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# what is considered a small sample size

However, all else being equal, large sized sample leads to increased precision in estimates of various properties of the population. − N Sample size in qualitative research. Ha is true (i.e. SurveyMonkey. the larger the required confidence level, the larger the sample size (given a constant precision requirement). That is, it represents a threshold above which the sample size is no longer considered small. What is an adequate sample size? 6 ∑ = , which yields {\displaystyle \sum {n_{h}}=n} p Rens van de Schoot, Milica Miočević (eds.). Over the years, researchers have grappled with the problem of finding the perfect sample size for statistically sound results. C K = normally distributed random sample with variance unknown. Using G*Power (a sample size and power calculator) a simple linear regression with a medium effect size, an alpha of .05, and a power level of .80 requires a sample size of 55 individuals. It has a “mean”, which is our mean of our sampling distribution. Z A sample size of 32 is quite small so I imagine your confidence intervals (for sensitivity, specificity, predictors of mortality etc.) ( [16][19][20][21] A tool akin to a quantitative power calculation, based on the negative binomial distribution, has been suggested for thematic analysis. However, always remember that the results reported may not be the exact value as numbers are preferably rounded up. The smaller the percentage, the larger your sample size will need to be. = There are many reasons to use stratified sampling:[7] to decrease variances of sample estimates, to use partly non-random methods, or to study strata individually. If you need to compare completion rates, task times, and rating scale data for two independent groups, there are two procedures you can use for small and large sample sizes. h . p 4 For example, if 45% of your survey respondents choose a particular answer and you have a 5% (+/- 5) margin of error, then you can assume that 40%-50% of the entire population will choose the same answer. using a target variance for an estimate to be derived from the sample eventually obtained, i.e. n = = W n The weights, . In considering the effect size of the outcome it is also necessary to ascertain if the â¦ {\displaystyle K} will form a 95% confidence interval for the true proportion. n The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. , where X is the number of 'positive' observations (e.g. For average populations (around 500 people) approx. Alternatively, sample size may be assessed based on the power of a hypothesis test. is a constant such that Determining sample size: how to make sure you get the correct sample size. (1965). If the population is small, and there are enough resources to obtain whatever information you want on the total population, then that is definitely enough â in fact, thatâs the best case scenario. {\displaystyle {\frac {4\times 1.96^{2}\times 15^{2}}{6^{2}}}=96.04} may be used in place of 0.25. In all the calculations presented above, that confidence interval was 95%. But the question remains, why? = Sample Populations vs. Target Populations Conversely, a small population variance means we don't have to take as many samples. Discover how many people you need to send a survey invitation to obtain your required sample. For example, if we are interested in estimating the proportion of the US population who supports a particular presidential candidate, and we want the width of 95% confidence interval to be at most 2 percentage points (0.02), then we would need a sample size of (1.962)/(0.022) = 9604. In this case, our sample average will come from a Normal distribution with mean μ*. Finally, the adjusted range with a specific % confidence will be equal to the mean +/- the range, as calculated above. Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. And in particular, the expectation is that this is going to be a particularly bad estimate when we have a really small sample size. (Note: W/2 = margin of error.). Researchers frequently cite statistician Jacob Cohen, who defined an effect size of +0.20 as âsmall,â +0.50 as âmoderate,â and +0.80 as âstrong.â However, Bloom, Hill, Black, & Lipsey (2008) claim that Cohen never really supported these criteria. p In both cases, however, we are bound by the fact that comparing effectiveness across treatments will probably be best related to the size of the sample (cohort) itself, the closest metric being the utilization factor. Let Xi, i = 1, 2, ..., n be independent observations taken from a normal distribution with unknown mean μ and known variance σ2. During this treatment, the doctors routinely monitor the level of virus in the patient’s blood – a measurement known as viral load – typically in terms of International Units per milliliter (IU/mL). {\displaystyle {\hat {p}}} [1] Using this and the Wald method for the binomial distribution, yields a confidence interval of the form, If we wish to have a confidence interval that is W units total in width (W/2 on each side of the sample mean), we would solve, n . A proportion is a special case of a mean. h An "optimum allocation" is reached when the sampling rates within the strata k The medical clinic has one staff member known â¦ . The right one depends on the type of data you have: continuous or discrete-binary.Comparing Means: If your data is generally continuous (not binary), such as task time or rating scales, use the two sample t-test. k proportional to the standard deviation within each stratum: {\displaystyle \Phi } Knowing that the value of the n is the minimum number of samples needed to acquire the desired result, the number of respondents then must lie on or above the minimum. Sandelowski, M. (1995). Consider two hypotheses, a null hypothesis: for some 'smallest significant difference' μ* > 0. The larger the sample size is the smaller the effect size that can be detected. 2 image created with: Flyer Maker ^ When I was slicing and dicing the data using different criteria such as age, sex, genotype etc., to report effectiveness of treatment, sometimes the sample sizes of these cohorts were becoming too small. ∑ A typical question faced is how much data is considered enough. Sample size calculator. In 1954 Hodges and Lehmann considered the following problem: given is an i.i. p {\displaystyle \sum {n_{h}}=n} This can result from the presence of systematic errors or strong dependence in the data, or if the data follows a heavy-tailed distribution. and W By using this site, you agree to this use. n {\displaystyle n={\frac {4Z^{2}p(1-p)}{W^{2}}}} S Calculating Sample Size To determine a sample size that will provide the most meaningful results, researchers first determine the preferred margin of error (ME) or the maximum amount they want the results to deviate from the statistical mean. Shamanism as Statistical Knowledge: Is a Sample Size of 30 All You Need, The opportunities and challenges of using…, Big Data, Machine Learning and Healthcare –…. Examples can be found in Is 30 the magic number issues in sample size estimation? = Theoretical Case Study: Dangers of Small Sample Size . Surveys. Typically, if there are H such sub-samples (from H different strata) then each of them will have a sample size nh, h = 1, 2, ..., H. These nh must conform to the rule that n1 + n2 + ... + nH = n (i.e. For the calculated values within each category, however, we should be able to report the numbers with a prescribed confidence interval. x We then find the range either from the t-table or Z-score, as mentioned above. using a confidence level, i.e. {\displaystyle C_{h}} All the parameters in the equation are in fact the degrees of freedom of the number of their concepts, and hence, their numbers are subtracted by 1 before insertion into the equation. The problem is that, firstly, according to Andrew Messing of the Center for Brain Science, Harvard University, like a lot of Rule of Thumb commonsense measures, this assumption does not have a solid theoretical basis to prove its veracity. 4 Itâs been shown to be accurate for smalâ¦ ¯ In a recent piece with You Do You titled 'What Is Sample Size?' In complicated studies there may be several different sample sizes: for example, in a stratified survey there would be different sizes for each stratum. Journal of Building Engineering, 1:2–12. if a high precision is required (narrow confidence interval) this translates to a low target variance of the estimator. For example, if we are interested in estimating the amount by which a drug lowers a subject's blood pressure with a 95% confidence interval that is six units wide, and we know that the standard deviation of blood pressure in the population is 15, then the required sample size is Weâve broken the process into 5 steps, allowing you to easily calculate your ideal sample size and ensure accuracy in your surveyâs results. / Our calculator shows you the amount of respondents you need to get statistically significant results for a specific population. Var X ) While researchers generally have a strong idea of the effect size in their planned study it is in determining an appropriate sample size that often leads to an underpowered study. For example, if we wish to know the proportion of a certain species of fish that is infected with a pathogen, we would generally have a more precise estimate of this proportion if we sampled and examined 200 rather than 100 fish. The best rationale I have come across of why this is such a popular number was given by Christopher C. Rout, of the University of KwaZulu-Natal, Department of Anesthetics and Critical Care, Durban, KwaZulu-Natal, South Africa. where h Enter sample size. p Engineering response surface example under. In this specific case, we assumed a T-distribution of our data. {\displaystyle n=\sum n_{h}} Someone cites a stat and then another person says it doesnât matter because the sample size is too small. When the target population is less than approximately 5000, or if the sample size is a significant proportion of the population size, such as 20% or more, then the standard sampling and statistical analysis techniques need to be changed. Overview Population and sample effect sizes. [15][16][17][18], There is a paucity of reliable guidance on estimating sample sizes before starting the research, with a range of suggestions given. / Selecting these nh optimally can be done in various ways, using (for example) Neyman's optimal allocation. For small populations (under 100 persons), the sample size is approximately equal to the population. h : where 2 Ha is true. The graph below shows the results where the sample size was 1,720 patients. [4] The parameters used are: Mead's resource equation is often used for estimating sample sizes of laboratory animals, as well as in many other laboratory experiments. h Z test) to be valid. Funny thing is that there is no formal proof that any of these numbers are useful because they all rely on assumptions that can fail to hold true in one or more ways, and as a result, the adequate sample size cannot be derived using the methods typically taught (and used) in the medical, social, cognitive, and behavioral sciences. A small sample size can also lead to cases of bias, such as non-response, which occurs when some subjects do not have the opportunity to participate in the survey. In both of these cases, it appears that sample size around 30 gives us enough statistical confidence in the results we are presenting. a power of 1 − β), and (2) reject H0 with probability α when H0 is true, then we need the following: If zα is the upper α percentage point of the standard normal distribution, then, is a decision rule which satisfies (2). It may not be as accurate as using other methods in estimating sample size, but gives a hint of what is the appropriate sample size where parameters such as expected standard deviations or expected differences in values between groups are unknown or very hard to estimate.[5]. {\displaystyle Z{\sqrt {\frac {p(1-p)}{n}}}=W/2} {\displaystyle k} Here we shed light on some methods and tools for sample size â¦ p ( [13] One approach is to continue to include further participants or material until saturation is reached. Perhaps you were only able to collect 21 participants, in which case (according to G*Power), that would be enough to find a large effect with a power of .80. Let's look at some fairly simple mathematical model now. For a population of 100,000 this â¦ n − n h Operationalising data saturation for theory-based interview studies. Let’s set the background first. Is 30 the magic number issues in sample size estimation? [8], In general, for H strata, a weighted sample mean is. N = = With a range that large, your small survey isn't saying much. #healthcare, #modeling, #confidence_intervals, #statistical_reporting, #sample_size, #Hepatitis_C, #viral_load, This website uses cookies to improve service and provide tailored ads. Therefore, we require, Through careful manipulation, this can be shown (see Statistical power#Example) to happen when. However if you are doing one sided t test, with confidence level of 99% (alpha = .01), or have a â¦ {\displaystyle n_{h}/N_{h}=kS_{h}} In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. In complicated studies there may be several different sample sizes: for exaâ¦ When the observations are independent, this estimator has a (scaled) binomial distribution (and is also the sample mean of data from a Bernoulli distribution). Bottom line is that if you are doing a 1 sample two sided t test with 95% confidence level (alpha=.05), and small to moderate skewness, n=30 should be ok. W Z n , in the case of using .5 as the most conservative estimate of the proportion. , which would be rounded up to 97, because the obtained value is the minimum sample size, and sample sizes must be integers and must lie on or above the calculated minimum. = 1 2 In practice, since p is unknown, the maximum variance is often used for sample size assessments. 15 It is generally a subjective judgment, taken as the research proceeds. The estimator of a proportion is ∑ {\displaystyle W_{h}} ( S Do qualitative interviews in building energy consumption research produce reliable knowledge? For sufficiently large n, the distribution of If the size of the sample is more than a cut off, say 30, we have used Z-scores; otherwise we have used t-table for calculation. The sample size assessment also depends on HOW the sample was collected? If your population is less than 100 then you really need to survey all of them. h Another factor to consider is the size of your sample; larger samples will tend to be more representative (assuming you are conducting random sampling). One of my domains is healthcare data analytics, a field that is perpetually inundated with data. (I am assuming t-tables and Z-scores are outside the scope of this article.) Otherwise, the formula would be Z Several fundamental facts of mathematical statistics describe this phenomenon, including the law of large numbers and the central limit theorem. p 20%. For example, if a study using laboratory animals is planned with four treatment groups (T=3), with eight animals per group, making 32 animals total (N=31), without any further stratification (B=0), then E would equal 28, which is above the cutoff of 20, indicating that sample size may be a bit too large, and six animals per group might be more appropriate.[6]. Z or Shamanism as Statistical Knowledge: Is a Sample Size of 30 All You Need?, for example. Select Accept cookies to consent to this use or Manage preferences to make your cookie choices. The next graph shows the results where the sample size is 28. W h 2 [14] The number needed to reach saturation has been investigated empirically.  The data I was looking at centers around the treatment of Hepatitis C.  The goal of Hepatitis C therapy is to clear the patient’s blood of the Hepatitis C virus (HCV). W A study that has a sample size which is too small may produce inconclusive results and could also be considered unethical, because exposing human subjects or lab animals to the possible risks associated with research is only justifiable if there is a realistic chance that the study will yield useful information. In practice, the sample size used in a study is usually determined based on the cost, time, or convenience of collecting the data, and the need for it to offer sufficient statistical power. Like so many others before me, this got me thinking. A good maximum sample size is usually 10% as long as it does not exceed 1000 A good maximum sample size is usually around 10% of the population, as long as this does not exceed 1000. ), Now we wish for this to happen with a probability at least 1 − β when If you increase the sample size to â¦ We chose 750 in each group. For example, we may wish to estimate the proportion of residents in a community who are at least 65 years old. within the strata, So a small sample is one that is << 30 A moderate sample is one that is around 30 and a â¦ Alternatively, voluntary response bias occurs when only a small number of non-representative subjects have the opportunity to participate in the survey, usually because they are the only ones who know about it. For example, if a proportion is being estimated, one may wish to have the 95% confidence interval be less than 0.06 units wide. Learn how many responses you need. Sample sizes may be chosen in several ways: Larger sample sizes generally lead to increased precision when estimating unknown parameters. If the p is equal to 0.65, the value of N is 25000 whereas the sample size is 50 then the value of standard deviation of sample proportion is The conditions such as large sample size to represent population and samples must be drawn randomly are included in This page was last edited on 4 December 2020, at 21:03. n can be solved for n, yielding[2][3] n = 4/W2 = 1/B2 where B is the error bound on the estimate, i.e., the estimate is usually given as within ± B. So, if we don't know that, the best thing we can put in there is our sample standard deviation. 2 At about 30 (actually between 32 and 33) this difference becomes less than 0.001, so in a way, the intuitive sense is that at or around that number of the sample size, the difference between samples of larger size may not contribute too much to the probability distribution calculation and a measure of estimated error goes down to acceptable levels. In an article on sample size in qualitative research, a marketing research consultant gives the example of a study conducted on patient satisfaction in a medical clinic. Sample size clothing worn by models on the catwalk tends to vary from a US size 0-4 which equates to a UK size 4-8. So normally what we can do is that we find the estimate of the true standard deviation, and then we can say that the standard deviation of the sampling distribution is equal to the true standard deviation of our population divided by the square root of n, which is the sample size. When estimating the population mean using an independent and identically distributed (iid) sample of size n, where each data value has variance σ2, the standard error of the sample mean is: This expression describes quantitatively how the estimate becomes more precise as the sample size increases. W 2 p A relatively simple situation is estimation of a proportion. = n Glaser, B. If a reasonable estimate for p is known the quantity Francis, J. J., Johnston, M., Robertson, C., Glidewell, L., Entwistle, V., Eccles, M. P., & Grimshaw, J. M. (2010). 2 For larger populations (it is 5000 pers), about 400 pers, but also a sample size of 1% can be significant. 2 The answer is it depends. Healthcare data is often sparse, making reporting results with confidence very challenging. / That is, it represents a threshold above which the sample size is no longer considered small. Products. n These numbers are quoted often in news reports of opinion polls and other sample surveys. Letâs say you do your research and find out your population of shark biologists are 80% women. h For example, for a population of 10,000 your sample size will be 370 for confidence level 95% and margin of erro 5%. This is where the trade-offs usually occur. (This is a 1-tailed test. Galvin R (2015). is the normal cumulative distribution function. In a census, data is sought for an entire population, hence the intended sample size is equal to the population. First and f oremost, we need to know what comprises the total population. , or, more generally, when, Sample size determination in qualitative studies takes a different approach. 1 It may have to do with the difference between the square roots of 1/n and 1/(n-1). / For education surveys, we recommend getting a statistically significant sample size that represents the population.If youâre planning on making changes in your school based on feedback from students about the institution, instructors, teachers, etc., a statistically significant sample size will help you get results to lead your school to success. 1 For two means, width of the 95% confidence interval for the difference = ±1.96Ïâ(2/n).If we put n = 740, we can calculate this for the chosen sample size: ±1.96Ïâ(2/750) = ±0.10Ï.This was thought to be ample for cost data and any other continuous variables. For â¦ 2 {\displaystyle {\hat {p}}=X/n} The margin of error in this case is 1 percentage point (half of 0.02). As in statistical estimation, the true effect size is distinguished from the observed effect size, e.g. You can change your cookie choices and withdraw your consent in your settings at any time. she modelled sample size clothing as a way to highlight the ridiculousness of the size of the clothing. For more information, see our Cookie Policy. Small sample size and SPSS. The constant comparative method of qualitative analysis. For a fixed sample size, that is When it comes to surveys in particular, sample size more precisely refers to the number of completed responses that a survey receives. This article looks at some "Rule of Thumb" guidelines and tries to verify it with real data. × But it also has fatter tails. h Should I test this rule of thumb and see if there is any truth to it? The reverse is also true; small sample sizes can detect large effect sizes. 2 A sample size that is too small increases the likelihood of a Type II error skewing the results, which decreases the power of the study. It has been a fairly well known assumption in Statistics that a sample size of 30 is a so-called magic number in estimating distribution or statistical errors. So, for B = 10% one requires n = 100, for B = 5% one needs n = 400, for B = 3% the requirement approximates to n = 1000, while for B = 1% a sample size of n = 10000 is required. With more complicated sampling techniques, such as stratified sampling, the sample can often be split up into sub-samples. {\displaystyle W_{h}=N_{h}/N} . − Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. ( You could then make sure that 80% of your sample consists of women, such as by quota sampling. that the total sample size is given by the sum of the sub-sample sizes). 2020. It is reasonable to use the 0.5 estimate for p in this case because the presidential races are often close to 50/50, and it is also prudent to use a conservative estimate. ^ This is the smallest value for which we care about observing a difference. A useful, partly non-random method would be to sample individuals where easily accessible, but, where not, sample clusters to save travel costs. If this interval needs to be no more than W units wide, the equation. How many interviews are enough? T-distribution is almost engineered so it gives a better estimate of our confidence intervals especially since we have a small sample size. ) 96.04 Practicality: Of course the sample size you select must make sense. ) {\displaystyle S_{h}={\sqrt {\operatorname {Var} ({\bar {x}}_{h})}}} is a constant such that , frequently, but not always, represent the proportions of the population elements in the strata, and h In some situations, the increase in precision for larger sample sizes is minimal, or even non-existent. Not be the exact value as numbers are preferably rounded up before,. 500 people ) approx it doesnât matter because the sample size is the Normal cumulative function.: given is an i.i simple situation is estimation of a mean n_ { h } } that total., and some textbooks give alternative magic numbers of 50 or 20 cookie choices should be to... % of your sample size ( given a constant precision requirement ) include further participants or material until saturation reached..., sample size determination is the Normal cumulative distribution function often sparse, reporting. Smallest value for which we care about observing a difference ”, which occurs when the true effect is. Question faced is how much data is sought for an estimate to be well approximated by Gaussian... Different treatment groups, there may be different sample sizes can detect effect... Of our sampling distribution precision requirement ) know what comprises the total population be derived from the presence systematic! Law of large numbers and the central limit theorem required confidence level, the maximum variance this. The calculated values within each category, however, all else being equal, large sized leads! Size was 1,720 patients no exact sample size clothing as a way highlight. To get statistically significant results for a specific % confidence will be equal the! Do n't know that, the increase in precision for larger sample sizes is minimal, even! So, if we do n't know that, the adjusted range with a probability at least −... Null hypothesis: for some 'smallest significant difference ' μ * > 0 =.! Your sample consists of women, such as stratified sampling, the true proportion is n't saying much requirement.... Or material until saturation is reached a special case of what is considered a small sample size mean the maximum variance this! It appears that sample size â¦ Enter sample size is no longer support the Internet Explorer 11 browser this especially... With you do your research and find out your population of 100,000 this â¦ the size of resulting..., the maximum variance of the sample was collected finding the perfect size! We then find the range either from the presence of systematic errors or strong in. Positive integer data follows a heavy-tailed distribution to obtain your required sample sizes for each group statistics this. For some 'smallest significant difference ' μ * statistical sample US size 0-4 which equates to a size... A Gaussian your required sample sizes is minimal, or we seldom know the true standard deviation sized sample to... Below shows the results we are presenting sizes ) groups, there may be in... Our mean of our confidence intervals with the difference between the square of. By the quality of the n sampled people what is considered a small sample size are at least 1 − β when Ha true... Your cookie choices and withdraw your consent in your settings at any time news of. Which is our mean of our sampling distribution here and it can vary in different settings! To it sampling techniques, such as stratified sampling, the sample size â¦ Enter size! Is a special case of a mean needed to reach saturation has been investigated empirically obtain your required sample may... Be able to report the numbers with a range that large, your small is! Sampling distribution how the sample size arbitrary, and some textbooks give alternative numbers... At 21:03 approach is to continue to include in a survey receives small sample size seldom know the true.! Distribution is 0.25n, which is our mean of our data our sampling distribution from. Find the range either from the observed effect size is typically denoted by n and it can in. Hypothesis test two hypotheses, a weighted sample mean is distribution is 0.25n, which occurs when the effect. To increased precision when estimating unknown parameters my sample set before I confidently. Is minimal, or we seldom know the true effect size, that n. There is our sample average will come from a Normal distribution with mean μ * > 0 should I this... It comes to surveys in particular, sample size may be assessed on... Of various properties of the clothing gives a better estimate of our confidence intervals especially since we a! Is n't saying much ( Note: W/2 = margin of error. ) cites. Sample set before I can confidently report that result problem of finding the perfect sample size,. Us enough statistical confidence in the results reported may not be the exact as.  rule of thumb '' guidelines and tries to verify it with real data different treatment groups, there be. Is, it appears that sample size â¦ Enter sample size of 30 all you need be! A US size 0-4 which equates to a low target variance for an entire population hence!: Flyer Maker the larger the sample size is distinguished from the t-table or Z-score, as calculated.... The proportion of residents in a census, data is often sparse making... Methods and tools for sample size â¦ Enter sample size was 1,720 patients +/- the range either from the or... Φ { \displaystyle \Phi } is the act of choosing the number of out. If we do n't know that, the true standard deviation women, such by... = ∑ n h { \displaystyle \Phi } is the Normal cumulative distribution function with a specific % interval... Ender rather than a conversation ender rather than a conversation starter can your. Sure you get the correct sample size? chosen in several ways: larger sizes. To continue to include further participants or material until saturation is reached intervals with the of. Sample can often be split up into sub-samples the true effect size that can be (! That can be detected rather than a conversation ender rather than a conversation starter statistics and usually. Catwalk tends to vary from a Normal distribution with mean μ * what is considered a small sample size mathematical... As in statistical estimation, the larger the sample size assessments data or... Two hypotheses, a null hypothesis: for some 'smallest significant difference μ... Outside the scope of this article. ) taken as the research proceeds the years when about. As the research proceeds both of these cases, it appears that sample size may evaluated. Textbooks give alternative magic numbers of 50 or 20 we require, Through careful manipulation, this be. At 21:03 to continue to include further participants or material until saturation is reached letâs say you you. = 0.5 being equal, large sized sample leads to increased precision in estimates of various properties the.


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 Solution Find (x^3-12*x^2-42)*(x^2+x-3)Solution:Your problem -> (x^3-12*x^2-42)*(x^2+x-3)(x3-12x2-42) (x2+x-3) = x3(x2+x-3) -12x2(x2+x-3) -42(x2+x-3) = x5 + x4 - 3x3 - 12x4 - 12x3 + 36x2 - 42x2 - 42x + 126 = x5 + x4 - 3x3 - 12x4 - 12x3 + 36x2 - 42x2 - 42x + 126 = x5 + x4 - 12x4 - 3x3 - 12x3 + 36x2 - 42x2 - 42x + 126 = x5 - 11x4 - 15x3 - 6x2 - 42x + 126 Solution provided by AtoZmath.com Any wrong solution, solution improvement, feedback then Submit Here Want to know about AtoZmath.com and me

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My Math Forum radius of convergence with indeterminate form

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 January 10th, 2017, 08:05 AM #1 Member   Joined: Jan 2017 From: California Posts: 62 Thanks: 5 radius of convergence with indeterminate form 20170110_080156.jpg The limit of this ratio test for this series suggests that for any x except zero the series diverge. The thing I cannot get over is the following: What if I choose an x that is so infinitely close to zero such as it dominates infinity? Such as that the product of the two factors is less than 1. I believe this was the case of an indeterminate form when L'Hôpital's rule was necessary in limits calculations. Thanks in advance. Last edited by skipjack; January 10th, 2017 at 08:46 AM.
January 10th, 2017, 08:57 AM   #2
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 Originally Posted by dthiaw Attachment 8341 The limit of this ratio test for this series suggests that for any x except zero the series diverge. The thing I cannot get over is the following: What if I choose an x that is so infinitely close to zero such as it dominates infinity?
Where did you get the idea that such a number exists? x is a fixed real number. It sounds like you are talking about x being an "infinitesimal" or you are taking a limit. Neither of those is true for x being a fixed real number.

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 Such as that the product of the two factors is less than 1. I believe this was the case of an indeterminate form when L'Hôpital's rule was necessary in limits calculations. Thanks in advance.

 January 10th, 2017, 09:02 AM #3 Member   Joined: Jan 2017 From: California Posts: 62 Thanks: 5 What if I choose an x so small such as 0.00000.... with the 1 all the way to the north pole? Can I do that? I assume I could since the value of x could be continuous and not discrete. Couldn't that small number overpower infinity (0 vs infinity) type of fight? Last edited by skipjack; January 10th, 2017 at 10:50 AM.
 January 10th, 2017, 10:01 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,778 Thanks: 2195 Math Focus: Mainly analysis and algebra Two approaches to put you on the right trackWhat can you say about the sequence $a_n=cn$ where $c \ne 0$? In particular, what can you say about the expression $a_{k+1}-a_k$? How many zeroes does it take to get to the 1 to the North Pole? A million? A billion? A googol? Let's say that the answer is $m$ zeros. $m$ is clearly a finite number. And $0.000\ldots01$, with $m$ zeroes (including the one to the left of the decimal point) is equal to $\frac1{10^m}$. What is the limit $$\lim_{n \to \infty} \frac{n}{10^m}$$ Alternatively, what does it mean in terms of the definition of the limit for $$\lim_{n \to \infty} \frac{n}{10^m}=1$$ Thanks from dthiaw
January 10th, 2017, 10:25 AM   #5
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 Originally Posted by dthiaw What if I choose an x so small such as 0.00000.... with the 1 all the way to the north pole? Can I do that? I assume I could since the value of x could be continuous and not discrete. Couldn't that small number overpower infinity (0 vs infinity) type of fight?
Can you quantify the number of "0"s required to go "all the way to the north pole"? You can say a very large, but specific, number of "0"s. However, that would not "overpower" infinity. Any number "times infinity" is "infinity". I put "times infinity" and "infinity" in quotes because "infinity" is NOT a real number and multiplication of any real number with "infinity" is not defined.

Last edited by skipjack; January 10th, 2017 at 10:52 AM.

January 10th, 2017, 11:21 AM   #6
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 Originally Posted by Country Boy Can you quantify the number of "0"s required to go "all the way to the north pole"? You can say a very large, but specific, number of "0"s. However, that would not "overpower" infinity. Any number "times infinity" is "infinity". I put "times infinity" and "infinity" in quotes because "infinity" is NOT a real number and multiplication of any real number with "infinity" is not defined.
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 Originally Posted by v8archie Two approaches to put you on the right trackWhat can you say about the sequence $a_n=cn$ where $c \ne 0$? In particular, what can you say about the expression $a_{k+1}-a_k$? How many zeros does it take to get to the 1 to the North Pole? A million? A billion? A googol? Let's say that the answer is $m$ zeros. $m$ is clearly a finite number. And $0.000\ldots01$, with $m$ zeros (including the one to the left of the decimal point) is equal to $\frac1{10^m}$. What is the limit $$\lim_{n \to \infty} \frac{n}{10^m}$$ Alternatively, what does it mean in terms of the definition of the limit for $$\lim_{n \to \infty} \frac{n}{10^m}=1$$
Ok I understand what you guys are saying. (that any finite number no matter how small it is multiplied by infinity = infinity)

However, there was an indeterminate case that forced the use of L'Hôpital's in limits calculation. It was stating that 0+ multiplied by infinity is not equal to infinity and that's why we had to rearrange things and present them in the form 0/0 or infinity over infinity. I feel like this is contradicting the stuff we're discussing.

Last edited by skipjack; January 10th, 2017 at 11:29 AM.

 January 10th, 2017, 12:06 PM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,778 Thanks: 2195 Math Focus: Mainly analysis and algebra If $f(x) \to 0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x)$$ is the indeterminate case $0 \cdot \infty$. This is the case you are thinking of. But that is because the limit of $f(x)$ is equal to zero, not because $f$ is very small. But if $f(x)=c \ne 0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x)$$ is the undefined (but not indeterminate) case $c \cdot \infty = \pm\infty$. And if $f(x)=0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x) = \lim_{x \to a} 0 \cdot g(x) = \lim_{x \to a} 0 = 0$$ Thanks from dthiaw
January 10th, 2017, 12:28 PM   #8
Member

Joined: Jan 2017
From: California

Posts: 62
Thanks: 5

Quote:
 Originally Posted by v8archie If $f(x) \to 0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x)$$ is the indeterminate case $0 \cdot \infty$. This is the case you are thinking of. But that is because the limit of $f(x)$ is equal to zero, not because $f$ is very small. But if $f(x)=c \ne 0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x)$$ is the undefined (but not indeterminate) case $c \cdot \infty = \pm\infty$. And if $f(x)=0$ and $g(x) \to \infty$ then $$\lim_{x \to a} f(x)g(x) = \lim_{x \to a} 0 \cdot g(x) = \lim_{x \to a} 0 = 0$$
Thank you so much V8archie. it s finally clear. You are the man

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# It is a peculiar result. 191/7 = 27.2857142857142857. Why???

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It is a peculiar result. 191/7 = 27.2857142857142857. "285714" repeats in the result. Is this not peculiar? What is this called as in maths??

Guest Aug 20, 2017
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This is called a $$"Repeating\space Decimal"$$, just like how the results are in fractional loops, the cause of it is because the remainder always cease to disappear in divisions, like how grass never go away no matter how many times you cut it.

In your case, $$\frac{191}{7}=27.285714285714...$$

The number $$\frac{191}{7}$$ can also be written as $$27+\frac{2}{7}$$

Since we know that $$\frac{1}{7}=0.142857142857...$$

The number  $$\frac{2}{7}=2\cdot\frac{1}{7}=0.285714285714...$$

Adding the previously deattached $$27$$

$$\frac{191}{7}$$ can be written in decimal form as: $$27.285714285714...$$

(Source: https://en.wikipedia.org/wiki/Repeating_decimal)

Hope this helps :)

Jeffes02  Aug 20, 2017


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# Why does a tree have latex?

It's the sap found naturally within the tree. Consider it to be the blood of the tree if you will. However latex is only found in some species of trees.
Thanks for the feedback!

# Where does Latex come from?

Latex comes from the Brazilian Rubber Tree milk..... Laytex is made from latex (rubber tree milk) from which proteins, sugars and water solubles are removed.
Thanks for the feedback!

# What is latex?

  Latex is natural ruber such as you may find condoms are made of or ruber gloves.
Thanks for the feedback!

# Where do you get latex?

i can get latex from many online stores! Latex is a number of things, including a kind of paint, a naturally occurring substance, and a kind of type set. See the related lin (MORE)

matex,hatex

# How do you get your latex skirt on?

If you are having trouble pulling it over your skin, try powdering  your skin or the inside of your skirt. It will help it slide over  easier.
Thanks for the feedback!

# What is latex balloon?

Latex balloon is a balloon made up of a rubbery type of material from all types of angiosperms called latex.33423
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In Uncategorized

# Is a wristband latex?

There are probably some types of wristbands that do contain latex. I did a quick search and they have latex wristbands that match latex clothing that you find in adult novelty (MORE)

# DO balloons have latex in them?

Yes, balloons are made of latex. Correction: Most balloons contain latex. However, some do not as some people are allergic to latex.
Thanks for the feedback!

# What is a latex?

In the context of birth control "a latex" is a condom made of latex and intended to cover the penis and prevent sperm from traveling into the female.
Thanks for the feedback!


text:
# Angular momentum of a system about the center of mass

Let $$\mathbf{R}$$ be the center of mass of a system of particles. Then the angular momentum of the system is \begin{align} \mathbf{L} &= \sum \mathbf{r}_i \times \mathbf{p}_i\\\\ &= \sum \left(\mathbf{R} + \mathbf{r}_i^\prime\right) \times \mathbf{p}_i\\\\ &= \mathbf{R} \times \sum \mathbf{p}_i + \sum \mathbf{r}_i^\prime \times \mathbf{p}_i\\\\ &= \mathbf{R} \times \mathbf{P} + \sum \mathbf{r}_i^\prime \times \mathbf{p}_i, \end{align}

yet Goldstein claims that $$\mathbf{L} = \mathbf{R} \times \mathbf{P} + \sum \mathbf{r}_i^\prime \times \mathbf{p}_i^\prime,\tag{1.28}$$ which is identical to what I derived with the exception that $$\mathbf{p}_i$$ gets replaced with $$\mathbf{p}_i^\prime.$$ How does one reconcile these similar yet different expressions for $$\mathbf{L}$$?

• You did not define p in your expression, m*v, relative to some coordinate system. v = Vcm + v', then you have more terms in your derivation. – ggcg Nov 12 '18 at 11:24
• To add to what @ggcg is saying, $p_i=p_{cm}+p'_i$ – Aaron Stevens Nov 12 '18 at 11:48

Note that $$\boldsymbol p'_i:=\boldsymbol p_i-m_i\dot{\boldsymbol R}$$. Therefore, $$\sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i-\sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i^\prime\equiv\sum_im_i\boldsymbol r'_i\times\dot{\boldsymbol R}$$ which is easily seen to vanish, inasmuch as $$\sum_im_i\boldsymbol r'_i\equiv0$$ by definition.

Thus, the expression in the OP and the one in Goldstein are identical.

AccidentalFourierTransform's answer is correct and complete. I would add that there really are two sensible ways of defining "the angular momentum about a moving point". It can mean the instantaneous angular momentum about that point in the lab frame, or the instantaneous angular momentum about that point in the point's frame. Your result is written in the first way, and Goldstein's is written in the second. Both notions are useful in different contexts.

Of course, the two coincide when the point is the center of mass, so there's nothing to worry about.

The answer by AccidentalFourierTransform is incorrect in that Goldstein doesn't define $$\sum_i m_i {\bf r}'_i = 0$$. Between 1.27 (which states: $$\bf{r_i}=\bf{R}+\bf{r'_i}$$) and 1.28 there is an argument (he doesn't put everything in symbols!),

[T]he factor $$\sum m_i \bf r'_i \bf$$, which, it will be recognized, defines the radius vector of the center of mass in the very coordinate system whose origin is the center of mass and is therefore a null vector.

but it isn't by definition that the sum vanishes, as some say. The straight forward proof of it is just a smidgen of algebra along with the definition of center of mass a few might have glossed over. Using double index as a implicit sum: $$m_i {\bf r}_i = M \bf R$$

Now, multiply through 1.27 by $$m_i$$, and use the definition of $$\bf R$$: $$m_i {\bf r}_i = m_i {\bf R} + m_i {\bf r}'_i$$ There, that's why $$m_i {\bf r}'_i = 0$$.


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# rocblas build wiki¶

## Home¶

### Building rocBLAS¶

1. For instructions to build rocblas library and clients, see Build rocBLAS libraries and verification code.

2. For an example using rocBLAS see Example C code calling rocBLAS function.

3. For instructions on how to run/use the client code, see Build rocBLAS libraries, verification-code, tests and benchmarks.

### Functionality¶

rocBLAS exports the following BLAS-like functions at this time.

### Rules for obtaining the rocBLAS API from Legacy BLAS¶

1. The Legacy BLAS routine name is changed to lower case, and prefixed by rocblas_.

2. A first argument rocblas_handle handle is added to all rocBlas functions.

3. Input arguments are declared with the const modifier.

4. Character arguments are replaced with enumerated types defined in rocblas_types.h. They are passed by value on the host.

5. Array arguments are passed by reference on the device.

6. Scalar arguments are passed by value on the host with the following two exceptions:

• Scalar values alpha and beta are passed by reference on either the host or the device. The rocBLAS functions will check to see it the value is on the device. If this is true, it is used, else the value on the host is used.

• Where Legacy BLAS functions have return values, the return value is instead added as the last function argument. It is returned by reference on either the host or the device. The rocBLAS functions will check to see it the value is on the device. If this is true, it is used, else the value is returned on the host. This applies to the following functions: xDOT, xDOTU, xNRM2, xASUM, IxAMAX, IxAMIN.

1. The return value of all functions is rocblas_status, defined in rocblas_types.h. It is used to check for errors.

### Additional notes¶

• The rocBLAS library is LP64, so rocblas_int arguments are 32 bit and rocblas_long arguments are 64 bit.

• rocBLAS uses column-major storage for 2D arrays, and 1 based indexing for the functions xMAX and xMIN. This is the same as Legacy BLAS and cuBLAS. If you need row-major and 0 based indexing (used in C language arrays) download the CBLAS file cblas.tgz. Look at the CBLAS functions that provide a thin interface to Legacy BLAS. They convert from row-major, 0 based, to column-major, 1 based. This is done by swapping the order of function arguments. It is not necessary to transpose matrices.

• The auxiliary functions rocblas_set_pointer and rocblas_get_pointer are used to set and get the value of the state variable rocblas_pointer_mode. This variable is not used, it is added for compatibility with cuBLAS. rocBLAS will check if your scalar argument passed by reference is on the device. If this is true it will pass by reference on the device, else it passes by reference on the host.

## Build¶

Download the master branch of rocBLAS from github using:

git clone -b master https://github.com/ROCmSoftwarePlatform/rocBLAS.git
cd rocBLAS

Note if you want to contribute to rocBLAS, you will need the develop branch, not the master branch, and you will need to read .github/CONTRIBUTING.md.

Below are steps to build either (dependencies + library) or (dependencies + library + client). You only need (dependencies + library) if you call rocBLAS from your code, or if you need to install rocBLAS for other users. The client contains the test code and examples.

It is recommended that the script install.sh be used to build rocBLAS. If you need individual commands, they are also given.

Common uses of install.sh to build (library dependencies + library) are in the table below.

The client contains executables in the table below.

executable name

description

rocblas-test

runs Google Tests to test the library

rocblas-bench

executable to benchmark or test individual functions

example-sscal

example C code calling rocblas_sscal function

Common uses of install.sh to build (dependencies + library + client) are in the table below.

install.sh_command

description

./install.sh -h

Help information.

./install.sh -dc

Build library dependencies, client dependencies, library, and client in your local directory. The -d flag only needs to be used once. For subsequent invocations of install.sh it is not necessary to rebuild the dependencies.

./install.sh -c

Build library and client in your local directory. It is assumed the dependencies have been built.

./install.sh -idc

Build library dependencies, client dependencies, library, client, then build and install the rocBLAS package. You will be prompted for sudo access. It is expected that if you want to install for all users you use the -i flag. If you want to keep rocBLAS in your local directory, you do not need the -i flag.

./install.sh -ic

Build and install rocBLAS package, and build the client. You will be prompted for sudo access. This will install for all users. If you want to keep rocBLAS in your local directory, you do not need the -i flag.

Before building the library please install the library dependencies CMake, Python 2.7, and Python-yaml.

CMake 3.5 or later

The build infrastructure for rocBLAS is based on Cmake v3.5. This is the version of cmake available on ROCm supported platforms. If you are on a headless machine without the x-windows system, we recommend using ccmake; if you have access to X-windows, we recommend using cmake-gui.

Install one-liners cmake:

• Ubuntu: sudo apt install cmake-qt-gui

• Fedora: sudo dnf install cmake-gui

Python 2.7

By default both python2 and python3 are on Ubuntu. You can check the installation with python -V. Python is used in Tensile, and Tensile is part of rocBLAS. To build rocBLAS the default version of Python must be Python 2.7, not Python 3.

Python-yaml

PyYAML files contain training information from Tensile that is used to build gemm kernels in rocBLAS.

Install one-liners PyYAML:

• Ubuntu: sudo apt install python2.7 python-yaml

• Fedora: sudo dnf install python PyYAML

Build library

The rocBLAS library contains both host and device code, so the HCC compiler must be specified during cmake configuration to properly initialize build tools. Example steps to build rocBLAS:

# after downloading and changing to rocblas directory:
mkdir -p build/release
cd build/release
# Default install path is in /opt/rocm, use -DCMAKE_INSTALL_PREFIX=<path> to specify other install path
# Default build config is 'Release', define -DCMAKE_BUILD_TYPE=Debug to specify Debug configuration
CXX=/opt/rocm/bin/hcc cmake ../..
make -j$(nproc) #if you want to install in /opt/rocm or the directory set in cmake with -DCMAKE_INSTALL_PREFIX sudo make install # sudo required if installing into system directory such as /opt/rocm  Additional dependencies for the rocBLAS clients The unit tests and benchmarking applications in the client introduce the following dependencies: 1. boost 2. fortran 3. lapack • lapack itself brings a dependency on a fortran compiler 4. googletest boost Linux distros typically have an easy installation mechanism for boost through the native package manager. Ubuntu: sudo apt install libboost-program-options-dev Fedora: sudo dnf install boost-program-options  Unfortunately, googletest and lapack are not as easy to install. Many distros do not provide a googletest package with pre-compiled libraries, and the lapack packages do not have the necessary cmake config files for cmake to configure linking the cblas library. rocBLAS provide a cmake script that builds the above dependencies from source. This is an optional step; users can provide their own builds of these dependencies and help cmake find them by setting the CMAKE_PREFIX_PATH definition. The following is a sequence of steps to build dependencies and install them to the cmake default /usr/local. gfortran and lapack LAPACK is used in the client to test rocBLAS. LAPACK is a Fortran Library, so gfortran is required for building the client. Ubuntu apt-get update apt-get install gfortran Fedora yum install gcc-gfortran mkdir -p build/release/deps cd build/release/deps cmake -DBUILD_BOOST=OFF ../../deps # assuming boost is installed through package manager as above make -j$(nproc) install

Once dependencies are available on the system, it is possible to configure the clients to build. This requires a few extra cmake flags to the library cmake configure script. If the dependencies are not installed into system defaults (like /usr/local ), you should pass the CMAKE_PREFIX_PATH to cmake to help find them.

-DCMAKE_PREFIX_PATH="<semicolon separated paths>"

# after downloading and changing to rocblas directory:
mkdir -p build/release
cd build/release
# Default install location is in /opt/rocm, use -DCMAKE_INSTALL_PREFIX=<path> to specify other
CXX=/opt/rocm/bin/hcc cmake -DBUILD_CLIENTS_TESTS=ON -DBUILD_CLIENTS_BENCHMARKS=ON -DBUILD_CLIENTS_SAMPLES=ON ../..
make -j$(nproc) sudo make install # sudo required if installing into system directory such as /opt/rocm  The rocBLAS library uses Tensile, which supplies the high-performance implementation of xGEMM. Tensile is downloaded by cmake during library configuration and automatically configured as part of the build, so no further action is required by the user to set it up. rocBLAS is written with HiP kernels, so it should build and run on CUDA platforms. However, currently the cmake infrastructure is broken with a CUDA backend. However, a BLAS marshalling library that presents a common interface for both ROCm and CUDA backends can be found with hipBLAS. • Issue: “Tensile could not be found because dependency Python Interp could not be found”. Solution: Due to a bug in Tensile, you may need cmake-gui 3.5 and above, though in the cmakefiles it requires 2.8. • Issue: HIP (/opt/rocm/hip) was built using hcc 1.0.xxx-xxx-xxx-xxx, but you are using /opt/rocm/hcc/hcc with version 1.0.yyy-yyy-yyy-yyy from hipcc. (version does not match) . Please rebuild HIP including cmake or update HCC_HOME variable. Solution: Download HIP from github and use hcc to build from source and then use the build HIP instead of /opt/rocm/hip one or singly overwrite the new build HIP to this location. • Issue: For Carrizo - HCC RUNTIME ERROR: Fail to find compatible kernel Solution: Add the following to the cmake command when configuring: -DCMAKE_CXX_FLAGS=”–amdgpu-target=gfx801” • Issue: For MI25 (Vega10 Server) - HCC RUNTIME ERROR: Fail to find compatible kernel Solution: export HCC_AMDGPU_TARGET=gfx900 • Issue: Could not find a package configuration file provided by “ROCM” with any of the following names: ROCMConfig.cmake rocm-config.cmake Solution: Install ROCm cmake modules. ## Example¶ #include <stdlib.h> #include <stdio.h> #include <vector> #include <math.h> #include "rocblas.h" using namespace std; int main() { rocblas_int N = 10240; float alpha = 10.0; vector<float> hx(N); vector<float> hz(N); float* dx; float tolerance = 0, error; rocblas_handle handle; rocblas_create_handle(&handle); // allocate memory on device hipMalloc(&dx, N * sizeof(float)); // Initial Data on CPU, srand(1); for( int i = 0; i < N; ++i ) { hx[i] = rand() % 10 + 1; //generate a integer number between [1, 10] } // save a copy in hz hz = hx; hipMemcpy(dx, hx.data(), sizeof(float) * N, hipMemcpyHostToDevice); rocblas_sscal(handle, N, &alpha, dx, 1); // copy output from device memory to host memory hipMemcpy(hx.data(), dx, sizeof(float) * N, hipMemcpyDeviceToHost); // verify rocblas_scal result for(rocblas_int i=0;i<N;i++) { error = fabs(hz[i] * alpha - hx[i]); if(error > tolerance) { printf("error in element %d: CPU=%f, GPU=%f ", i, hz[i] * alpha, hx[i]); break; } } if(error > tolerance) { printf("SCAL Failed !\n"); } else { printf("SCAL Success !\n"); } hipFree(dx); rocblas_destroy_handle(handle); return 0; }  Paste the above code into the file rocblas_sscal_example.cpp Use hipcc Compiler: The recommend host compiler is hipcc. To use hipcc you will need to add /opt/rocm/bin to your path with the following: export PATH=$PATH:/opt/rocm/bin

The following makefile can be used to build the executable.

The Makefile assumes that rocBLAS is installed in the default location /opt/rocm/rocblas. If you have rocBLAS installed in your home directory in ~/rocBLAS/build/release/rocblas-install/rocblas then edit Makefile and change /opt/rocm/rocblas to ~/rocBLAS/build/release/rocblas-install/rocblas.

You may need to give the location of the library with

export LD_LIBRARY_PATH=/opt/rocm/rocblas/lib${LD_LIBRARY_PATH:+:$LD_LIBRARY_PATH}

Run the executable with the command

./rocblas_sscal_example

# Makefile assumes rocBLAS is installed in /opt/rocm/rocblas

ROCBLAS_INSTALL_DIR=/opt/rocm/rocblas
ROCBLAS_INCLUDE=$(ROCBLAS_INSTALL_DIR)/include ROCBLAS_LIB_PATH=$(ROCBLAS_INSTALL_DIR)/lib
ROCBLAS_LIB=rocblas
HIP_INCLUDE=/opt/rocm/hip/include
LDFLAGS=-L$(ROCBLAS_LIB_PATH) -l$(ROCBLAS_LIB)
LD=hipcc
CFLAGS=-I$(ROCBLAS_INCLUDE) -I$(HIP_INCLUDE)
CPP=hipcc
OBJ=rocblas_sscal_example.o
EXE=rocblas_sscal_example

%.o: %.cpp
$(CPP) -c -o$@ $<$(CFLAGS)

$(EXE) :$(OBJ)
$(LD)$(OBJ) $(LDFLAGS) -o$@

clean:
rm -f $(EXE)$(OBJ)

Use g++ Compiler:

Use the Makefile below:

ROCBLAS_INSTALL_DIR=/opt/rocm/rocblas
ROCBLAS_INCLUDE=$(ROCBLAS_INSTALL_DIR)/include ROCBLAS_LIB_PATH=$(ROCBLAS_INSTALL_DIR)/lib
ROCBLAS_LIB=rocblas
ROCM_INCLUDE=/opt/rocm/include
LDFLAGS=-L$(ROCBLAS_LIB_PATH) -l$(ROCBLAS_LIB) -L/opt/rocm/lib -lhip_hcc
LD=g++
CFLAGS=-I$(ROCBLAS_INCLUDE) -I$(ROCM_INCLUDE) -D__HIP_PLATFORM_HCC__
CPP=g++
OBJ=rocblas_sscal_example.o
EXE=rocblas_sscal_example

%.o: %.cpp
$(CPP) -c -o$@ $<$(CFLAGS)

$(EXE) :$(OBJ)
$(LD)$(OBJ) $(LDFLAGS) -o$@

clean:
rm -f $(EXE)$(OBJ)

## Running¶

This wiki describes running the examples, tests, and benchmarks in the client. Before reading this Wiki, it is assumed rocBLAS (dependencies + library + client) has been built as described in Build

The default for [BUILD_DIR] is ~/rocblas/build.

cd [BUILD_DIR]/release/clients/staging
./example-sscal
./example-scal-template
./example-sgemm
./example-sgemm-strided-batched

Code for the examples is at: samples

In addition see Example

Run tests with the following:

cd [BUILD_DIR]/release/clients/staging ./rocblas-test

To run specific tests, use –gtest_filter=match where match is a ‘:’-separated list of wildcard patterns (called the positive patterns) optionally followed by a ‘-‘ and another ‘:’-separated pattern list (called the negative patterns). For example, run gemv tests with the following:

cd [BUILD_DIR]/release/clients/staging ./rocblas-test –gtest_filter=*checkin*gemm*float*-batched:NaN

Run bencharmks with the following:

cd [BUILD_DIR]/release/clients/staging ./rocblas-bench -h

The following are examples for running particular gemm and gemv benchmark:

./rocblas-bench -f gemm -r s -m 1024 -n 1024 -k 1024 --transposeB T -v 1
./rocblas-bench -f gemv -m 9216 -n 9216 --lda 9216 --transposeA T

## Exported functions¶

### rocBLAS includes the following auxiliary functions¶

 Function name rocblas_create_handle rocblas_destroy_handle rocblas_add_stream rocblas_set_stream rocblas_get_stream rocblas_set_pointer_mode rocblas_get_pointer_mode rocblas_set_vector rocblas_get_vector rocblas_set_matrix rocblas_get_matrix

### rocBLAS includes the following Level 1, 2, and 3 functions¶

Level 1

 Function single double single complex double complex half rocblas_Xscal x x x x rocblas_Xcopy x x x x rocblas_Xdot x x x x rocblas_Xswap x x x x rocblas_Xaxpy x x x x x rocblas_Xasum x x x x rocblas_Xnrm2 x x x x rocblas_Xamax x x x x rocblas_Xamin x x x x

Level 2

 Function single double single complex double complex half rocblas_Xgemv x x x x rocblas_Xger x x x x

Level 3

 Function single double single complex double complex half rocblas_Xgemm x x rocblas_Xtrtri x x rocblas_Xtrtri_batched x x rocblas_Xtrsm x x

#### Rules for obtaining the rocBLAS API from Legacy BLAS¶

• The Legacy BLAS routine name is changed to lower case, and prefixed by rocblas_.

• A first argument rocblas_handle handle is added to all rocBlas functions.

• Input arguments are declared with the const modifier.

• Character arguments are replaced with enumerated types defined in rocblas_types.h. They are passed by value on the host.

• Array arguments are passed by reference on the device.

• Scalar arguments are passed by value on the host with the following two exceptions:

• Scalar values alpha and beta are passed by reference on either the host or the device. The rocBLAS functions will check to see it the value is on the device. If this is true, it is used, else the value on the host is used.

• Where Legacy BLAS functions have return values, the return value is instead added as the last function argument. It is returned by reference on either the host or the device. The rocBLAS functions will check to see it the value is on the device. If this is true, it is used, else the value is returned on the host. This applies to the following functions: xDOT, xDOTU, xNRM2, xASUM, IxAMAX, IxAMIN.

• The return value of all functions is rocblas_status, defined in rocblas_types.h. It is used to check for errors.

#### LP64 interface¶

The rocBLAS library is LP64, so rocblas_int arguments are 32 bit and rocblas_long arguments are 64 bit.

#### Column-major storage and 1 based indexing¶

rocBLAS uses column-major storage for 2D arrays, and 1 based indexing for the functions xMAX and xMIN. This is the same as Legacy BLAS and cuBLAS.

If you need row-major and 0 based indexing (used in C language arrays) download the CBLAS file cblas.tgz. Look at the CBLAS functions that provide a thin interface to Legacy BLAS. They convert from row-major, 0 based, to column-major, 1 based. This is done by swapping the order of function arguments. It is not necessary to transpose matrices.

#### Pointer mode¶

The auxiliary functions rocblas_set_pointer and rocblas_get_pointer are used to set and get the value of the state variable rocblas_pointer_mode. If rocblas_pointer_mode == rocblas_pointer_mode_host then scalar parameters must be allocated on the host. If rocblas_pointer_mode == rocblas_pointer_mode_device, then scalar parameters must be allocated on the device.

There are two types of scalar parameter:

scaling parameters like alpha and beta used in functions like axpy, gemv, gemm scalar results from functions amax, amin, asum, dot, nrm2

For scalar parameters like alpha and beta when rocblas_pointer_mode == rocblas_pointer_mode_host they can be allocated on the host heap or stack. The kernel launch is asynchronous, and if they are on the heap they can be freed after the return from the kernel launch. When rocblas_pointer_mode == rocblas_pointer_mode_device they must not be changed till the kernel completes.

For scalar results, when rocblas_pointer_mode == rocblas_pointer_mode_host then the function blocks the CPU till the GPU has copied the result back to the host. When rocblas_pointer_mode == rocblas_pointer_mode_device the function will return after the asynchronous launch. Similarly to vector and matrix results, the scalar result is only available when the kernel has completed execution.

#### Asynchronous API¶

Except a functions having memory allocation inside preventing asynchronicity, most of the rocBLAS functions are configured to operate in asynchronous fashion with respect to CPU, meaning these library functions return immediately

## Logging¶

Four environment variables can be set to control logging:

• ROCBLAS_LAYER

• ROCBLAS_LOG_TRACE_PATH

• ROCBLAS_LOG_BENCH_PATH

• ROCBLAS_LOG_PROFILE_PATH

ROCBLAS_LAYER is a bitwise OR of zero or more bit masks as follows:

• If ROCBLAS_LAYER is not set, then there is no logging

• If (ROCBLAS_LAYER & 1) != 0, then there is trace logging

• If (ROCBLAS_LAYER & 2) != 0, then there is bench logging

• If (ROCBLAS_LAYER & 4) != 0, then there is profile logging

Trace logging outputs a line each time a rocBLAS function is called. The line contains the function name and the values of arguments.

Bench logging outputs a line each time a rocBLAS function is called. The line can be used with the executable rocblas-bench to call the function with the same arguments.

Profile logging, at the end of program execution, outputs a YAML description of each rocBLAS function called, the values of its arguments, and the number of times it was called with those arguments.

The default stream for logging output is standard error. Three environment variables can set the full path name for a log file:

• ROCBLAS_LOG_TRACE_PATH sets the full path name for trace logging

• ROCBLAS_LOG_BENCH_PATH sets the full path name for bench logging

• ROCBLAS_LOG_PROFILE_PATH sets the full path name for profile logging

If a path name cannot be opened, then the corresponding logging output is streamed to standard error.

Note that performance will degrade when logging is enabled.

When profile logging is enabled, memory usage will increase. If the program exits abnormally, then it is possible that profile logging will not be outputted before the program exits.

## Train Tensile for rocBLAS¶

Below are 10 steps that can be used to build Tensile and rocBLAS for the sizes specified in rocblas_sgemm_asm_miopen.yaml

git clone -b develop https://github.com/ROCmSoftwarePlatform/Tensile.git
cd Tensile
mkdir build
cd build
time python ../Tensile/Tensile.py ../Tensile/Configs/rocblas_sgemm_asm_miopen.yaml ./ 2>&1 | tee Tensile.out
cd ../..
git clone -b develop https://github.com/ROCmSoftwarePlatform/rocBLAS.git
cd rocBLAS
cp ../Tensile/build/3_LibraryLogic/*.yaml library/src/blas3/Tensile/Logic/asm_miopen
./install.sh -idc -l asm_miopen

Notes on each step

• Clones the develop branch of Tensile

• Change to Tensile directory

• Make a directory for building Tensile

• Change to build directory

• Time the build of Tensile, and echo output to Tensile.out. This builds Tensile for the kernel parameters and sizes specified in rocblas_sgemm_asm_miopen.yaml. The output are the .yaml files in 3_LibraryLogic. These output files contain all the information rocBLAS needs to build the assembly files for sgemm.

• Change directory

• Clone the develop branch of rocBLAS

• Change directory

• Copy the .yaml files for rocblas_sgemm_asm_miopen.yaml from Tensile to rocBLAS. These files contain the information to build the sgemm assembly kernels in rocBLAS

• Build and install rocBLAS using the specifications in the .yaml files in library/src/blas3/Tensile/Logic/asm_miopen. The install.sh flag d installs dependencies. You only need this the first time you run install.sh. For subsequent runs use ./install.sh -ic -l asm_miopen.

## Device and Stream management in rocBLAS¶

### HIP Device management¶

hipSetDevice() & hipGetDevice() are HIP device management APIs. They are NOT part of the rocBLAS API.

Before a HIP kernel invocation, users need to call hipSetDevice() to set a device, e.g. device 1. If users do not explicitly call it, the system by default sets it as device 0. Unless users explicitly call hipSetDevice() to set to another device, their HIP kernels are always launched on device 0.

The above is a HIP (and CUDA) device management approach and has nothing to do with rocBLAS. rocBLAS honors the approach above and assumes users have already set the device before a rocBLAS routine call.

Once users set the device, they create a handle with rocblas_status rocblas_create_handle(rocblas_handle *handle)

Subsequent rocBLAS routines take this handle as an input parameter. rocBLAS ONLY queries (by hipGetDevice) the user’s device; rocBLAS but does NOT set the device for users. If rocBLAS does not see a valid device, it returns an error message to users. It is the users’ responsibility to provide a valid device to rocBLAS and ensure the device safety as explained soon.

Users CANNOT switch devices between rocblas_create_handle() and rocblas_destroy_handle() (the same as cuBLAS requires). If users want to change device, they must destroy the current handle, and create another rocBLAS handle (context).

### Stream management¶

HIP kernels are always launched in a queue (otherwise known as a stream, they are the same thing).

If users do not explicitly specify a stream, the system provides a default stream, maintained by the system. Users cannot create or destroy the default stream. Howevers, users can freely create new streams (with hipStreamCreate) and bind it to the rocBLAS handle: rocblas_set_stream(rocblas_handle handle, hipStream_t stream_id) HIP kernels are invoked in rocBLAS routines. The rocBLAS handles are always associated with a stream, and rocBLAS passes its stream to the kernels inside the routine. One rocBLAS routine only takes one stream in a single invocation. If users create a stream, they are responsible for destroying it.

### Multiple streams and multiple devices¶

If the system under test has 4 HIP devices, users can run 4 rocBLAS handles (also known as contexts) on 4 devices concurrently, but can NOT span a single rocBLAS handle on 4 discrete devices. Each handle is associated with a particular singular device, and a new handle should be created for each additional device.

## Numerical Stability in TRSM¶

### Division¶

Most BLAS routines like GEMM, GEMV, GER only perform multiplication and add. As long as there is no overflow (e.g. very big number), the GPU always achieve bit-wise consistence with the CPU results. However, the TRSM and TRTRI routine perform division. Therefore, there are some precision differences between Netlib CBLAS (which we compare to) and rocBLAS.

For example, the gtest may falsely report such failure by using gtest macro ASSERT_FLOAT_EQ in STRSM:

Expected: hCPU[i+j*lda] Which is: -0.66039091

To be equal to: hGPU[i+j*lda] Which is: -0.66039127

However, since we achieve 5 digits consistency (0.66039), we think rocBLAS result is correct. Therefore, we use gtest ASSERT_NEAR in TRSM to replace ASSERT_FLOAT_EQ which is used for other routines.

### Well-conditioned Matrix¶

Generally, TRTRI and TRSM has matrix inversion involved (accessing half of the matrix). Matrix inversion generally is not numerical stable operation, especially when the matrix is [ill-conditioned] (https://en.wikipedia.org/wiki/Condition_number). A random matrix is very ill-conditioned typically. Such matrix will cause serious overflow.

In order to generate a well-conditioned matrix, we perform an LU factorization first on this random matrix. The factorized Lower and upper part overwrite the original matrix. The factorized matrix is a well-conditioned matrix and used as the input matrix in rocBLAS TRSM test.

## Profile rocBLAS kernels¶

### By environment variable¶

• In bash: “export HIP_TRACE_API=1” (reset by =0) Launch your application, then it profiles every HIP APIs, including rocBLAS kernels.

• “export HIP_LAUNCH_BLOCKING = 0”: make HIP APIs host-synchronous so they are blocked until any kernel launches or data-copy commands are complete (an alias is CUDA_LAUNCH_BLOCKING)

• For more profiling tools, see Profiling and Debugging HIP Code

The IR and ISA can be dumped by setting the following environment variable before building and running the app.

export KMDUMPISA=1

export KMDUMPLLVM=1

export KMDUMPDIR=/path/to/dump

#### By roprof¶

a tool very similar to nvprof, roprof is a command line tool to profile HIP kernels, roprof is located in /opt/rocm/profiler/bin

example usage

/opt/rocm/profiler/bin/rcprof -T -a profile.atp ./your_executable

it will dump several a bunch of profile.HSA*.html files, you can view it by any internet browser.

/opt/rocm/profiler/bin/rcprof --help for more options


text:
# show that $x_{-} < \frac{1}{2} < x_{+}$

If we have the equation $$f(x,y) = \left(\begin{matrix}x^2-y^2+\alpha\\2xy+\beta\end{matrix}\right)$$ where $\alpha$ and $\beta$ are constants. We wish to solve the equation $f(\boldsymbol{x}) = \boldsymbol{x}$

Show for all $\alpha$ and $\beta$ that it exists two solutions $\boldsymbol{x}_{\pm}$ that are like: $$\boldsymbol{x}_{\pm} = \left(\begin{matrix}x_{\pm}\\y_{\pm}\end{matrix}\right), x_{-} < \frac{1}{2} < x_{+}, y_{\pm} = \frac{\beta}{1-2x_{\pm}}$$

I managed to show that $y_{\pm} = \frac{\beta}{1-2x_{\pm}}$ but I don't understand how I can show that $x_{-} < \frac{1}{2} < x_{+}$

Hint: Take $z=x+iy$ and $w=\alpha + i\beta$ then you get $f = (Re(z^2 + w) , Im(z^2+w))$. Finally the equation becomes

$$z^2 + w = z\to z_{\pm}=\frac{1\pm \sqrt{1-4w}}{2}.$$

Now substitute for $z_{\pm}$ and $w$ and decompose into real and imaginary parts. Notice that in square root you need to write polar representation of complex numbers, i.e. $a+ib \equiv R \exp[i\theta]$.

$\textbf{Proof of$x_- <0.5<x_+$}$

You can write $$\sqrt{1-w}= \sqrt{1-4\alpha -4i\beta }= \sqrt{R e^{i\theta}}=\sqrt{R} \exp\left(\frac{i\theta}{2}\right) , \quad 0 \le \theta < \pi$$ Notice that $\theta$ is the principal argument and is in the above interval. We don't need the exact form of $R$ and $\theta$, but notice that for all $\alpha,\beta\in \mathbb{R}$, then $R \ge 0$. Now roots are

$$z_{\pm}= x_{\pm} + iy_{\pm}=\frac{1}{2} \pm \frac{\sqrt{R}}{2}\exp\left(\frac{i\theta}{2}\right)$$ therefore

$$x_{\pm} = \frac{1}{2} \pm \frac{\sqrt{R}}{2}\cos\left(\frac{\theta}{2}\right)$$

Now for $0 \le \theta < \pi$ you have $0\le \theta/2 < \pi/2$ and in this interval $0 < \cos(\theta/2) \le 1$ therefore the second term is a positive constant that implies $x_+ > 1/2$ and $x_-< 1/2$.

• Can I just ask how exactly that proves: $x_{-} < \frac{1}{2} < x_{+}$? It might just be me, but I can't quite see how this would work. May 8, 2017 at 13:34
• I edit the answer to include it. May 8, 2017 at 14:02
• I see now, thank you so much May 8, 2017 at 14:21


text:
# How can I find the determinant of the conjugate matrix?

#### ptolema

1. The problem statement, all variables and given/known data

2. Relevant equations

complex conjugate of a+bi is a-bi

3. The attempt at a solution

I defined M = A+Bi, where A and B contain real number entries. So that means that $$\bar{}M$$ = A-Bi. Past that point, I don't know what to do. How can I find the determinant of the conjugate matrix? Or the complex conjugate of det (M)? Could someone give me a hand?

Last edited:
Related Calculus and Beyond Homework News on Phys.org

#### I like Serena

Homework Helper
Re: determinant

I'd say that the determinant of a matrix is a bunch of additions on products of complex numbers which make up the matrix.

Note that when you multiply or add 2 conjugate numbers, the result is the same when you multiply or add the original numbers and then take the conjugate.

So the determinant of a conjugated matrix has to be the same as the conjugate of the determinant of a matrix.

#### spamiam

Re: determinant

I think using induction and then doing a cofactor expansion would work, though I haven't checked the details.

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text:
# Thread: Integer roots over Z_p

1. ## Integer roots over Z_p

So for awhile we've been talking about finding polynomial ladders. This is an interesting topic since a given ladder of degree k, has $\displaystyle 2^k$ roots, and yet the polynomial itself can be calculated in only k multiplications (or k squaring). Unfortunately, it doesn't appear that polynomial ladders can be used to create any kind of fast Integer factoring algorithm (and this is really what inspired the original question in the Crandall/Pomerance book), since it is likely that we cannot find such polynomials for k larger than 4.

On the positive side though, it is worth noting that for the purpose of integer factoring, one doesn't necessarily need to have a polynomial with a large # of integer roots, but rather just has many roots over $\displaystyle Z_p$ (where p could be the unknown prime factor we wish to find).

For example, consider the polynomial $\displaystyle x^{60} - 1$. This polynomial has only two roots, namely $\displaystyle \pm 1$. On the other hand, over $\displaystyle Z_p$, this polynomial can have up to and including 60 roots.

For example, when $\displaystyle p = 61$, $\displaystyle x^{60} - 1$ has exactly 60 roots over Z_61 by Fermat's test. As a result, if we wish to factor an integer like $\displaystyle 5429$ ( $\displaystyle = 61*89$), an obvious choice would be to use the polynomial $\displaystyle x^{60} - 1$.

Another advantage of working over Z_p, is that we can even use polynomials that have negative exponents. For example, we can calculate integer values of a polynomial like $\displaystyle x^2 + x^{-1} + 1$ over $\displaystyle Z_p$.

Anyway, since were working at squaring ladders at the moment, the first question I'd like to pose is this; Find a polynomial squaring ladder with k = 5, such that this polynomial has exactly $\displaystyle 2^5$ roots over the prime number $\displaystyle Z_{97}$.

2. I've been thinking about this problem somewhat. I realize its actually probably better to not consider using a polynomial ladder at all in this case (at least not if we wish to optimize the factoring algorithm). Generally, what we want is a polynomial (or map) function $\displaystyle f$ from $\displaystyle Z_n$ to $\displaystyle Z_n$ such that the range of $\displaystyle f$ is much smaller than its domain.

For example, if we use the function $\displaystyle f(x) = x^{lcm(1,M)} - 1 mod(n)$, then for large M, this will have a small range regardless of what $\displaystyle Z_n$ is.

A factoring algorithm which is actually a combination of the Pollard Strassen Polynomial Evaluation Method (PSPEM) and the
P - 1 Method (both of these factoring algorithms can be found in chapter 5 of Pomerance book Prime Numbers: A Computational Perspective) can be done as follows:

Consider the polynomial map $\displaystyle P(x;a) = (a^dx - 1).....(a^2x - 1)(ax - 1)$ where $\displaystyle a = 2^{lcm(1,M)} mod(n)$ and evaluate it for values of $\displaystyle x = k^i$, where $\displaystyle k = a^{d} mod(n)$, for consecutive values of $\displaystyle i$ starting at 1.

The polynomial $\displaystyle P(x)$ would take the same amount of time to compute as any other polynomial (and we'd use Horner's Scheme to evaluate it most likely). However the value of $\displaystyle i$ for which $\displaystyle gcd(P(k^i),n) = p$, maybe fairly small if $\displaystyle p - 1$ (where p is the unknown factor of n) is such that $\displaystyle L = gcd(p-1,lcm(1,M))$ is large, as $\displaystyle p-1 / L$ would be much smaller.

There are of course other maps from $\displaystyle Z_n$ to $\displaystyle Z_n$ which have small range (for example, consider $\displaystyle f(x) = (x^2 + x^{-2} + 1)^2 mod(n))$, however in general it is hard to find a function that has uniformly small ranges over arbitrary $\displaystyle Z_n$, that aren't also increasingly more time consuming to compute. If we could find a polynomial ladder for very large k, this would be good, since we would know that many values of x in the domain would go to 0.

3. ## Clarifying a point

I should probably clarify something with regards to the above post.

Any function $\displaystyle f$ from $\displaystyle Z_n$ to $\displaystyle Z_n$ must satisfy the condition that for any divisor $\displaystyle d$ of $\displaystyle n$, we have that if $\displaystyle u \equiv v mod(d)$, then $\displaystyle f(u) \equiv f(v) mod(d)$, where $\displaystyle f$ is our function from $\displaystyle Z_n$ to $\displaystyle Z_n$.

The reason the above must hold is because otherwise we cannot use this map to obtain a factorization of $\displaystyle n$, regardless of how small the range of $\displaystyle f$ is over $\displaystyle Z_n$.

The only type of functions $\displaystyle f$ that satisfy the above are polynomials (note that we can include polynomials which have negative exponents).

Another type of function we should avoid is the constant function $\displaystyle f(x) = c$. This satisfies the above condition, but clearly we cannot utilize it to obtain any information regarding the factors of $\displaystyle n$.


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# Featured member - Alan Enrique Ontiveros Salazar

We will be featuring different members of the Brilliant community, so that you can get to know them better. For the tenth issue, we are featuring Alan Enrique Ontiveros Salazar, who despite only discovering his love of mathematics late, has developed greatly in the past 3 years.

Alan has posted more than 50 problems, and he has produced numerous gems. Two of this problems that I like ask you to solve a system of equations. They are Tricky system and Newton's Sum, and you can view them here:

$\begin{array} { l l c | c l l } a^2 + b^2 + \sqrt{3} ab & = 727 & \quad \quad & \quad \quad & x + y + z & = 3 \\ a^2 + c^2 + \sqrt{3} ac & = 829 & & & x^3 + y^3 + z^3 & = 15 \\ b^2 + c^2 - bc & = 39 & & & x^4 + y^4 + z^4 & = 35 \\ \end{array} \$/extract_itex] He has posted nearly 200 solutions (oh wow!). I enjoy reading them, as they often offer a deeper insight into understanding the problem, and present the motivation for considering such an approach. Check out Incremental Complex Products, where you have to solve: $\begin{array} { l l l } (a+1)(b+1)(c+1) & = & 1 \\ (a+2)(b+2)(c+2) & = & 2\\ (a+3)(b+3)(c+3) & = & 3\\ (a+4)(b+4)(c+4) & = & ? \end{array}$ Tell us more about yourself. Well, I am just another guy who loves math and all its applications. I'm 16 now, almost 17. Apart from math, I play tennis and go swimming. I am a little introvert, even though I always have a topic to talk about. I speak Spanish and English, which I'm learning principally through my activity on Brilliant. I learnt to program as a child, when I thought that I broke my computer and tried to fix it. The games that I used to play didn't open, and in the process of poking around, I reached the point where I was doing small programs in HTML and Visual Basic, and have grown to love programming. I'm from Mexico, one of the most interesting and rich countries around the world. Unfortunately, most people here hate maths. Everything I know about maths I learnt since 2011. Before then, I didn't know anything about it. I realized that I had a powerful tool to solve other types of problems, and my love for mathematics developed from there. Since then, I've been challenging myself at algebra, then at geometry, and more recently in Calculus. At the start, I used the internet to solve my doubts. When I joined Brilliant, I got to know many fantastic people, and I learnt how to present my solutions formally (and even improve my English). What would you recommend for someone who is new to learning math? First, get rid of stereotypes that people believe about math. It is not true that only certain people can be good at math, or that math is very difficult and useless. Second, practice math as a way to exercise your brain and get smarter in many aspects of daily life. If you focus only on "What is it for? How am I gonna use it in my life?", then you will not learn enough for it to become useful. Third, Math is not about memorization of complicated and long formulas. Unfortunately, the way that schools teach math is the reason why most people hate math. Math is actually very beautiful, and you should explore it for fun. Fourth, if you want to improve, try problems that are slightly more difficult than you are used to. Be creative, explore new ways and ask around for help. Also, you can read articles on the internet, and find communities (like Brilliant) where people share their love for math. Finally, do not be intimidated by people who can solve very hard problems that seem impossible to you. They spent their time developing their skills, and you do not know how much effort they have put in. Even I don't know many things, and everyone has to go through this process of learning. What were the greatest lessons that you learnt? I have 7 notebooks which record down various work that I have done, and what I have learnt from it. Let me share some of what challenged me (at that point in time). 1. When I first saw the general formula for the quadratics, I tried to do the same with cubics. I had no success initially, so I looked it up on the internet. I saw a very ugly formula, and then looked at the process of how to solve it, and I finally found a better method / solution. I next did it with quartics, and was again successful, which made me happy. I then tried it with quintics (without knowing that it was impossible), and obviously had no success. Then, I found a theorem and started reading about it. 2. I like to make short algorithms for simple tasks. One day, when I was studying a way to interpolate a polynomial given its initial values, I discovered an algorithm that could do this! It took me one afternoon to code it up. After I joined Brilliant, I learnt that it was the Method of Differences, and was very proud of myself. 3. It requires a lot of hard work to improve. You have to be willing to struggle and think for yourself, and even accept it when things do not go well at times. Constantly put in the effort to improve, and you will develop over time. What do you want to accomplish in the next few years? I am in 12th grade, which is the 5th semester of Preparatoria in Mexico. My school is Instituto Politécnico Nacional (IPN), one of the best high schools in Mexico. I'm currently studying a technical speciality about computer networks, in CET 1, which is one of the 17 high schools of IPN. In one year I'll be finishing high school. Then, I'll go to ESCOM (also a school of IPN) to study computer engineering. That will last 4 years, and I plan to find a job while studying. After that, I'd like to create my own company about technology, and innovate in some aspects. For now I've developed some projects, like a website that is like Wolfram Alpha (but less advanced), and a music searcher. What do you wish for Brilliant? First I'd like that it would be also in Spanish, because most people who doesn't speak English but likes math wouldn't be able to participate here, in this great community. Second, I hope that this community grows up everyday faster, because, for example, I was looking for a website around 2013 where I could find harder challenges about math, but I didn't find Brilliant until December 2013. Finally, I just want to thank you all for making this place a great community where I can talk with people with the same interests of me, learn from others and teach what I know. Note by Calvin Lin 5 years, 6 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or \[ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

@Alan Enrique Ontiveros Salazar Congrats for being featured! You are probably the one of the guys having the longest names on Brilliant (may be in the world too! :P). Your methods , solutions are quite innovative and I love them. Its quite awesome that inspite of having a math-hating background , you managed to progress so much in maths! Finally I got to see the face of such wonderful talent! :)

- 5 years, 6 months ago

Oh, thanks :) I also appreciate your working here, in a short time you have contributed with many wonderful problems and solutions, and I wish I could be more active in this days, because of school. Also, the names here in Mexico are as long as mine, or even more :D

- 5 years, 6 months ago

You seems to be fond of watching different faces..with the irony you have setup dp of someone else..!!

- 5 years, 6 months ago

Lol , you want to see my face?

- 5 years, 6 months ago

I want you to be featured

- 5 years, 6 months ago

Yeah I wanna see your face!

- 5 years, 6 months ago

Your resonate with me a lot. I too found my love for math pretty late. I also grew up surrounded by practical-minded people, meaning that they only care about whether something they learn would be used in the future(I actually have a lot to say about this topic, but I don't want to spew such a rant here).

- 5 years, 6 months ago

I can let you rant in a "Featured Member" post. Interested?

Staff - 5 years, 6 months ago

Yes, that is very common in my country. It is not bad to try to find practical uses for every thing we learn, but if we don't find that utility, we simply don't learn it.

- 5 years, 6 months ago

Congratulation!!!!!

- 5 years, 6 months ago

Yeah waited for Alan to be featured!

- 5 years, 6 months ago

Congrats @Alan Enrique Ontiveros Salazar !!

I am also surrounded with many "practical minded" people.... In my school maths is taught just for scoring marks.....

- 5 years, 6 months ago

So true. Personally, I don't like the way that school teaches maths I hope it elucidates like Brilliant or Khan Academy

- 2 years, 9 months ago

Hi, Alan ! You're so great in the photo . Great to see you featured !

- 5 years, 6 months ago

Congrats for being featured ... When I read your name I do remember Salazar Slytherin :P I love your solutions . You posses a Beautiful Mind.....

- 5 years, 6 months ago

Congratulations Alan for being featured

- 5 years, 6 months ago

@Alan Enrique Ontiveros Salazar , Hello sir and congrats on being featured, first of all I thought you were a kid about my age! but its really good to have a nearly adult like you on brilliant, so everyone could understand.

- 5 years, 6 months ago

- 2 years, 9 months ago

Congratulations on being featured! :D

- 5 years, 6 months ago

Congratulations

- 5 years, 6 months ago

@Alan Enrique Ontiveros Salazar I was waiting for many days for you to get featured. You solved most of my Level 5 Problems. You are awesome!! Congrats for being featured.

- 5 years, 6 months ago

Congratulations!

- 5 years, 6 months ago

Congrats ! for being featured and yes does your name mean anything in particular?

- 5 years, 5 months ago

Congrats @Alan Enrique Ontiveros Salazar

You are amazing and a great inspiration to me and the rest...

- 2 years, 9 months ago

Alan, soy de ESCOM gracias por estas palabras que dices =), es hermoso lo que has escrito para mortales como yo. Gracias señor DIOS Alan.

- 1 year, 7 months ago


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# Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev

$$\sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}}$$

$\sigma$ and $\mu$ being the "normal" Stdev and Mean, respectively. (Both in absolute values, resulting from a simulation of the pay-offs.)

Since it is about real options, it sounds reasonable to have the volatility decrease approaching the execution date of a project, but why design the volatility like this? I have plotted the function here via Wolframalpha.com. Even though the volatility should be somewhere around 10% in this example, it never assumes that value. Why does that make sense?

I've run a simulation and compared the values. Since the volatility changes significantly, the option value changes, of course, are significant.

Here some equivalent expressions. Maybe it reminds somebody of something that might help?

$\Longleftrightarrow t\sigma'^{2}=ln(1+\frac{\sigma}{\mu})^{2}$

$\Longleftrightarrow\sqrt{exp(t\sigma'^{2})}-1=\frac{\sigma}{\mu}$

$\Longleftrightarrow\sigma=\mu\left[\sqrt{exp(t\sigma'^{2})}-1\right]$

It somehow looks similar to the arithmetic moments of the log-normal distribution, but it does not fit 100%.

-
real options... have they started to take into account that one is short one's competitor's options ? –  nicolas May 22 '11 at 18:52


text:
Lemma 29.14.8. The following properties of ring maps are stable under base change.

1. (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$.

2. (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi (f) \not\in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_ f \to A_ f$.

3. Add more here as needed1.

Proof. Omitted. $\square$

[1] But only those properties that are not already dealt with separately elsewhere.

Comment #2246 by JuanPablo on

Something seems to be wrong with 3 and 4.

On 3 maybe $A\otimes_R \kappa(\mathfrak p)$ is meant to be geometrically reduced over $\kappa(\mathfrak p)$?.

Because one can take $R=K$ a field of characteristic $p>0$, $A=L=K(\alpha)$ field extension obtaining from adjoining a $p$ root of an element in $K$ ($\alpha^p=a\in K$) and $R'=L$, so that $L\otimes_K L=L[x]/(x^p-a)=L[x]/(x-\alpha)^p$ is not reduced.

On 4 one can take $n=0$, $R=K$ a field and $R'=A=K(x)$ field of rational functions, so that $A\otimes_K R'=S^{-1}K[x,y]$, with $S$ elements of the form $f(x)g(y)$, which is a domain but not a field (so has Krull dimension $\geq 1$)

Comment #2281 by on

OMG!!! Yes, terrible! Thanks very much. The change is here.

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text:
# VCI2022 - The 16th Vienna Conference on Instrumentation

Feb 21 – 25, 2022
Vienna University of Technology
Europe/Vienna timezone

## A 4D diamond detector for HL-LHC and beyond

Feb 25, 2022, 10:35 AM
20m
Vienna University of Technology

#### Vienna University of Technology

Gusshausstraße 27-29, 1040 Wien
Live Presentation Semiconductor Detectors

### Speaker

Michele Veltri (Universita e INFN, Firenze (IT))

### Description

The unprecedented density of charged particles foreseen at the next generation of experiments at future hadronic machines poses a significant challenge to the tracking detectors, that are expected to stand extreme levels of radiation as well as to be able to efficiently reconstruct a huge number of tracks and primary vertices. To meet this challenge new extremely radiation hard materials and sensor designs will be needed, to build high granularity and excellent time resolution tracking detectors. In particular, the availability of the time coordinate ("4D-tracking") significantly simplifies the track and vertex reconstruction problem. Diamond 3D pixel sensors, with thin columnar resistive electrodes orthogonal to the surface,specifically optimised for timing applications may provide an optimal solution to the above problems.The 3D geometry enhances the well known radiation hardness of diamond and allows to exploit its excellent timing properties, possibly improving the performances of the extensively studied planar diamond sensors.
We report on the timing characterization, based on beta-source and particle beam tests, of innovative 3D diamond detectors optimised for timing applications, fabricated by laser graphitisation of conductive electrodes in the bulk of 500 $\mu$m thick single-crystal diamonds. Preliminary results on the simulation of the full chain of signal formation in the sensor will also be presented and plans for further optimisation briefly discussed.

Primary experiment TIMESPOT

### Primary authors

Chiara Corsi (LENS, Firenze (IT)) Chiara Lucarelli (Universita e INFN, Firenze (IT)) Giovanni Passaleva (INFN Florence (IT)) Lucio Anderlini (Universita e INFN, Firenze (IT)) Marco Bellini (INO and CNR, Firenze (IT)) Michele Veltri (Universita e INFN, Firenze (IT)) Silvio Sciortino (Universita e INFN, Firenze (IT)) Stefano Lagomarsino (INFN, Firenze (IT))