Lecture note 1 Numerical Analysis ·Why1+f? The first nonzero bit is always 1.To save bits,we don't store it. 101101..00 is to represent1.101101..00. underflow and overflow -Special numbers 土0 s=0or1,c=0,f=0 土00 8=0or1,c=2047,f=0 NaN (not a number)c=2047,f is anything but 0 -The Smallest positive number is s=0.c=1 and f=0 2021-1023(1+0)≈0.2225×10-307 Underflow if a number occurs with a magnitude less than it. The largest positive number is s=0,c=2046 and f=1- 22016-10232-2a）≈01797×109 1 Overflow if a number occurs with a magnitude greater than it. -Machine precision:女≈2.2204×10-16Type“eps”in Matlab,you will get this number. There are various kinds of errors that we encounter when using a computer for computation. Truncation Error:Caused by adding up to a finite number of terms,while we should add infinitely many terms to get the exact answer in theory. Errors depending on the numerical algorithms,step size,and so on. Overflow/Underflow:Caused by too large or too small numbers to be represented/stored properly in finite bits. Negligible Addition:Caused by adding two numbers of magnitudes differ- ing by over 52 bits,as can be seen in the last section. Round-off Error:Caused by representing/storing numeric data in finite bits. Loss of Significance:Caused by a bad subtraction,which means a subtrac- tion of a number from another one that is almost equal in value. Error Magnification:Caused and magnified/propagated by multiplying/dividing a number containing a small error by a large/small number. 8Lecture note 1 Numerical Analysis • Why 1 + f? The first nonzero bit is always 1. To save bits, we don’t store it. 101101 . . . 00 is to represent 1.101101 . . .00. • underflow and overflow – Special numbers ±0 s = 0 or 1, c = 0, f = 0 ±∞ s = 0 or 1, c = 2047, f = 0 NaN (not a number) c = 2047, f is anything but 0 – The Smallest positive number is s = 0, c = 1 and f = 0 2 0 2 1−1023(1 + 0) ≈ 0.2225 × 10−307 Underflow if a number occurs with a magnitude less than it. – The largest positive number is s = 0, c = 2046 and f = 1 − 1 2 52 2 0 2 2046−1023(2 − 1 2 52 ) ≈ 0.17977 × 10309 Overflow if a number occurs with a magnitude greater than it. – Machine precision: 1 2 52 ≈ 2.2204× 10−16 Type “eps” in Matlab, you will get this number. There are various kinds of errors that we encounter when using a computer for computation. • Truncation Error: Caused by adding up to a finite number of terms, while we should add infinitely many terms to get the exact answer in theory. • Errors depending on the numerical algorithms, step size, and so on. • Overflow/Underflow: Caused by too large or too small numbers to be represented/ stored properly in finite bits. • Negligible Addition: Caused by adding two numbers of magnitudes differ￾ing by over 52 bits, as can be seen in the last section. • Round-off Error: Caused by representing/storing numeric data in finite bits. • Loss of Significance: Caused by a bad subtraction, which means a subtrac￾tion of a number from another one that is almost equal in value. • Error Magnification: Caused and magnified/propagated by multiplying/dividing a number containing a small error by a large/small number. 8