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Research Papers

Leibniz Rule and Fractional Derivatives of Power Functions

[+] Author and Article Information
Vasily E. Tarasov

Skobeltsyn Institute of Nuclear Physics,
Lomonosov Moscow State University,
Moscow 119991, Russia
e-mail: tarasov@theory.sinp.msu.ru

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 14, 2015; final manuscript received August 17, 2015; published online October 23, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(3), 031014 (Oct 23, 2015) (4 pages) Paper No: CND-15-1215; doi: 10.1115/1.4031364 History: Received July 14, 2015; Revised August 17, 2015

Abstract

In this paper, we prove that unviolated simple Leibniz rule and equation for fractional-order derivative of power function cannot hold together for derivatives of orders $α≠1$. To prove this statement, we use an algebraic approach, where special form of fractional-order derivatives is not applied.

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References

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Ben Adda, F. , and Cresson, J. , 2001, “ About Non-Differentiable Functions,” J. Math. Anal. Appl., 263(2), pp. 721–737.
Liu, C.-S. , 2015, “ Counterexamples on Jumarie's Two Basic Fractional Calculus Formulae,” Commun. Nonlinear Sci. Numer. Simul., 22(1–3), pp. 92–94.
Tarasov, V. E. , 2016, “ On Chain Rule for Fractional Derivatives,” Commun. Nonlinear Sci. Numer. Simul., 30(1–3), pp. 1–4.
Ortigueira, M. D. , and Tenreiro Machado, J. A. , “ What is a Fractional Derivative?,” J. Comput. Phys., 293, pp. 4–13.
Tarasov, V. E. , 2015, “ Local Fractional Derivatives of Differentiable Functions are Integer-Order Derivatives or Zero,” Int. J. Appl. Comput. Math. (in press).
Tarasov, V. E. , 2008, Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, Elsevier Science, New York.
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Figures

Fig. 1

Plot of the function Z(x) (Eq. (24)) for the range x=α∈[0;1.5]

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text:
## Function that finds the quotient and remainder of the Division Algorithm applied to two integers a and b

Can someone help guide me on writing the code this problem? I am so lost.

Write a function procedure which finds the quotient and remainder of the Division Algorithm applied to two integers a and b.

## Long Division when dividend and divisor are of same size (same number of digits)

I want to know if there is some optimized method (constant complexity preferred) to obtain the quotient when the dividend and the divisor contain the same number of digits. I know about algorithm D of “The Art of Computer Programming” but there Knuth describes how to find quotient digit when the dividend is $$n+1$$ in length and divisor is $$n$$. Is there any method of finding the quotient digit when they both are of the same length? Obviously I can search linearly from $$1$$ to $$base$$ or use Binary search for the same but I was looking for something more optimal as my Base is very large $$(\approx 10^{18})$$

## Factoring after polynomial long division

I have gotten the code to do polynomial long division, but I want my students to also factor the solution if possible.

## Floating point arithmetic on division

I am trying to figure out how $$(x/y)$$ in floating point arithmetic

$$fl(fl(x) / fl(y))$$ where $$fl(x) = x(1-\delta_1)$$, $$fl(y) = y(1-\delta_2)$$, $$fl = (1-\delta_3)$$

I have:

$$= x/y \cdot ((1-\delta_1)/(1-\delta_2))(1-\delta_3)$$

after arithmetic

$$= x/y \cdot (\delta_3 \delta_1 – \delta_1 – \delta_3 + 1)/(1-\delta_2)$$

Not sure how to fully write the rest I believe i have to shorten them using the info that $$\delta < \epsilon$$ and $$\delta^2 \leq \delta$$

## Finding square root without division and initial guess

I am trying to look for a division-free algorithms which finds square-root using only addition and multiplication operations, with the initial guess as the number itself. Is there any such algorithms ?

## Division erronea (Javascript)

Soy estudiante de Javascript, y estaba armando una calculadora del IMC (indice de masa corporal) para practicar un poco.

Estoy teniendo un problema con la funcion del IMC().

Les paso el codigo:

    function PuntuarMetros(numero){     n = numero.toString();     a = n.slice(0,1);     b = a + "." + n.slice(1,3);     return parseFloat(b); }  function NumeroAlCuadrado(numero){     //Agregar condicional si tiene o no "." o ","     a = PuntuarMetros(numero);         b = a * a;     c= b.toFixed(2);     return parseFloat(c); }  function DividirNumero(dividendo,divisor){     a = dividendo;     b = divisor;     c = a / b;     return c; }  function IMC(peso,altura){     a = peso;     console.warn(a)     b = NumeroAlCuadrado(altura);     console.warn(b)     c = DividirNumero(a,b);     console.warn(c)     return c; }

Consola de Chrome:

DividirNumero(64,2.92) 21.91780821917808 64/2.92 21.91780821917808  IMC(64,171) 64 2.92 0.5856164383561644

Como bien se puede ver, cuando realizo la division de dos numeros mediante la funcion me da como resultado “21.9”, luego pruebo dividirlo fuera de la funcion y me da el mismo resultado.

Ahora cuando lo hago dentro de la funcion de IMC, la division me la realiza de forma erronea.

Espero haya sido claro con mi problema.

Muchas gracias por la ayuda 😀

## how to divide div part in multiple division

how to divide div part in multiple division Hi

Does anyone know how can divide a div section into two and display content in it side by side a bit like two table cells in a table row.

When I do content A content B

The content B appears below content A and not next to it. How do I get it to display next to it?

## Is it good practice to replace division with multiplication when possible?

Whenever I need division, for example, condition checking, I would like to refactor the expression of division into multiplication, for example:

Original version:

if(newValue / oldValue >= SOME_CONSTANT)

New version:

if(newValue >= oldValue * SOME_CONSTANT)

Because I think it can avoid:

1. Division by zero

2. Overflow when oldValue is very small

Is that right? Is there a problem for this habit?

## Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

I asked this a week ago at math.stackexchange, without success, so I hope it will be appropriate here.

Let $${\mathcal C}$$ be a symmetric closed monoidal category, and let me denote the internal hom-functor by a fraction $$(X,Y)\mapsto\frac{Y}{X},$$ so that we have an isomorphism of functors $$\operatorname{Mor}(A\otimes B,C)\cong \operatorname{Mor}\left(A,\frac{C}{B}\right).$$ As is known, $${\mathcal C}$$ is an enriched category over itself. For each objects $$A,B,C$$ let me denote by $$\bullet_{A,B,C}$$ the “inner composition” in $${\mathcal C}$$ as in an enriched category, i.e. the morphism $$\bullet_{A,B,C}:\frac{C}{B}\otimes\frac{B}{A}\to\frac{C}{A}$$ with the corresponding properties.

I wonder if the following identity always holds $$\bullet_{A,C,D}\circ\left(1_{\frac{D}{C}}\otimes\frac{\varphi}{1_A}\right)= \bullet_{A,B,D}\circ\left(\frac{1_D}{\varphi}\otimes1_{\frac{B}{A}}\right)$$ (for arbitrary objects $$A,B,C,D$$ and for arbitrary morphism $$\varphi:B\to C$$).

This is strange, I can prove this only in the case when the unit $$I$$ is a separating object in $${\mathcal C}$$ (what does not always hold). Is it possible that there is a counterexample?

## How to do arbitrary non-negative powered polynomial division in Mathematica?

Suppose I have a simple polynomial in $$\{a,b\}$$, defined as $$a^k-b^k \ \forall \ k\in \mathbb{Z}^{\geq0}$$. If I know one of the factor is $$(a-b)$$, is there a way to get a representation of its remaining factors in Wolfram Language?

I know one of its representation is $$\sum_{j=0}^{k-1}{(a^{(k-1)-j}\ b^j)}$$ and Mathematica recognizes that

Sum[a^((k-1)-j) b^j,{j,0,k-1}]

gives

(a^k - b^k)/(a - b)

FullSimplify[(a^k-b^k)/(a-b),Assumptions->k\[Element]NonNegativeIntegers]

It is unable to do anything.

Also what is the proper way to give assumptions?

Is the above expression intepreted differently if I give as

Assuming[k\[Element]NonNegativeIntegers,FullSimplify[(a^k-b^k)/(a-b)]]

Also tried factoring directly without giving a single factor to no success,

Assuming[k\[Element]NonNegativeIntegers,Factor[a^k-b^k]]

format: plain


text:
Proof 24: Newton’s 2nd law in rotation

Get some friends to help you push the stuck carousel with your daughter sitting in it. Each applies a force on her. Friction and other force may counteract. Many forces could be involved – let’s denote them $F_1$, $F_2$, $F_3$ and so on – and we use Newton’s 2nd law (see the Newton’s 2nd law skill) on all these forces that are acting at this position to find the acceleration of your daughter.

\begin{align} \sum F&=ma\quad\Leftrightarrow\quad \\ \underbrace{F_1+F_2+F_3+\cdots}_\text{all forces}&=ma \quad\Leftrightarrow\quad \\ \frac{\tau_1}r+\frac{\tau_2}r+\frac{\tau_3}r+\cdots&=ma \quad\Leftrightarrow\quad \qquad\leftarrow \text{Torque }\tau=Fr\Leftrightarrow F=\tau/r\\ \underbrace{\tau_1+\tau_2+\tau_3+\cdots}_\text{all torques}&=rma\\ \sum \tau &=rma \quad\Leftrightarrow\quad \\ \sum \tau &=rm\underbrace{r\alpha}_a \quad\Leftrightarrow\quad \qquad \leftarrow \text{Geometric bond } a=r\alpha\\ \sum \tau &=mr^2\alpha \end{align}

The mass $m$ is just the daughter. But actually not only her is being set in motion, so not only her mass must be involved. The entire carousel structure is being set in motion and the mass of every “particle” of it is involved. The farther away from the centre, the longer it must be moved per second, so the “tougher”.

Each “particle” can be accounted for my splitting the carousel arm (and other parts of the structure) into infinitely many “particles” with each “particle” being pushes/pulled in by its neighbour particles. Due to Newton’s 3rd law, the force from one neighbour “particle” is also felt by that neighbour “particle” itself, but oppositely. So, when summing up all forces for the entire situation, all these “particle”-“particle” force-pairs cancel out. We end with only all the “toughnesses”, all the inertias” $mr^2$ adding up. So, let’s add a sum symbol to the “toughness” group of parameters:

\begin{align} \sum \tau &=\underbrace{\sum (mr^2)}_I\alpha\quad\Leftrightarrow\\ \sum \tau &=I\alpha \qquad\blacksquare\\ \end{align}

This “toughness” chunk is called the moment-of-inertia:

$$I=\sum mr^2$$

Note, this result is also often written as an integral in case of an infinite amount of “particles” being infinitely small (a continuous body):

$$I=\int r^2\;\mathrm dm$$

Proof 14: Motion equations

In motion along a path, $a$ is acceleration, $v_0$ and $v$ are initial and final (current) speed, and $t_0$ and $t$ are initial and final…

Proof 31: Elastic-collision speed “conservation”

We call a collision elastic if the total kinetic energy is conserved (same before as after the collision) and there are no other energies involved.…

Proof 22: Geometric bonds

Angle $\theta$ measured in radians is ‘the number of radius-lengths along the periphery’. Multiply that number with the radius $r$ to get the total length…

Proof 35: Kinetic energy

Linear version of kinetic energy A spaceship in outer space starts its rocket engines to propel itself forwards with a rocket force. That force does…

Proof 32: Momentum conservation law

The moment conservation law and Newton’s 3rd law as well as the idea of impulse seem equivalent (they all describe impacts/collisions). And in fact the…


text: EN|RU Volume 18, No 2, 2011, P. 75-94 UDC 519.95 I. P. Chukhrov On kernel and shortest complexes of faces in the unit cube Abstract: Studying extreme kernel complexes of faces of given dimension, we obtain lower estimates of the number of shortest complexes of faces in the unit $n$-dimensional cube. It is shown that the number of shortest complex $k$-dimensional faces is of the same logarithm order as the number of complexes consisting of no more than $2^{n-1}$ different faces of dimension $k$, with $1\le k\le c\cdot n$ and $c<0.5$. This implies similar lower bounds for the maximum length of kernel and the number of shortest DNF Boolean functions. Bibliogr. 15. Keywords: face, interval, kernel face, complex of faces in n-dimensional unit cube, boolean function, shortest covering. Chukhrov Igor Petrovich 1 1. Institute of Automatization of Designing, RAS, 19/18 2nd Brestskaia str., 123056 Moscow, Russia e-mail: chip@icad.org.ru © Sobolev Institute of Mathematics, 2015


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## Tuesday, October 06, 2009

### "congrats" I guess

Scott quips, "Well, if that's the way its going to be, does that mean I can 'pay' back the student loans?" As a side note, this is the second example of congrats in quotation marks from UMD. I guess they don't think much of managing to finish their bachelors degree.

Alissa said...

Unnecessary quotes and unnecessary capitalization. Let's hope whoever is writing these letters is not a graduate.

supersonicjan said...

Actually, this was for my MD degree. Yes, 8 years of education (and tuition) beyond high school and all I get is a "congratulations."

bethany said...

worse!

E Broderick Noonan said...

Well, maybe they were using the quotation marks because they were *saying* "congratulations", as a quote. Maybe?

Arnold said...

Or, as I was told when I graduated (not from UMD), congrat's!


text:
Para a média e o desvio padrão fornecidos, a calculadora encontrará várias probabilidades para a variável aleatória e vice-versa: para a probabilidade especificada, ela encontrará os valores da variável aleatória.

If $X$ is a normally distributed variable with the mean $\mu=$ and the standard deviation $\sigma=$, then
$P(X \leq$ $)=?$
$P(X \geq$ $)=?$
$P($ $\leq X \leq$ $)=?$
$P\left ( X \leq ? \right) =$
$P\left ( X \geq ? \right) =$
$P \left( -? \leq X \leq ? \right) =$
Skip input, if you don't need it.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your input: calculate $P\left( -? \leq X \leq ? \right)=0.95$ with $\mu=0$ and $\sigma=1$

$P\left(-0.975 \leq X \leq 0.975\right)=0.95$


text:
# Philocrat's theory of time

1. Sep 25, 2004

### Philocrat

THE STANDARD UNIVERSAL NOW (SUN): A Philosophical Analysis of Time (PART I)

I have always suspected that the scepticism about the reality of time and its perceived dimensions (past, present and future) has something to do with the way we are physically designed and configured. This problem, like all other gaps in the human knowledge, may remain irresolvable for a very long time to come, unless perhaps the human reality is subsequently reviewed and re-engineered. Now, the problem has regressed further to a sceptical height where it becomes possible to begin to deny one of the dimensions of time - the present. The suggestion that there may not be any such thing as the present, therefore we are always either in the past or in the future, seems to me to be very radical requiring an equally radical philosophical review. My argument is that if there is no present, along with the uncertainty as to whether we are in the past or future, then we need to look at some mathematical formulation involving the notion ‘OMNIPRESENCY’ already been imagined in scriptures and in science fiction novels and movies. The beauty of it is that we have already imagined it in one form or the other. Here I am going to investigate and state the quantitative and logical implications of the concept of omnipresency. The derivation of this investigation is a ‘superstructure’ that I call the ‘STANDARD UNIVERSAL NOW’ (SUN), otherwise relabelled ‘THE PSYCHOLOGY OF THE SUN’, and which I will link to the notions of a ‘SUPER BEING’ and a ‘SUPER OBSERVER’. It is a formulation that assumes Progressive existence. My arguments therefore will end by claiming that (1) the SUN is the ‘true’ present that is at least in principle possible, if not in actuality, and (2) this conceptual possibility, or possibility in principle, does not rule out actuality or physical realisation. The physical realisation or actuality of this conceptual possibility remains potentially active.

OMNIPRESENCY: Physical or Devine?

When it comes to the notion of ‘OMNIPRESENCY’, there are two fundamental types: (1) ‘MECHANICAL (PHYSICAL) OMNIPRESENCY’ and (2) ‘DIVINE OMNIPRESENCY’. Both are equally problematic. Einstein’s universal constant in his theory of relativity has imposed a physical limitation upon the possibility of (1). So, in physics, physicists would just straightforwardly deny it. With regards to (2), many philosophers don’t even believe in the existence of God or Divine Agency, let alone its possibility. This denial has its origin in the so-called ‘ONTOLOGICAL ARGUMENT’ (http://www.iep.utm.edu/o/ont-arg.htm) (http://www.faithnet.org.uk/A2 Subjects/Philosophyofreligion/ontologicalargument.htm) (http://www.philosophyofreligion.info/ontological.html) for the existence of God. As far as they are concerned there is no such thing as a divine omnipresency. Another problem is the vagueness of the term itself. It is not clear whether by the term we mean:

a) One thing being in every place at the same time
b) Or many of the same thing being everywhere at the same time.

In this piece I am going to assume (a), that is, one thing being everywhere at the same time. However, since physics has denied (1) and philosophy analytically disposed of (2), I am going to systematically ignore (1) and (2) and just take (a) as a purely imaginary but quantitative and logical exercise. But immediately after making this decision, another serious question arises: do I have to assign a specific ‘PURPOSE’ to anything wanting to be everywhere? Why would anything want to reference every position in space at the same time or at time t = 0? Why be everywhere at the same time? In other to make sense of this imaginary quantitative and logical exercise, I decided to assign a purpose to an imaginary being that is separately spaced and timed but who is nevertheless able to perform our phantom feat of being omnipresent. Let’s call this being ‘THE FINAL BEING’ or ‘TIMELESS TRAVELLER’. This being was once a structurally and functionally defective being that has progressed to a final perfect state of being through constant self-revaluation and structural self-reengineering. But this final being needs a witness in form of an observer. Let us call this witness ‘THE FINAL OBSERVER’. This final onlooker is by its whole nature structurally and functionally identical to the Final being. He/she too has progressed through time to state of Visual Perfection. What he/she sees stays seen. The stability in his/her visual faculty is unshakeable by any device of doubt. So, quantitatively and logically, the Final Being stands relative to the Final Observer.

THE PSYCHOLOGY OF THE ‘SUN’ AND THE ‘FINAL MAN’

The notion of omnipresency is such that both the final being and the final observer must progress to a point where time will be perceived as nothing more than a supercritical ‘NOW’ and space as a supercritical ‘HERE’. This, by implications, means that ‘Here’ is neither there nor everywhere, but a permanent and non-reversible ‘Here’, which holds and conserves the final good. The same is true of ‘Now’ for it is neither then nor ordinary occasions, nor past nor future, but a permanent and non-reversible ‘Now,’ which participates in the measure, maintenance of final things in the supercritical Here. Both when combined disposes of the Einstein’s defective SPACETIME and forms a new superstructure called ‘HERENOW’. I was also considering the possibility of reversing the two words and renaming it ‘NOWHERE’, but I quickly spotted the perceptual danger of being erroneously perceived and construed by the future observers and innocent bystanders or spectators as ‘NO WHERE’. It would have been an absolute psychological disaster to perceive an ideal superstructural NOW and HERE as a supercritical NO-WHERE or NOTHINGNESS.

Although physical space distances and time differences may still be physically installed between things, such as a thousand miles between separately spaced events still remaining a thousand miles or the physical time duration between them still remaining at, say, six hours, but new abilities in visually led super-structured things will cause them to act and perceive timelessly and spacelessly on their causal pathways, given that the principle of the ‘SUN’ is causally and relationally viable in the first place.

In terms of purpose, the final aim of all causal possibilities on the universal causal pathways is to create and maintain a ‘Standard Universal NOW (SUN)’ in the visual faculty of the observer. It is the final frontier in the human perception and understanding of Reality. That is, the ability of a self-proclaimed entity to be everywhere at time t = 0. To freeze diverse senses of time into a CRITICAL, NON-REVERSIBLE NOW in the ever curious eye of the ‘FINAL’ onlooker or observer, will be the highest point in the human visualisation and understanding of reality.

The mathematical equations for achieving omnipresency may be stated as follows (and this is also the mathematics for disposing of time in spatio-temporal relations):

$$\alpha = \frac{\sum y}{z^y}$$

Where $$\alpha$$= infinity or zero time travelled;
Where $$\sum y$$ = the sum of distances between events referenced at once, and;
Where $$z^y$$ = the speed of light times a scalar squared. A scalar is a special number that when applied to the speed of light and square, helps shrink both the distance and time travelled to zero in the observer’s or final man’s imagination or visual frame of reference, in a full compliance with the principle of the SUN above.

It must be appreciated that a scalar is not some number that would be cooked up by some mad mathematician or conjured into existence by some unknown divine agency. I do not know what this scalar or special number is going to be in future, whether it will be expressed as a variant of energy required by the final traveller in response to the fluctuating Sg, but in the ‘Book of Nature’, a scalar would be a number of unique value with proscribed physical consequences: it must be a number quantitatively derived, perhaps, from a structural and physical re-design of reality. All structurally and functionally improved or re-designed parts of a whole must effectively subscribe to this number, since it is logically and quantitatively consistent that the final traveller is able to physically adjust its speed according to the fluctuating distance, but for the purpose of this equation I have decided to use an imaginary number.

The problem with using a scalar in this equation is that it can never lead to an absolute zero in the anticipated value of $$\alpha$$, rather it always leads to an approximate fractional zero value. To help push the fractional zero value as close to an absolute zero as possible, I have decided to use the value of $$\sum y$$ as my own scalar. It must be realised that the application of a scalar to this equation in this manner is an imaginary violation of Einstein’s universal constant, the speed of light, which is currently claimed to be the fastest material substance known to man. So physics says, but I am personally sceptical about this.This is very necessary in the equation to demonstrate and display for the whole world to see what is expected of the final being of any kind, if he/she/it is to stay indefinitely OMNIPRESENT in the final observer’s visual frame of reference.

Continue in part II on the next link.......

Last edited: Sep 26, 2004
2. Sep 25, 2004

### Philocrat

THE STANDARD UNIVERSAL NOW (SUN): A Philosophical Analysis of Time(PART II)

To simplify the above equation so as to take account of the scalar mentioned above, it may re-expressed as follows:

$$\alpha = \frac{\sum y}{z^y}$$

Where $$\sum y = (d1+d2+d3+...dn)$$ = Distance d1 right through to
distance dn
Where $$z^y = (\mu \times \sum y)$$ = Universal constant $$\mu$$ (speed of light) times the sum of distance $$\sum y$$, the scalar

Why a being of any kind would like to do many things, let alone everything, in no time, is still scientifically unclear. However, should there in the end still be a strong cosmic desire for this possibility, then for it to reference all these things under its care at their different space and time locations, there must be some form of MOTION, however the means. The final superstructured being, in its unique form, size, time and place, can only reference different points in space-time if it moves itself through a well-defined and quantifiable geometrical path.

It is by implication also an action fulfilable by satisfying not only a clearly self-evident geometrical condition of motion, but also the inevitable visual condition of omnipresency within the final observer’s visual frame of reference. If this final superstructured being is divine, there is inevitably an extra demand: the duty requires that when this final superstructured being reaches bad events by motion prevents them from happening, if they had already happened reverses them, and if they are good events acknowledges, permits and conserves them.

Or we may even argue that, since it is the final divine being’s ambition to still govern the final moments of existence, it is immaterial what relationship it may still desire to keep with the final those under its care, provided it can satisfy one most important person, the FINAL OBSERVER. The eye of this important final individual is so sharp and critical that, during the convergence era, it may no longer be able to admit circularities, randomness, ephemerality, magic, miracles fantasies, or even sentiments, rather his eye would admit only that with unqualified degree of permanency, stability, irreversibility and indestructibility.

The Final Being is the ‘Timeless Traveller’. Regardless of the space distances between events, a timeless traveller all ways does everything at time $$t = 0$$ . The Timeless Traveller always stands relative to the ‘Final Observer’. The critical eye of the final observer must at all angles of visualisation, and for always, see no less in the thoughts and deeds of the final being than that which may be construed as truly omnipresent. In his eye the final being must finally and permanently travel at time $$t = 0$$ while in action. This clearly means that if, for example, there is an earthquake engulfing a village at position A, hurricane flattening a city at position B, flood ruining a farm at position C, a bank robbery at position D, a motor accident at position E and a war at position F, not only must the final being (if divine) contemporaneously reference these events and prevent them from happening, but also must actually do so in no time in the glaring eye of the final observer.

Applying the above equation to this example, if the final traveller, be it divine or material, is timed and spaced at location Z, then the Sum of all the Distances between the six events (Earthquake, Hurricane, Flood, Bank robbery, Motor accident and War) may be estimated as follows:

$$\sum y = (d0 = Z \Longrightarrow A; d1= A \Longrightarrow B; d2 = B \Longrightarrow C; d3 = C \Longrightarrow D; d4 = D \Longrightarrow E; d5 = E \Longrightarrow F; d6 = F \Longrightarrow Z)$$

$$\sum y = (d0+d1+d2+d3+d4+d5+d6)$$

It must be noticed that in the above estimation the final (divine) traveller only travels once. I honestly could not imagine why the final traveller from its own space and time location may desire to travel and reference each event at a time. The mathematics of economy must apply. At this very instance, it effectively and economically makes only one journey, otherwise it would had to had to wastefully or exhaustively travel twice or make a double journey if it decided to visit each event at time. Based on the amount of energy and time available, it may have had to waste a great deal of energy and time to do so.

If for example, the summed up $$\sum y = 149,600,000 km$$, then we may do the actual calculations as follows:

$$\alpha = \frac{\sum y}{z^y}$$

$$z^y = (\mu \times \sum y)^2$$
$$\sum y = 149,600,000 km$$
$$\mu = 300,000$$

$$\alpha = \frac{149,600,000 }{(300,000 \times 149,600,000)^2}$$

$$= 0.00000000$$

This result as we have already predicted above is not an absolute zero. It has a fractional part which extends into infinity. But it does demonstrate that the final traveller had by approximation referenced all the six events and back at time t = 0. I have used a scalar ($$\sum y$$) which guarantees the shrinking of the travelling time to zero. When the actual scalar is found in future by the structural engineers, this imaginary scalar ($$\sum y$$) that I have used above must be replaced by the new one in the sum of $$z^y$$. But whatever number replaces it, as I have already made it clear above, must be not only a definite number of some physical origin, but must also be a number that when applied to the speed of light guarantees the shrinking of the travelled time to zero.

Finally, one other issue that the future investigators of superstructural phenomenon will have to resolve is the premonitional ability of the final superstructured man with regards to the above equation. As you can see in the above example, we do not know how he managed to know that all these six events were going to happen or when they were going to happen. My prediction is that the only way the above equation is going to come alive is when the final traveller’s Premonitional ability is linked to his physical ability linked to time and linked to the scalar in it. Then not only will the scalar have a definite physical origin and makes sense, so will the whole equation make a great deal of sense and manifest into a physical reality.

However, if the final traveller is visually loaded with a premonitional ability as well as being physically equipped with a special ability to reverse events in space-time after they have occurred, then the occurrence times of events are irrelevant in the above equation. All that is necessary in the above equation is his overall visual ability to predict events and the physical ability to reference and resolve events at time $$t = 0$$ and nothing else.

NOTE: If the Principle or the Psychology of the sun is possible, then the following observations are my conclusions:

1) There is no REAL structural relation between space and time. If they both exist, then one is structurally without the other.
2) Time is an illusion, at least from the human point of view. It is forced upon the both the ACTOR and the OBSERVER by action. A world without physical action keeps space and time permanently devoid of one another.
3) An actor and an observer who can act and see omnipresently are said to be both physically and psychologically devoid of time.

I know that these are startling conclusions that would not go down well with the faint hearted.

Last edited: Sep 26, 2004


text:
### B Spline Tutorial

The orange point at x=4 corresponds the point I wish to directly determine the y value of the b-spline at. Example 6: Subgraphs Please note there are some quirks here, First the name of the subgraphs are important, to be visually separated they must be prefixed with cluster_ as shown below, and second only the DOT and FDP layout methods seem to support subgraphs (See the graph generation page for more information on the layout methods). A curve is times differentiable at a point where duplicate knot values occur. I will try to explain the best I can. A note on the relationship between B-splines and the Riemann's Point Lattices in Computer Graphics and Visualization Tutorial at IEEE Reproducible Research. Older Cubic Spline Library Bezier Cubic Splines are an excellent and preferred method to draw the smooth continuous curves often found in typography, CAD/CAM, and graphics in general. Using the B-Spline Tool, start with a line that curves to the right and up. Kauermann, Krivobokova, and. Hence, m = 4 and u 0 = 0, u 1 = 0. FORMul[a]RCH is a great blog containing personal explorations and examples of popular generative design techniques. The toolbox lets you perform exploratory data analysis, preprocess and post-process data, compare candidate models, and remove outliers. However, the conversion won't be 1:1 because there are more heuristics working to decide which curves will be merged into one when the algorithm reaches a T-junction among the visible colors. i want to test the spline with any three points in x[0] \">\/a> \/small> \/span> DLs\/span> \/small> \/li> \/ul> \/div> folder\/matIcon> \"> \/a> \/small> \">. Could anyone help me about B-spline Curve error? I want to draw B-spline Curve in c++, but even though all coordinates are positive, the segment's coordinate is negative. What is a SPLINE. How to Design Curves Curves in CorelDRAW can be as simple as a single straight line, or complex open or closed shapes comprised of curved or straight segments. It features two orientation modes: NODE : It directly takes the orientation from the control point TANGENT : Calculates the orientation based on the tangent to the curve in. B-spline interpolation of control points of any dimensionality using de Boor's algorithm. Below are a series of tutorials that should help you become familar with Matlab syntax. B-splines 3. The positions of control points can be specified either in ordinary coordinates as {x, y} or {x, y, z}, or in scaled coordinates as Scaled [{x, y. It is not designed to be a full lesson; it is more suited to those who have already learnt the basics and would like to reinforce their knowledge. …The Bezier is the curve widely used in 2D applications like Illustrator and…Photoshop, and let's you draw or edit using points connected to handlebars,…which are then used to adjust the shape of the. An introduction to modeling for statistical/machine learning via smoothing splines. What is a SPLINE. • You can find the Bézier control points from the B-spline control points • In the case of a quadratic B-Spline: P 0 is an interpolation between v i-2 and v i-1, P 1 = v i-1 P 2 is an interpolation between v i-1 and v i Relation between 3rd-degree Bézier curves and B-Splines • Constructing Bézier points from B-Spline points v j-3 f(t j. The attached source code project draws spline curves between two points. and Sappa A. Follow these steps to draw a free-form curve by using the SPLine command: Start a new drawing. bs: Evaluate a Spline Basis (splines) splineDesign: Design Matrix for B-splines (splines) splineKnots: Knot Vector from a Spline (splines) splineOrder: Determine the Order of a Spline (splines) Robust regression. I am having a data and am trying to interpolate using b-spline toolbox in Matlab. com for more free tutorials. Multilevel B-splines are introduced to compute a C2 continuous surface through a set of. Demonstrating a basic area chart, also known as a mountain chart. Description. The Basics of Basis Functions Previous - 1 - 2 - Next: How do I know what basis system to use?. spalde (x, tck) Evaluate all derivatives of a B-spline. Basic illumination models, polygon rendering methods. The positions of control points can be specified either in ordinary coordinates as {x, y} or {x, y, z}, or in scaled coordinates as Scaled [{x, y. A mesh modified by a curve modified by a curve ensures constant length. Lagrange interpolating polynomial of 2nd order; (b) Newton’s interpolating polynomial. Viewed 1k times 1 $\begingroup$ I am using the de boors algorithm to generate B splines in python. Interpolating values on a spline based on a normalized distance between each sample: Points on Spline, Points and Tangents on Spline, Transforms on Spline, Transforms on Spline with Guide. Check out the other tools in the B-spline tools toolbar for more B-spline editing tools. Download it once and read it on your Kindle device, PC, phones or tablets. 11 with its control polygon. 5, Double_t b=31) {// array. Look up spline curves in the AutoCAD online Help system if you need precision splines. …The Bezier is the curve widely used in 2D applications like Illustrator and…Photoshop, and let's you draw or edit using points connected to handlebars,…which are then used to adjust the shape of the. B-spline Polynomial Let p 0;:::;p n be the control points. Schoenberg who introduced them in 1946 [123] for the case of uniform knots. Scattered data fitting has always been a challenging problem in the fields of geometric modeling and computer-aided design. This leads to the conclusion that the main use of non-uniform B-splines is to allow for multiple knots, which adjust the continuity of the curve at the knot values. Mathematics of Computer Graphics and Virtual Environments 59,457 views. void splines_test(Int_t np=23, Double_t a=-0. We also go over how to control the interpolation of the spline points within the vectorizer object. With this tool, you select a series of points, and Fusion will create a smooth curve that passes through them. b spline basis function. If the knot-points are equally spaced with spacing , then the B-spline approximation to a 1-dimensional function is the finite-basis expansion. b-spline In computer graphics, a curve that is generated using a mathematical formula that assures continuity with other b-splines. The final product will look like the one below. F or this reason, the B-splines are de ned via the recurrence. Examples of use of the spline classes. Lectures by Walter Lewin. B-Splines and Smoothing Splines B-Spline Properties. Mathematics of Computer Graphics and Virtual Environments 59,457 views. B-spline curves have the following properties − The sum of the B-spline basis functions for any parameter value is 1. For this area we don't really have to play with it initially, because this is a more advanced field. It allows you to move your GameObject smoothly through an easily defined path, interpolating both position and orientation. That is it sets the number of line segments in the control frame. Spline is a drawing method that enables you to draw basic lines or shapes in the same way that you would use the Pen Tool in Adobe Photoshop or Macromedia Fireworks, while Lathe is a feature that enables you to create a 3D object by rotating a shape or a line around an axis. A spline is simply a curve that connects two or more specific points. The order of the B-spline basis functions. curves-and-splines-02. Using this Guide (p. Conceptually, the sample points extrude to the height of their magnitude. edu is a platform for academics to share research papers. Polynomial regression is computed between knots. This site is maintained by Erico Tjoa (visit my Github) - Recent Highlights and others- Review on Interpretable….  In this case, we chose to do a goat but you can pick any animal you like. Power BI suite provides multiple software, connector, and services - Power BI desktop, Power BI service based on Saas. FORMul[a]RCH is a great blog containing personal explorations and examples of popular generative design techniques. The algorithm used for generating is for cubic Bspline. Now instead of stopping and holding at Pose B, it gets almost all the way to Pose B and then eases in. (Note: This tutorial was created using Unreal Engine v. The performance of the identification depends on an optimization algorithm for the training of the BSNN in order to avoid any possible local minima. Find(interpolate) and draw the B-spline curve that go trough plist points and or in other words a curve fitting using a cubic B-spline curve. You will start with the same pawnshop model featured in the Composite Map tutorial, and use a spline to map a brick-like material onto the building’s arched entrance. If the knot-points are equally spaced with spacing Δ x, then the B-spline approximation to a 1-dimensional function is the finite-basis expansion. As shown in the last example, the main problem with Bezier curves is their lack of local control. To begin, set your active project to tutorial_files. However, I now need to reproduce a b-spline model in an external (non-R) code. The attached source code project draws spline curves between two points. It gets even trickier to work with when you deal with NURBS, or Non-Uniform Rational B-Splines. This is a draft so feel free to point out any mistakes. As I said, there is no direct inverse for the mapping from knot space to physical space. The BSpline tool specifies the exact points through which the curve will go; on. The options here are all essential to spline modeling, stuff like Create Line, Attach, Weld and Refine are used ALL the time. In mathematics, a spline is a special function defined piecewise by polynomials. The order of the B-spline basis functions. This addendum does not apply to major diameter fit splines. Multilevel B-splines are introduced to compute a C2 continuous surface through a set of. With this piece of advice this tutorial ends here. Medical Image Registration Using B-Spline Transform Zufeng WU*, Tian LAN, Jiang WANG, Yi DING, Zhiguang QIN School of Information and Software Engineering , University of Electronic Science and Technology of China, Chengdu, Sichuan 610054. Evaluate a B-spline or its derivatives. Drawing a cubic B-Spline curve where plist is the control polygon. Medical Image Registration Using B-Spline Transform Zufeng WU*, Tian LAN, Jiang WANG, Yi DING, Zhiguang QIN School of Information and Software Engineering , University of Electronic Science and Technology of China, Chengdu, Sichuan 610054. This tutorial builds on the foundation laid by previous tutorials. 1-6) Outlines the organization of this user’s guide. For nonperiodic B-splines, the first knots are equal to 0 and the last knots are equal to 1. edu The present note clariﬁes some of the underlying facts which are used in the calculation of the basis functions of B-spline using R. A B-spline (or basis spline) can achieve high orders of continuity, but the tradeoff is that it's a bit harder to calculate things than with cubic Hermite splines or Bezier curves. Please wash your hands and practise social distancing. A B-spline curve made from 4 points (in white). B-Spline Curves B-Spline Curves (Two Advantages) 1. Essentially, they are made up of a series of 'from' and 'to' co-ordinates. B-spline based repetitive controller using iterative swarm learning for CACF VSI version 1. The below image shows the guide polygon and the b-spline generated by the above code. In this paper we analyze relevance of vectorization for evaluation of Non-Uniform Rational B-Spline (NURBS) surfaces broadly used in Computer Aided Design (CAD) industry to describe free-form surfaces. This is then extended to the bivariate case and to bivariate irregular grids, where the principal schemes are outlined. 1) E [f]= 1 n X (y i x)) 2 + Z 00) dx in the space of functions with square in tegrable second deriv ativ e. A tutorial. This is a tutorial about interpolation in Unity with C# code. basis-open - an open B-spline; may not intersect the start or end. Rational B-spline curves Homogeneous coordinates are used to represent points in 3D space in terms of points in 4D space16-19. Welcome to SolidWorks tutorials for beginners and you are going to learn how to use helix feature command to make. I wrote this tutorial to give some kind of help, even if I repeat that I never used the splines tool, or curved lines. …So to take a look at that let's go get another clip. Select the "Spline Tool" in the Toolbox Panel or use keyboard the shortcut Alt B to enable it. So, what's the meaning of the matrix generated by bs? Example:. are B-spline basis functions of degree k and knots t. NET Nov 08, 2012. B-spline basis functions Recurrence relations. the B-spline surface defined by these information is the following: where N i,p (u) and N j,q (v) are B-spline basis functions of degree p and q, respectively. For ease of calculation take the knot vector to be [0,0,0,1,1,1]. The SPLINE command creates a parti cular type of spline known as a nonuniform rational B-spline (NURBS) curve. By changing t from 0 to 1 one will build a cubic B-spline between points ControlPoint2 and ControlPoint3. See Smith for an excellent introduction to splines. Is there any help/tutorial/info on how to use the B-spline tool in the sketcher workbench. pp = bspline( t ) plots nothing and returns the ppform of the B-spline. Properties of B-spline Curve. Simple Knots Suppose the knot vector is U = { 0, 0. splint (a, b, tck[, full_output]) Evaluate the definite integral of a B-spline between two given points. Using Implicit B-Splines for Surface Reconstruction out of 3D point clouds. In the mathematical subfield of numerical analysis, a B-spline, or basis spline, is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. It’s the process in which you convert the interpolation of the keys from stepped to spline. This is the original Ellipse. Up to now I've never needed to understand the output of bs in detail: I would just choose the model I was interested in, and fit it with lm. The algorithm consists of the following steps: * Initialization of the B-spline surface by using the Principal Component Analysis (PCA). B-splines over nonuniform knots go back to a review article by H. 75 and u 4 = 1. Formally the estimate is the curve that minimizes the criterion: (1/n) sum(k=1,n) w. Using the B-Spline Tool, start with a line that curves to the right and up. B-Spline Basis: Motivation The computation of the coefficients is perhaps the most complex step because they must ensure certain continuity conditions. The BSpline tool specifies the exact points through which the curve will go; on. Does anyone out there know of a good spline tutorial. The basis functions of PHT-splines have the same important properties as B-splines do, such as nonnegativity, local support and partition of unity. They are a very simple thing, worth to study once and then feel comfortable in the world of vector graphics and advanced animations. B(t) = P 0 + t(P 1 - P 0) = (1 - t)P 0 + tP 1, tε[0,1] What is B-Spline Curve? B-Spline curves are considered as a generalization of Bezier curves and as such share many similarities with it. Is a B-spline just a set of Bezier curves that have been tacted together in a continuous fashion? Pretty much exactly right. Up to now I've never needed to understand the output of bs in detail: I would just choose the model I was interested in, and fit it with lm. seperti tool baru bernama B-Spline, desain ulang color manajemen, corel power trace, kecepatan dan performa, dan banyak lagi di coreldraw x5, pada versi sebelumnya yaitu X4 belum terdapat tool bernama b-spline. The splines of a newer class, the ap-. B-Spline Basis: Motivation The computation of the coefficients is perhaps the most complex step because they must ensure certain continuity conditions. xvg (for A-B interactions), table_A_A. B(asic)-Spline Basics Carl de Bo or. B-spline knots. A NURBS curve produces a smooth curve between control points. I need a favor on that part. The next step is the spline tool. A B-spline (or basis spline) can achieve high orders of continuity, but the tradeoff is that it's a bit harder to calculate things than with cubic Hermite splines or Bezier curves. b-spline B-spline interpolation. Power BI Tutorial. Cubic spline (spline) First derivative. I'm sure you think it is the other way around, but not me. Pictured are standard output splined shaft sizes. sproot (tck[, mest]) Find the roots of a cubic B-spline. Slideshow Tutorial. 12, but most of this material does not appear in the text If we assume that a series of segments from cubic functions will give a nice curve, then we can use the following general equation for the curve:. I understand the B-spline basis function calculations for zeroth and first degree but I have no idea on how to calculate for the 2nd degree. In this note, we use u 0 = 0 and u m = 1 frequently so that the domain is the closed interval [0,1]. -> Delete the spline. The black point is the value of the b-spline when the value of 4 is passed as a parameter. However, non-uniform B-splines are the general form of the B-spline because they incorporate open uniform and uniform B-splines as special cases. It is considerably 'stiffer' than a polynomial in the sense that it. B spline curve generation in Python. This kind of spline morphing animation is awesome for using in conjunction your 2D workflow in After Effects by applying a Cel Shader material to your splines. However, I now need to reproduce a b-spline model in an external (non-R) code. 1 De nition of B-Spline Curves A B-spline curve is de ned for a collection of n+ 1 control points fQ i gn i=0 by X(t) = Xn i=0 N i;d(t)Q i (1). WaveJSON is a format that describes Digital Timing Diagrams. Properties of B-spline Curve. b-spline B-spline interpolation. B-Splines using open knots vector : The drawback of the Cardinal B-Splines is the lack of interpolatory functions that allow us to impose, in a strong form, Dirichlet boundary conditions. 5-7 STATS 202: Data mining and analysis Jonathan Taylor Nov 5, 2018 Slide credits: Sergio Bacallado 1/1. CubicSpline InterpolateBoundariesSorted ( Double[] x, Double[] y, SplineBoundaryCondition leftBoundaryCondition, double leftBoundary. 6 B-SPLINE CURVES AND NURBS. When you release your dragging motion the new shape will appear. Then once saved, write some Swift code and Command+B and it should magically works. Splines in MATLAB (p. If you are looking for a way to work with CAD on Linux, Windows or Mac OS X and replace AutoCAD, then you should consider looking into FreeCAD. SPLINE S AND B -SP LINES AN INTR ODUCTION c1 c2! 0. t j + k), the linear system for the B-spline coefficients of the spline to be determined, by interpolation or least squares approximation, or even as the approximate solution of some differential equation, is banded, making the solving of that linear system particularly easy. curves-and-splines-03. This leads to the conclusion that the main use of non-uniform B-splines is to allow for multiple knots, which adjust the continuity of the curve at the knot values. An n-degree B-spline curve produces n-degree Bézier curves that are automatically C n-1 with each other. Drawing a cubic B-Spline curve where plist is the control polygon. curves-and-splines-02. xvg (for A-A interactions), and table_B_B. Digital Toolbox has over 38 tutorials for all levels of experience. A basis spline is a specific category of spline that supports some types of computer graphics animation. To convert it into true spline type SPE on the command line and press Enter. B-spline Basis Functions: Computation Examples. Building complex B-spline curves. in reverse engineering (RE) area, to employ B-spline curves for non-trivial cases that include curves with discontinuous points, cusps or turning points from. Drawing B-Spline Curves using GDI+ in VB. Many rookie 3D animators (and even some veterans) are uncomfortable, perhaps even intimidated, by the spline editor in their animation software. The nonrational form of a B-spline is. The algorithm given in w:Spline interpolation is also a method by solving the system of equations to obtain the cubic function in the symmetrical form. There may be 2, 3, 4 or more. The shape will be a square in U and a line in V. To define B-spline basis functions, we need one more parameter, the degree of these basis functions, p. An order k open uniform B-spline with n +1= k points is the Bezier curve of order k. Jan 18, 2015 - This is CorelDRAW X7 Class NO 24, in this video tutorial you can learn how to working with B-Spline Stay safe and healthy. Regression with restricted cubic splines in SAS This section provides an example of using splines in PROC GLMSELECT to fit a GLM regression model. b spline basis function. com/drawplus and ask questions in our friendly communi. unitypackage The project after Splines. Use features like bookmarks, note taking and highlighting while reading R Tutorial with Bayesian Statistics Using OpenBUGS. So, what's the meaning of the matrix generated by bs? Example:. However, non-uniform B-splines are the general form of the B-spline because they incorporate open uniform and uniform B-splines as special cases. Cubic B-Spline Interpolation. B-Splines Convex hull property For a B- Spline curve of order k (degree k-1 ) a point on the curve lies within the convex hull of k neighboring points All points of B- Spline curve must lie within the union of all such convex hulls. It is considerably 'stiffer' than a polynomial in the sense that it. Among their many advantages is a very sparse data set allowing a mere eight values ( or four x,y points ) to completely define a full and carefully controlled and. In the following we consider approximating between any two consecutive points and by a linear, quadratic, and cubic polynomial (of first, second, and third degree). The BSpline tool specifies the exact points through which the curve will go; on. These curves can be scaled indefinitely. Hitchhiker's Guide to the WaveDrom WaveDrom is a JavaScript application. the degree of function is (m-1), where m is an order of the function. The best way to think of a spline is a machine that generates Bézier curves. Use Draft WireToBSpline to convert one to the other. The essential parts of the B-form are the knot sequence t and the B-spline coefficient sequence a. Hence, m = 4 and u 0 = 0, u 1 = 0. Create sphere. Splinophilia Part 1. 102 videos Play all Computer Graphics Tutorials Point (India) Ltd. splrep returns an array of tuples containing the coefficients. cardinal - a Cardinal spline, with control point duplication on the ends. SPLINE creates curves called nonuniform rational B-splines (NURBS), referred to as splines for simplicity. BSplineCurve is also known as basis spline curve or nonuniform rational B-spline (NURBS) curve. 12, but most of this material does not appear in the text If we assume that a series of segments from cubic functions will give a nice curve, then we can use the following general equation for the curve:. This is the source distribution for the NCAR/EOL BSpline C++ library. Introduction. Could anyone help me about B-spline Curve error? I want to draw B-spline Curve in c++, but even though all coordinates are positive, the segment's coordinate is negative. Therefore, a B-spline surface is another example of tensor product surfaces. Cubic Spline Regression Restricted Cubic Spline Regression To force linearity when X a k, the last two s are redundant, i. B(t) = P 0 + t(P 1 - P 0) = (1 - t)P 0 + tP 1, tε[0,1] What is B-Spline Curve? B-Spline curves are considered as a generalization of Bezier curves and as such share many similarities with it. Let's focus on the ease-in to Pose B first. Essentially, they are made up of a series of 'from' and 'to' co-ordinates. Interpolation of an N-D curve¶ The scipy. Uniform B-splines. So well done!! One last thing that would be great to add is creasing in spline modeling. Splines are defined either with fit points, or with control vertices. 4 B-spline curves and Contents Index 1. Editor handles for points on the path. Entity b_spline_surface definition. Cubic Spline Interpolation of a Circle Hey there - Thanks for the great tutorials - they really helped me! I'm trying to duplicate your results for cubic interpolation of a circle with 4 points and I got the same solution for the 2nd derivatives in the x and y directions. E Displays the point editor dialog in the current display view, or sets keyboard focus to point value controls in the spline and timeline editors. , the rth derivative at τ of the jth B-spline, for all j, for some r and some site τ. K Creates a keyframe on the selected spline at the current mouse position. It features two orientation modes: NODE : It directly takes the orientation from the control point TANGENT : Calculates the orientation based on the tangent to the curve in. Note that the variable SPLINESEGS controls the fineness of the b-spline. Do you want a B-spline curve that smoothly connect (i. Navigate to the Splines folder and open the file spline_1_start. …First, a Bezier, there's a beautiful mathematical formula,…and, just kidding, let's go with some images. By default, the splines are created as renderable objects, but you can change that at the “Display / Options” rollout, at the bottom of the interface. As shown in the last example, the main problem with Bezier curves is their lack of local control. However I am getting spikes in the final. So, what's the meaning of the matrix generated by bs? Example:. Until now, this requirement has been met by the existing Spline command. A curve is times differentiable at a point where duplicate knot values occur. T YouTube video about T-Splines and Other Stuff @noyb Personally I would like to thank you for putting in the effort to even make the video, and I would like to commend you for its presentation as I found it to be very upbeat and far from boring like 99% of all the other videos out there. com for more free tutorials. As shown in the last example, the main problem with Bezier curves is their lack of local control. [Because MATLAB's spcol accepts the order as an input, you need to specify order=4 rather. 12, but most of this material does not appear in the text If we assume that a series of segments from cubic functions will give a nice curve, then we can use the following general equation for the curve:. B-splines: Setting the Options Specified by – – m+1 control points, P 0 … P m – m-2 cubic polynomial curve segments, Q 3…Q m – m-1 knot points, t 4 … t m+1 – segments Q i of the B-spline curve are defined over a knot interval defined by 4 of the control points, P i-3 … Pi – segments Q i of the B-spline curve are blended. You need to be clearer about what you want. It is not designed to be a full lesson; it is more suited to those who have already learnt the basics and would like to reinforce their knowledge. t j + k), the linear system for the B-spline coefficients of the spline to be determined, by interpolation or least squares approximation, or even as the approximate solution of some differential equation, is banded, making the solving of that linear system particularly easy. China Abstract — Hierarchical B-spline has been widely used for shape modeling in many fields. edu The present note clariﬁes some of the underlying facts which are used in the calculation of the basis functions of B-spline using R. Bondioli & Pavesi 35mm x 12 spline 40mm x 14 spline 45mm x 16 spline Walterscheid 30mm x 16 spline 35mm x 18 spline 40mm x 20 spline 52mm x 25 spline Weasler 1. bspline(t) This B-spline consists of 4 polynomial pieces: the red, green, purple, and black curves in the plot, each of degree 3. A basis spline is a specific category of spline that supports some types of computer graphics animation. Two things are going wrong in your code. When drawing lines by using the curve tools, you can hide the bounding box that is displayed around the lines once they are drawn. Examples of use of the spline classes. Their basic constituents are piecewise power functions of degree α. Convert to Editable Mesh, then Editable Poly. This kind of spline morphing animation is awesome for using in conjunction your 2D workflow in After Effects by applying a Cel Shader material to your splines. Simply increasing the number of control points adds little local control to the curve. To define B-spline basis functions, we need one more parameter, the degree of these basis functions, p. Edit in jsFiddle Edit in CodePen Edit in Highcharts Cloud. ) If you use splines to create shapes, try to use a very small number of vertexes. Drawing a cubic B-Spline curve where plist is the control polygon. The algorithm used for generating is for cubic Bspline. B-Spline curves. curves-and-splines-finished. Could anyone help me about B-spline Curve error? I want to draw B-spline Curve in c++, but even though all coordinates are positive, the segment's coordinate is negative. Number of Points Specifies the number of interpolated points. Mathematics of Computer Graphics and Virtual Environments 59,457 views. 1-8) Describes the intended audience for this product. I often use B-splines for regression. bSpline: Evaluate a Spline at New Values of x (splines) predict. To leave that mode just select another tool from the toolbox. A bezier curve is defined by control points. The higher the order is, the more smooth the spline becomes. Knot vectors. Generally you would also supply a value at each knot, and either a control point between each consecutive pair or a first derivative. Each single click creates a new "Vertex". A B-spline with no internal knots is a Bézier curve. An Introduction to Bezier Curves, B-Splines, and Tensor Product Surfaces with History and Applications Benjamin T. If necessary, you must build the appropriate index groups for A and B before running grompp. Univariate spline in the B-spline basis. Demonstrating a basic area chart, also known as a mountain chart. Consider the function f x xexp( ) sampled at points x,0 0. Splines in MATLAB (p. B-splines over nonuniform knots go back to a review article by H. Using the B-Spline Tool, start with a line that curves to the right and up. The algorithm given in w:Spline interpolation is also a method by solving the system of equations to obtain the cubic function in the symmetrical form. B-spline Curves: Definition. Example 12-5 renders a NURBS surface in the shape of a symmetrical hill with control points ranging from -3. That is it sets the number of line segments in the control frame. China Abstract — Hierarchical B-spline has been widely used for shape modeling in many fields. the B-spline surface defined by these information is the following: where N i,p (u) and N j,q (v) are B-spline basis functions of degree p and q, respectively. Recently, there exist some demands, e. 2-D spline representation: Procedural (bisplrep) ¶For (smooth) spline-fitting to a 2-D surface, the function bisplrep is available. 65) where the knot vector is obtained by dropping the first and last knots from (1. 1940 1950 1960 1970 1980 1990 2000 2010 0k 10k 20k 30k 40k 50k Highcharts. B-spline Polynomial Let p 0;:::;p n be the control points. 2 An Example The default R function for tting a smoothing spline is called smooth. 64) are tangent to the control polygon at their endpoints. Splines are the most used objects in any normal animation done with Synfig, and the Spline Tool allows you to create them. Let's focus on the ease-in to Pose B first. b spline basis function. T YouTube video about T-Splines and Other Stuff @noyb Personally I would like to thank you for putting in the effort to even make the video, and I would like to commend you for its presentation as I found it to be very upbeat and far from boring like 99% of all the other videos out there. You will start with the same pawnshop model featured in the Composite Map tutorial, and use a spline to map a brick-like material onto the building’s arched entrance. The BSpline package provides an implementation of a Cubic B-Spline method devised by Vic Ooyama, and brought to our attention by James Franklin. Fbsp wavelet (302 words) exact match in snippet view article find links to article. B-Spline curves. Many types of constraints are not supported at this time. Control point splines - also known as CV splines, NURBS curves or style splines - provide a way for defining complex curves in a Sketch. Now instead of stopping and holding at Pose B, it gets almost all the way to Pose B and then eases in. Please refer to the wiki for details on using this class member. Here is another tutorial about FreeCAD, which is an open-source alternative to AutoCAD.  In this case, we chose to do a goat but you can pick any animal you like. Up to now I've never needed to understand the output of bs in detail: I would just choose the model I was interested in, and fit it with lm. b-spline B-spline interpolation. 5, Double_t b=31) {// array. Polynomial regression is computed between knots. Now instead of stopping and holding at Pose B, it gets almost all the way to Pose B and then eases in. de Boor started to work for the General Motors Research labs and began. Recently, there exist some demands, e. Use Draft WireToBSpline to convert one to the other. spalde (x, tck) Evaluate all derivatives of a B-spline. The B-spline basis is non-global. In tro duction This essa y reviews those basic facts ab out (univ ariate) B-splines whic h are of in terest in CA GD. Each basis function has precisely one maximum value, except for k=1. As shown in the last example, the main problem with Bezier curves is their lack of local control. A mesh modified by a curve modified by a curve ensures constant length. B-spline curves with a knot vector (1. Entity b_spline_surface definition. B-spline Basis Functions: Computation Examples. Limitations. The curve exhibits the variation diminishing property. From this, you will have the knowledge to create any spline. The basis functions of degree. B-splines (short for Basis Splines) go back to I. Here is what you need to do. I often use B-splines for regression. WaveDrom renders the diagrams directly inside the browser. E Displays the point editor dialog in the current display view, or sets keyboard focus to point value controls in the spline and timeline editors. Then plot the B-spline with knot sequence t, as well as its polynomial pieces, by using the bspline function. This will teach you how to combine several splines to form a single object, use edit spline extortion modifiers. Each such curve segment is controlled by a couple of consecutive control points. As shown in the last example, the main problem with Bezier curves is their lack of local control. You will start with the same pawnshop model featured in the Composite Map tutorial, and use a spline to map a brick-like material onto the building’s arched entrance. are B-spline basis functions of degree k and knots t. Help retrieves a point on a B-spline curve Author vvvv group. Make your Curve a path via a Path Constraint controllers for the sphere. The orange point at x=4 corresponds the point I wish to directly determine the y value of the b-spline at. A few tips: Make sure you zoom to the layer extent of the spline interpolation map first before adding it as a map to the print composer, text editing for the 'label' is done in a text editing box on the right. PDF Version Quick Guide Resources Job Search Discussion. I think there is some confusion here between degree and order. With this tool, you select a series of points, and Fusion will create a smooth curve that passes through them. The package comprises of a graphical utility to place uniform B-spline control points and see how the B-spline is redrawn as control points or control point weights are adjusted, and functions to estimate B-splines with known knot vector, given a set of noisy data points either with known or unknown associated parameter values. Tutorial for Spline Rational B-splines have a weight for every control point, which can raise or lower the influence of the control point, default weight = 1, to lower the influence set a weight < 1 to raise the influence set a weight > 1. , the rth derivative at τ of the jth B-spline, for all j, for some r and some site τ. So we're gonna leave this area alone. Bruce and Bruce 2017). Splining is a 3D animation term. In function BSplineCurve the t parameter should take values in [0, 1] range. Technical Note: If the stoke is not active you can re-activate it while in the Pen tool mode by holding the Command key while clicking anywhere along the stroke. Splinophilia Part 1. Knot vectors. 1-8) Describes the intended audience for this product. Steiger (Vanderbilt University) An Introduction to Splines 17 / 23. 12 | MarinStatsLectures - Duration: 6:48. Make your Curve a path via a Path Constraint controllers for the sphere. Great tutorials! There are so many tutes on the net which are just do A, then B, then C. MATLAB is an interactive program for numerical computation and data visualization; it is used extensively by control engineers for analysis and design. The basis functions of degree. Drawing a cubic B-Spline curve where plist is the control polygon. sproot (tck[, mest]) Find the roots of a cubic B-spline. In AutoCAD, a SPLINE is a smooth curve that passes through or near a set of fit points. The advantage to B-splines is that we can represent m Bézier curves. Linear spline: with two parameters and can only satisfy the following two equations required for to be continuous:. The places where the pieces meet are known as knots. Cubic Spline Regression Restricted Cubic Spline Regression To force linearity when X a k, the last two s are redundant, i. You will start with the same pawnshop model featured in the Composite Map tutorial, and use a spline to map a brick-like material onto the building’s arched entrance. I don't see spline handles if I draw a B-spline. This is the first of a few tutorials that demonstrate useful functionality of Unity that is not directly associated with shaders; thus, no shader programming is required and all the presented code is in C#. Interpolating cubic B-splines. …We'll compare it to it's more well known cousin,…the Bezier Curve, for which it is often mistaken. Josef Pirkl. B-spline Polynomial Let p 0;:::;p n be the control points. 1 B-splines. B-spline curves with a knot vector (1. Blogger viral that we have collected from various sites, and of course what we recommend is the most excellent of image for B Spline Tutorial. The novel method models the background and foreground using the B-Spline method followed by a tracking-by-detection algorithm. 1-8) Describes the intended audience for this product.  In this case, we chose to do a goat but you can pick any animal you like. k))**2 + lambda R(f). You can adjust the thickness of this tube structure by increasing or decreasing the radius of the N-Side Spline. The active anchor point (B) is solid and the inactive anchor point (A) is an empty box. The basis functions of degree. What is a SPLINE. It would be a useful exercise for you to prove this for k =3. curves-and-splines-02. Splines conforming to B92. Essentially, they are made up of a series of 'from' and 'to' co-ordinates. As shown in the last example, the main problem with Bezier curves is their lack of local control. With this piece of advice this tutorial ends here. In function BSplineCurve the t parameter should take values in [0, 1] range. 2 B-spline curve Up: 1. The splines of a newer class, the ap-. Please refer to the wiki for details on using this class member. B-Splines and Smoothing Splines B-Spline Properties. They are a very simple thing, worth to study once and then feel comfortable in the world of vector graphics and advanced animations. 00" x 15 spline 1. One constructs the corresponding B-splines through a localization process similar to the classical one, replacing finite differences by fractional differences (c. , the matrix that, in each row, contains the sequence of numbers D r B j,k (τ), i. Given a 3D cloud of points accompanied by normal vectors an implicit b-spline surface will be reconstructed. Video Tutorials. Unreal Engine 4 Marketplace Smart Spline Generator This tool allows to generate various assets that automatically align/react to any surface, in highly customizable way. Because B j,k is nonzero only on the interval (t j. A B-spline neural network (BSNN) consists of the piecewise polynomials with a set of local basis functions to model an unknown function for which a finite set of process samples are available. Because the functionality is contained in the EFFECT statement, the syntax is the same for other procedures. It features two orientation modes: NODE : It directly takes the orientation from the control point TANGENT : Calculates the orientation based on the tangent to the curve in. I select the 3rd key and lower it a little, then tilt its tangent handle so that the spline flows smoothly through it. B-Spline path. Splines conforming to B92. It gets even trickier to work with when you deal with NURBS, or Non-Uniform Rational B-Splines. Cara Menggunakan B-Spline di CorelDraw, pada setiap versi corel dengan rilisan terbaru, beberapa fitur tambahan telah ditambahkan. and Sappa A. In this note, we use u 0 = 0 and u m = 1 frequently so that the domain is the closed interval [0,1]. b-spline In computer graphics, a curve that is generated using a mathematical formula that assures continuity with other b-splines. -splines are implemented in the Wolfram Language as BSplineCurve[pts]. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. Test dimensions over pins can be seen in the picture below. B-spline knots. Help retrieves a point on a B-spline curve Author vvvv group. Here is another basic Spline Modeling Tutorial for Beginners. Find a, b and c so that S (x) is a natural cubic spline. Control frames provide a convenient method to shape the spline. com/drawplus and ask questions in our friendly communi. edu is a platform for academics to share research papers. Corresponding t intervals (in the right window) are painted in the same colors. A set of non-descending breaking points defines a knot vector. B-spline basis functions Recurrence relations. The other method used quite often is w:Cubic Hermite spline, this gives us the spline in w:Hermite form. Until now, this requirement has been met by the existing Spline command. The basis function is a cubic B-spline, but the knot sequence is nonuniform, with a multiplicity of 4 at each endpoint, causing the basis function to behave like a Bezier curve in each direction. hello, I would like to display a cylinder in openGL. the degree of function is (m-1), where m is an order of the function. vdwtype = User coulombtype = User energygrps = A B energygrp_table = A A B B Now mdrun will expect to find three files: table. B-Spline curves. This section provides two examples of NURBS curves: a B-spline curve and a uniform B-spline curve that passes through the end control points. Learn B-spline skills by watching tutorial videos about Advanced Music Theory for DAW Musicians, Launchpad Pro: The Video Manual, Getting Started With Illustrator, Daniel Wyatt's Mastering Tips, DAW Studio Setup and Design, & more. Parametric spline curves. There are many different toolboxes available which extend the basic functions of MATLAB into different application areas; in these tutorials,. Drawing B-Spline Curves using GDI+ in VB. By default, the splines are created as renderable objects, but you can change that at the “Display / Options” rollout, at the bottom of the interface. Each such curve segment is controlled by a couple of consecutive control points. Therefore, a B-spline surface is another example of tensor product surfaces. How to Design Curves Curves in CorelDRAW can be as simple as a single straight line, or complex open or closed shapes comprised of curved or straight segments. Now instead of stopping and holding at Pose B, it gets almost all the way to Pose B and then eases in. I am trying to construct a toy example but I can't seem to get my example to work. Power BI is a Data Visualization and Business Intelligence tool that converts data from different data sources to interactive dashboards and BI reports. Scattered data fitting has always been a challenging problem in the fields of geometric modeling and computer-aided design. However, they have more desired properties than Bezier curves. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. Here is my sample output. Two things are going wrong in your code. It's my mistake to have proposed for programming because I'm the collab who had already explained the do loop without programming and tutored. Please visit the Tutorial page here: 5. Here is an image of a SPLINE. 1-6) Outlines the organization of this user's guide. Schoenberg who introduced them in 1946 [123] for the case of uniform knots. This is due to the nature of the bleanding used for Bezier curves. A table of the variables and the formula that calculates its value is included. Cubic Splines Cubic […] Related Post Chi-Squared Test - The Purpose, The Math, When and How. In tro duction This essa y reviews those basic facts ab out (univ ariate) B-splines whic h are of in terest in CA GD. B-spline curves, or basis spline curves, are shapes that consists of several Bézier curves. The curve generally follows the shape of defining polygon. Exceedingly Simple B-spline Code; by Jan de Leeuw; Last updated almost 5 years ago; Hide Comments (-) Share Hide Toolbars. , the matrix that, in each row, contains the sequence of numbers D r B j,k (τ), i. Each single click creates a new "Vertex". where N i,p (u)'s are B-spline basis functions of degree p. Button to add new point mid way between points. Creating splines is quite a simple process once you understand the basic logic behind them. Thus, a change in the position of a control point only proapagates upto a predictable range. ) If you use splines to create shapes, try to use a very small number of vertexes. I'm thinking this is where I need to go on my next POV Learning journey :) I got a spline editor. B-spline Basis Functions: Computation Examples. The algorithm given in w:Spline interpolation is also a method by solving the system of equations to obtain the cubic function in the symmetrical form. Suppose we want to construct the basis functions. 1) E [f]= 1 n X (y i x)) 2 + Z 00) dx in the space of functions with square in tegrable second deriv ativ e. The orange point at x=4 corresponds the point I wish to directly determine the y value of the b-spline at. Polynomial regression is computed between knots. Cubic Spline Regression Restricted Cubic Spline Regression To force linearity when X a k, the last two s are redundant, i. Since there are n intervals, there are 4ncoe cients to determine. In order to do this we have to go to Edit – Current Drawing Preferences – Splines and define the number of line segment for the spline in the current drawing, the higher the number, the more accurate and smoother the line: notice that into the image below 8 segment are setted, this set is fair enough for our tutorial. What is a SPLINE. Thank you for any. The model is reprojecting the wgs84-points to UTM32N (EPSG:25832). B spline curve generation in Python. Each WaveLane has two mandatory fields: "name" and "wave". We will show how to construt a Spline space where all basis functions vanish on the boundary. k))**2 + lambda R(f). Tips and Tricks. 1-7) Compares spline approximation using the MATLAB® spline command with the capabilities of the Spline Toolbox. Please wash your hands and practise social distancing. It would be a useful exercise for you to prove this for k =3. For ease of calculation take the knot vector to be [0,0,0,1,1,1]. basis-closed - a closed B-spline, as in a loop. In other words, splines are series of polynomial segments strung together, joining at knots (P. These 4 points control the shape of the curve. B(t) = P 0 + t(P 1 - P 0) = (1 - t)P 0 + tP 1, tε[0,1] What is B-Spline Curve? B-Spline curves are considered as a generalization of Bezier curves and as such share many similarities with it. \) The default output is a list $$\left[tx,ty,c,kx,ky\right]$$ whose entries represent respectively, the components of the knot. Select a B-spline and apply. Expected Background (p. B(asic)-Spline Basics Carl de Bo or 1. A very short note on B-splines By Samiran Sinha Department of Statistics, Texas A&M University, College Station, Texas 77843-3143 [email protected] Element "signal" is an array of WaveLanes. Viewed 1k times 1 $\begingroup$ I am using the de boors algorithm to generate B splines in python. Scattered data fitting has always been a challenging problem in the fields of geometric modeling and computer-aided design. Procreate's 10 Best Features (draw straight lines, paint bucket tool, gradients and more) - Duration: 7:51. Curry in 1947 [38]. This tutorial discusses one way to render quadratic Bézier curves and splines in Unity. Create sphere. When drawing lines by using the curve tools, you can hide the bounding box that is displayed around the lines once they are drawn. Interpolation of an N-D curve¶ The scipy. Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions. Thus PHT-splines are a generalization of B-splines over hierarchical T-meshes. A B-spline (or basis spline) can achieve high orders of continuity, but the tradeoff is that it's a bit harder to calculate things than with cubic Hermite splines or Bezier curves. The sole tutorials valid to read are the Matlab and completed Mathcad treasury, passed many times in this collab. In this video, we'll take a look at the foundation of organic modeling, the…B-spline, and compare to it's far more well known cousin, the Bezier curve for…which it is often mistaken. The interpolator can take an optional weight vector, making the resulting curve a Non-Uniform Rational B-Spline (NURBS) curve if you wish so. Modeling a Key with Splines. 3 Choosing a Curve Fit Model 1. Click the Spline Fit button on the Draw panel slideout of the Home tab, or type SPL and. An Introduction to Splines 1 Linear Regression Simple Regression and the Least Squares Method Least Squares Fitting in R Polynomial Regression 2 Smoothing Splines Simple Splines B-splines Over tting and Smoothness. NURBS in FreeCAD come from Open Cascade, which is the geometry kernel the application uses at its core. Here is my sample output. In this note, we use u 0 = 0 and u m = 1 frequently so that the domain is the closed interval [0,1]. Examples of use of the spline classes. Package 'splines2' June 14, 2018 Title Regression Spline Functions and Classes Version 0. It is easy to use this tool. Reuse Animations. Learn B-spline skills by watching tutorial videos about Advanced Music Theory for DAW Musicians, Launchpad Pro: The Video Manual, Getting Started With Illustrator, Daniel Wyatt's Mastering Tips, DAW Studio Setup and Design, & more. Disclaimer: This tutorial assumes that the reader has a basic understanding of 3d modeling concepts and scripting with blueprints in UE4. f definitions ). i+ b ix+ c ix2 + d ix3: To determine the spline we must determine the coe cients, a i, b i, c i, and d i for each i. Spline Curves and Surfaces Introduction. It features two orientation modes: NODE : It directly takes the orientation from the control point TANGENT : Calculates the orientation based on the tangent to the curve in. The deviation in the response for any observation does not depend on any other observation. METHOD: NATURAL CUBIC SPLINE. @misc{_nurbscurves, author = {}, title = {nurbs curves and surfaces tutorial introduction to curves and surfaces 2 introduction to b-splines. B-spline curve showing knot multiplicity. …We'll compare it to it's more well known cousin,…the Bezier Curve, for which it is often mistaken. B-spline Basis Functions: Computation Examples. Disseminates high level research results and engineering developments in all aspects of signal, image, and video processing ; Sets forth practical solutions for current signal, image, and video processing problems in engineering and science ; Features reviews, tutorials, and accounts of practical developments, in addition to original research work. In the Motion Panel/Trajectories, use the Convert To button to create spline. B-Spline Curve Example 8-1 creates and displays a B-spline curve. 2 B-spline curve Up: 1. gaona これカモ 凍結防止ヒーター サーモスタットつき 20m ga-ke023 1ヶ. The model is reprojecting the wgs84-points to UTM32N (EPSG:25832). 4 B-spline curves and Contents Index 1.
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# Parallele Computing - 2 vs. 4 processor speed [closed]

I am evaluating a code which ends with Table having ParallelEvaluate of a function XXXX[phi, theta, si]. For a grid of 225 points, a normal 2 processor laptop is taking 7 h as compared to 8.30 h by a high end Xeon 4 processor computer. CPU and memory usage for laptop and computer are about 66% vs 99% and 700MB vs 900 MB respectively. Will be thankful for any suggestion on how to improve the evaluation speed on computer. Thanks

• Okay, having skimmed through the code, I can say that your actual problem is not parallelization but your coding style. You use far too much symbolic computation; you use unpacked arrays (in particular because you mix integer, symbolic and machine precision numbers in arrays); you recompute data over and over again (have a look at the many Sort and SortBy commands); instead of concise function calls with purely numerical input and output, you use replace rules;... – Henrik Schumacher Jan 19 '19 at 2:20
• Running XXXX[0, 0, 0] once takes 135 s on my machine. I guess this can be executed 100--1000 times faster with proper refactoring of your code (and probably by using Compile here and there). – Henrik Schumacher Jan 19 '19 at 2:22
• @HenrikSchumacher You are correct, there could be many ways to write code, and as someone writing such complicated Mathematica code for the first time, I might have not opted the most efficient sub-steps. Though my Q about parallelization remains. Let us consider for any code taking x sec to evaluate at one point, how can we scale it linearly with no of points and no of processors (using ParallelTable or ParallelEvaluate or any other method) ? Will be thankful for your suggestion on that. In the meantime, I will try to modify code to reduce time/point "x" by incorporating your suggestion. Thx – user49535 Jan 19 '19 at 4:49
• @user49535 I think that your replacement rules are killing you. If I run DSC[0, 0, 0, 1] by itself, I get output that's over 12 million bytes because the code is unable to multiply the numbers by your f values since they're one of the last things to be defined. If possible, I would try to store the f values as actual numbers in a matrix. It looks to me like the output of DSC is actually supposed to be a matrix with 36 rows and 3 columns, where the second 2 columns are just indices, so it should be on the order of 1000 bytes. – MassDefect Jan 19 '19 at 5:52
• @user49535 Out of curiosity, is there a particular resource (like an algorithm from a book or code in another language) that you're trying to emulate? I'm trying to figure out what f does, but it's difficult. XX1 calls XXXX which calls DSC -> SEM -> SE -> TrueStrain -> EigenStrain which contains these f variables, but the computer doesn't know what they are yet. Then we go all the way back up the stack to DSC, and some of the f variables are replaced with other f variables but still not assigned a value, until we go back up to XXXX and have dsc/.R[m-1] – MassDefect Jan 19 '19 at 7:39

Without knowing the exact function (I assume it's something fairly long, possibly involving integrals or differential equations), I can only make the following suggestions:

It looks like you're using exact numbers. If this is necessary for your application, then there's probably not a lot you can do, but exact numbers usually slow things down substantially. If you can, use Real numbers (just place a dot after the numbers like {phi, 0., Pi/4., Pi/56.}. If you need more precision than that but don't necessarily require the infinite precision of exact numbers, you can also do this: {phi, 050, Pi/450, Pi/5650}. This will give you 50 digits of precision to work with which should make your final answer pretty close to the exact answer.

The other thing I would try is:

XX1 = ParallelTable[
{XXXX[phi, theta, si]], phi, theta, si},
{phi, 0, Pi/4, Pi/56},
{theta, 0, ArcCot[Cos[phi]], ArcCot[Cos[phi]]/14},
{si, 0 Pi, 0 Pi, 0}
]

I think that ParallelTable is a better way to handle this than ParallelEvaluate. On a trial function, I see about a 100x speedup. ParallelEvaluate is simply evaluating your exact same function 4 times at each data point rather than splitting the task into multiple threads.

If you can, combine both things for the best speedup.

I hope this helps a bit! There are some people on here that are amazing at optimizing, perhaps they will be able to improve the speed even more. If it's possible, I would recommend posting your XXXX function unless it's insanely long.

• Thanks @LukasLang ! How do you type grave accents without it interpreting them as the inline code markers? I tried backslashes before them, but that didn’t help. – MassDefect Jan 18 '19 at 9:10
• You have to increase the amount of enclosing accents:   Codewithaccents . If you need double accents, you enclose the code with three, and so on (edit: for some reason, it doesn't work in the comment section - but you can edit your answer to see how it's done) – Lukas Lang Jan 18 '19 at 9:24
• @LukasLang Oh, I see! Thanks! – MassDefect Jan 18 '19 at 9:32
• Thanks both of you. three points 1. I do not necessary need to use exact values of (theta, phi) if it can speed up, can use ".". 2. I tried to use ParalleleTable first, but in contrast to your experience, it took 30h/48h for 4/2 processor computer as compared to 8h/7h for ParallelEvaluate. 4 - 8 times slower. 3. How can I combine both...you mean ParallelTable[ParallelEvaluate[. ?? – user49535 Jan 18 '19 at 11:03
• @user49535 As MassDefect already pointed out, using ParallelEvaluate here does not make sense at all. It enforces that the same value is computed on each of your CPU cores which is why you won't gain any speedup. It really depends on your actual function XXXX  whether ParallelTable can help at all. If it is a pure function then ParallelTable should help.But if XXXX has side effects (like modifying data that has to be used by another thread) then it is hard to parallelize the execution. In a nutshell, we cannot give any further suggestions without knowing XXXX`. – Henrik Schumacher Jan 18 '19 at 11:49