# Calculating Speedup using Amdahl’s Law

Speedup equation is a powerful tool used to calculate the expected speedup of a system when one part is improved. The formula is based on Amdahl’s law and is widely used in the field of computer science to estimate the speedup of parallel algorithms. In this article, we will discuss the speedup equation in detail and provide an example to help you understand how it works.

The speedup equation is expressed as follows:

S = 1 / ((1 – p) + (p / n))

Where S is the speedup, p is the fraction of the program that can be parallelized, and n is the number of processors used. The value of p ranges from 0 to 1, where 0 means that the program cannot be parallelized, and 1 means that the program can be fully parallelized.

The speedup equation is based on Amdahl’s law, which states that the speedup of a system is limited by the proportion of the program that cannot be parallelized. In oher words, if a program cannot be parallelized at all, then the speedup will be limited to 1 (no speedup at all). On the other hand, if a program can be fully parallelized, then the speedup can be theoretically infinite (but practically limited by factors such as communication overhead and load balancing).

The speedup equation takes into account the fraction of the program that can be parallelized (p) and the number of processors used (n). The first term in the denominator, (1 – p), represents the fraction of the program that cannot be parallelized. The second term, (p / n), represents the fraction of the program that can be parallelized but is limited by the number of processors used. The speedup is calculated by dividing 1 by the sum of these two terms.

Example

Let’s take an example to understand how the speedup equation works. Suppose a program spends 75% of its time in iterative computations, and a more robust iteration process is used, which speeds up the computations by a factor of 10. What is the effective speedup?

We can use the speedup equation to calculate the speedup as follows:

P = 0.75 (fraction of program that can be parallelized)
N = 10 (number of processors used)
S = 1 / ((1 – 0.75) + (0.75 / 10))
S = 3.077

Therefore, the effective speedup is 3.077, which means that the program runs 3.077 times faster with the more robust iteration process.

The speedup equation is a powerful tool used to estimate the speedup of a system when one part is improved. It takes into account the fraction of the program that can be parallelized and the number of processors used. By using this equation, programmers can estimate the speedup of their programs before implementing parallelization techniques. However, it is important to note that the speedup is limited by factors such as communication overhead and load balancing, which must be taken into account when designing parallel algorithms.

## What Is Speed Up Formula?

The speedup formula is a mathematical equation used to calculate the efficiency gain of a system after an improvement has been made. It is commonly used in computer science, engineering, and other technical fields to determine the effectiveness of a particular upgrade or optimization. The formula is expressed as follows:

Speedup = Original Execution Time / New Execution Time

In other words, the speedup is calculated by dividing the time it took to complete a task befre the improvement by the time it takes to complete the same task after the improvement. A speedup greater than 1 indicates that the system is performing faster after the improvement, while a speedup less than 1 indicates that the system is performing slower.

It is important to note that the speedup formula assumes that the task being performed is the same before and after the improvement. Additionally, the formula does not take into account any other factors that may affect the overall performance of the system, such as hardware limitations or external constraints.

To summarize, the speedup formula is a useful tool for measuring the efficiency gain of a system after an improvement, and is calculated by dividing the original execution time by the new execution time.

## What Is Speed Up Ratio?

Speedup ratio is a performance metric used in parallel computing to measure the improvement in execution time of a program when run on multiple processors or cores in parallel, as compared to running the program on a single processor or core. Mathematically, speedup ratio is defined as the ratio of the execution time on a single processor to the execution time on multiple processors. In other words, it measures how much faster a program runs when executed in parallel.

Speedup ratio is an important metric for assessing the effectiveness of parallelization, and is often used to evaluate the performance of parallel algorithms and implementations. It is typically denoted by the symbol S, and is calculated using the following formula:

S = T(1) / T(p)

Where T(1) is the execution time on a single processor, and T(p) is the execution time on p processors.

In addition to speedup ratio, parallel efficiency (E) is another commonly used metric for evaluating parallel performance. Parallel efficiency measures the degree to which the parallelization is able to utilize the available resources, and is defined as the ratio of the speedup ratio to the number of processors:

E = S / p

A high speedup ratio and parallel efficiency indicte effective parallelization, while a low speedup ratio and parallel efficiency indicate that the program may not be well-suited for parallel execution, or that the parallel implementation may have inefficiencies or bottlenecks.

## How Do You Calculate Speedup In Amdahl’s Law?

To calculate speedup in Amdahl’s law, you need to use the formula Smax = 1/(1-p) + p/s, where p represents the proportion of the system that is being improved, expressed as a decimal between 0 and 1. The formula calculates the expected speedup of the entire system if one part of it is improved.

To use the formula, you first need to subtract the part of the system that is not being improved from 1. This gives you the fraction of the system that is being improved, whch you can then divide by the speedup of that part after it has been improved.

For example, if you are improving 40% of the system and expect it to be twice as fast after the improvement, you would calculate the speedup as follows:

Smax = 1/(1-0.4) + 0.4/2
Smax = 1.67

This means that you can expect the entire system to be 1.67 times faster after the improvement has been made.

Amdahl’s law formula calculates the expected speedup of the system if one part is improved. It takes into account the proportion of the system being improved and the speedup of that part after the improvement.

## What Is Speed Up Algorithm?

Speedup algorithm is a measure of the efficiency of a parallel algorithm. This metric determines the ratio of the time required by the best sequential algorithm to solve a problem, T(1), to the time required by a parallel algorithm using p processors to solve the same problem, T(p). In other words, speedup algorithm is the improvement in the performance of a parallel algorithm compared to a sequential algorithm.

To put it simply, if a parallel algorithm can solve a problem in less time than the best sequential algorithm for the same problem, then the speedup is greater than one. A speedup of one means that the parallel algorithm is no faster than the sequential algorithm, wile a speedup of two means that the parallel algorithm is twice as fast as the sequential algorithm.

The speedup algorithm is an important measure in the field of parallel computing, as it allows us to evaluate the performance of parallel algorithms and determine the optimal number of processors to use for a given problem. It is also useful for comparing different parallel algorithms to determine which one is more efficient.

Speedup algorithm is a measure of the efficiency of a parallel algorithm and is defined as the ratio of the time required by the best sequential algorithm to solve a problem to the time required by a parallel algorithm using p processors to solve the same problem.

## Conclusion

The speedup equation provides a ueful tool for estimating the potential improvement in computational performance when implementing a parallel algorithm or improving a specific part of a system. The equation takes into account the proportion of time spent on the improved part and the number of processors used to solve the problem. A higher speedup ratio indicates a more significant improvement in performance, while a lower ratio indicates a less effective enhancement. The speedup ratio can also be used to calculate the parallel efficiency of the system, which is a measure of how well the system utilizes the available processors. Therefore, understanding and utilizing the speedup equation is crucial for optimizing computational performance and achieving efficient parallelization.

William Armstrong

William Armstrong is a senior editor with H-O-M-E.org, where he writes on a wide variety of topics. He has also worked as a radio reporter and holds a degree from Moody College of Communication. William was born in Denton, TX and currently resides in Austin.