative forms EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Derivative Mean-Value Theorem EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Alternating Series Theorem Alternating Series is special case of Taylor Series. This means that the error is less or equal to the magnitude of the first omitted term in the alternating series. EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Alternating Series Example 6 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Remark In this course all angles are assumed to be in radian unless you are told otherwise EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Maclurine series Find Maclurine Maclurine series expansion of cos (x) Maclurine series is a special case of Taylor series with the center of expansion c = 0 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 7 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 8 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Taylor Series Example 8 EE 3561_Unit_1 (c)Al-Dhaifallah 1435

Summary Taylor series expansion is very important in most numerical methods applications approximation remainder Remainder can be used to estimate approximation error or estimate the number of terms to achieve desirable accuracy EE 3561_Unit_1 (c)Al-Dhaifallah 1435