Renormalize files
This commit is contained in:
@ -1,10 +1,10 @@
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- What was in the exam details PDF posted
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- How to convert decimal to binary
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- No cheat sheet
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- BRING CALCULATOR!
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- Differentiation Methods
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- Which is slower
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- Which is easier to calculate
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- Whichever is more accurate
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- Whichever is easier to modify/add to after
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- What was in the exam details PDF posted
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- How to convert decimal to binary
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- No cheat sheet
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- BRING CALCULATOR!
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- Differentiation Methods
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- Which is slower
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- Which is easier to calculate
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- Whichever is more accurate
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- Whichever is easier to modify/add to after
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-
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@ -1,61 +1,61 @@
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## Convert numbers to binary
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Number in decimal: 53.7
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### Decimal conversion
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| Calculation | Remainder/Binary |
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| ----------- | ---------------- |
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| 53 / 2 = 26 | 1 |
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| 26 / 2 = 13 | 0 |
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| 13 / 2 = 6 | 1 |
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| 6 / 2 = 3 | 0 |
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| 3 / 2 = 1 | 1 |
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| 1 / 2 = 0 | 1 |
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So in binary 110101, as the order is in reverse of the decimals
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### Fraction Conversion
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| Calculation | Non Decimal Portion/Binary |
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| ------------- | -------------------------- |
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| 0.7 x 2 = 1.4 | 1 |
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| 0.4 x 2 = 0.8 | 0 |
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| 0.8 x 2 = 1.6 | 1 |
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| 0.6 x 2 = 1.2 | 1 |
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| 0.2 x 2 = 0.4 | 0 |
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And so on.. so the fraction would be .10110, with 0110 repeating infinitely
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So the final number would be 110101.10110...
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**Normalization** is the process is the process of adjusting a number so only 1 non zero digit on the left side of a number, i.e. the number is in scientific notation
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## Floating point number types
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| Precision | Sign | Exponent | Mantissa |
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| ----------- | ---- | -------- | -------- |
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| single | 1 | 8 | 23 |
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| double | 1 | 11 | 52 |
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| long double | 1 | 15 | 64 |
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Truncation types
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**Chopping**: Omit the numbers that we don't want, looking to the first bit that we want to erase
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**Rounding**: We should take care about the first digit that we want to omit and adjust the 52nd bit
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**Machine Epsilon**: Difference between the smallest floating point number greater than 1 and 1, i.e. the smallest number that when added to 1, will be different than 1
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IEEE rounding to nearest role:
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1. If we have zero in the 53rd bit, we will round down
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2. If we have one as the 53rd bit, we will round up
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1. If the 52nd bit is one we will round up
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2. If the 52nd bit is zero will round down
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Convert a real number to a floating point number:
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1. Decimal to binary number
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2. Justify step: Shift the radix to the right of the left most one, compensate with the exponent. 100.1 -> 1.0001 x_2^3
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3. We do it with respect to p.A d.p 52 numbers ??? I think she means IEEE rounding
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There is no need to represent the first bit of the mantissa, since it is always 1 with certain exceptions
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Overflow - exponent greater than 1023
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Underflow - exponent less than 2^-1074
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Normally set to zero
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Mean value theorem - *Info and graph in slides*
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Taylor's Theorem
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## Convert numbers to binary
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Number in decimal: 53.7
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### Decimal conversion
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| Calculation | Remainder/Binary |
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| ----------- | ---------------- |
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| 53 / 2 = 26 | 1 |
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| 26 / 2 = 13 | 0 |
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| 13 / 2 = 6 | 1 |
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| 6 / 2 = 3 | 0 |
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| 3 / 2 = 1 | 1 |
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| 1 / 2 = 0 | 1 |
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So in binary 110101, as the order is in reverse of the decimals
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### Fraction Conversion
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| Calculation | Non Decimal Portion/Binary |
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| ------------- | -------------------------- |
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| 0.7 x 2 = 1.4 | 1 |
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| 0.4 x 2 = 0.8 | 0 |
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| 0.8 x 2 = 1.6 | 1 |
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| 0.6 x 2 = 1.2 | 1 |
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| 0.2 x 2 = 0.4 | 0 |
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And so on.. so the fraction would be .10110, with 0110 repeating infinitely
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So the final number would be 110101.10110...
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**Normalization** is the process is the process of adjusting a number so only 1 non zero digit on the left side of a number, i.e. the number is in scientific notation
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## Floating point number types
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| Precision | Sign | Exponent | Mantissa |
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| ----------- | ---- | -------- | -------- |
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| single | 1 | 8 | 23 |
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| double | 1 | 11 | 52 |
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| long double | 1 | 15 | 64 |
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Truncation types
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**Chopping**: Omit the numbers that we don't want, looking to the first bit that we want to erase
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**Rounding**: We should take care about the first digit that we want to omit and adjust the 52nd bit
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**Machine Epsilon**: Difference between the smallest floating point number greater than 1 and 1, i.e. the smallest number that when added to 1, will be different than 1
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IEEE rounding to nearest role:
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1. If we have zero in the 53rd bit, we will round down
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2. If we have one as the 53rd bit, we will round up
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1. If the 52nd bit is one we will round up
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2. If the 52nd bit is zero will round down
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Convert a real number to a floating point number:
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1. Decimal to binary number
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2. Justify step: Shift the radix to the right of the left most one, compensate with the exponent. 100.1 -> 1.0001 x_2^3
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3. We do it with respect to p.A d.p 52 numbers ??? I think she means IEEE rounding
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There is no need to represent the first bit of the mantissa, since it is always 1 with certain exceptions
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Overflow - exponent greater than 1023
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Underflow - exponent less than 2^-1074
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Normally set to zero
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Mean value theorem - *Info and graph in slides*
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Taylor's Theorem
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@ -1,16 +1,16 @@
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# Definitions and Concepts
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- Floating point number
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- Convergence, conditioning, complexity
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- Why is it important? We make the numerical methods to approximate the real answer (Convergence). For condition, when an approximation is made, when we make minor modifications to the input, we expect not a lot of variance in the in output. Complexity is the amount of time (CPU?) that a computer will need to use a method (Big O?)
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- Main components of a floating point number?
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-
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- Machine epsilon
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- Loss of significant bit. Truncation
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Methods of root finding
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| Bisection | Final point | Newton | Secant |
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| --------- | ------------- | ------------- | ------------- |
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| Guarentee | Not Guarentee | Not guarentee | Not guarentee |
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| Linear | linear | quatdratic | super linear |
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| Slow | | Fast | Fast |
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# Definitions and Concepts
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- Floating point number
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- Convergence, conditioning, complexity
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- Why is it important? We make the numerical methods to approximate the real answer (Convergence). For condition, when an approximation is made, when we make minor modifications to the input, we expect not a lot of variance in the in output. Complexity is the amount of time (CPU?) that a computer will need to use a method (Big O?)
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- Main components of a floating point number?
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-
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- Machine epsilon
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- Loss of significant bit. Truncation
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Methods of root finding
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| Bisection | Final point | Newton | Secant |
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| --------- | ------------- | ------------- | ------------- |
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| Guarentee | Not Guarentee | Not guarentee | Not guarentee |
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| Linear | linear | quatdratic | super linear |
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| Slow | | Fast | Fast |
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