Good morning. Starting from this lecture, in a few lectures, we will cover the classical topic of numerical analysis, function interpolation, then numerical integration and numerical solution of ordinary differential equation. First, in this lecture, we discuss interpolation and function approximation. The, why just use interpolation of functions is through Poisson mill. So, first we discuss Poisson mill interpolation. The problem here is to find an analytical representation of a function from information at a finite number of data points. For example, suppose you have got a function of x for which you have data regarding the function at several data points, say here, here, here, here and here. And from that, you want to develop the function in the form and in the form of an expression. So, that with that expression, you can develop the function continuously and evaluate the function at any point in between. So, that is called function interpolation. Now, the purpose is to evaluate at arbitrary points at which your data is not there. And apart from that, you, if you have an analytical representation in the form of an expression, then you can differentiate or integrate the function as the need arises. Besides, once you have an expression for the function, you can draw certain conclusions regarding the trends of the function or nature of the function. So, with these purposes in mind, we try to develop analytical expressions, analytical representations of functions from information available at discrete data points. Perhaps, those data points, those discrete information can be obtained from experimentation or through some other computations, which cannot be conducted at infinite points. Now, interpolation which we are going to consider now, which we are going to discuss now is one of the ways of function representation. There are other ways also. And in the interpolatory approximation of a function, the sampled data, the data which is given are exactly satisfied. Now, polynomials offer a convenient class of basis functions as linear combinations of which we can express the function quite easily. So, in a particular problem, suppose y i as f of x y is available at these n plus 1 points, x 0, x 1, x 2, x 3 up to x n. With the help of these n plus 1 pieces of data, we want to fit a polynomial of the nth degree. Now, why nth degree and n plus 1 points? Because a polynomial of nth degree will have n plus 1 coefficient, the determination of which will require n plus 1 pieces of data. And that n plus 1 pieces of data you get from the function values at these n plus 1 points starting from x 0 up to x n. Now, the task of formulating or finding this polynomial is basically the determination of these a 1, a 2, a 0, a 1, a 2 coefficients which will match the given n plus 1 pieces of data exactly that is interpolation. After that at any other value of x in this interval, if we try to find out the value of p x from this polynomial, that will in a way represent the function f at that point. And this is what is called interpolation finding the value in at an in between point. If based on this same expression, if we try to find out the function value or estimate the function value at a point outside this interval, such an activity is called extrapolation which for all situations is not considered safe and reliable enough. Now, one question arises that for given values n plus 1 values of the function at n plus 1 points, can we always find a polynomial of the n s d v like this? And the other question that arises that if we can find then will a single set of values for this coefficient will be found or whether we can find multiple sets of values for this coefficient that is is this polynomial fitting this data unique. First of all whether it exists for all data of this sort and whether that is unique whether the polynomial exists which will interpolate this data and whether that polynomial is unique. Now, if we just try to insert the values of x 0 x 1 etcetera in this and equate the corresponding expressions to y 0 y 1 etcetera, then we get n plus 1 equations in these coefficients. These equations are here a 0 plus a 1 x 0 etcetera up to a n x 0 to the power n equal to function value at x 0 and so on. Now, the values of a 0 a 1 up to a n will exist and there will be unique if this coefficient matrix is non singular. Now, this particular matrix with different values of x 0 x 1 x 2 x 3 etcetera is a well known matrix known as the van der monder matrix and it is known to be invertible. You can analytically show that it is invertible and but then it is typically ill conditioned. Now, you can analytically show it is invertible, but here we will not take up that exercise because we are going to establish the invertibility of this matrix that is existence and uniqueness of this coefficient that is the polynomial through some other shortcut means. So, therefore, we will not spend our time on formally showing that it is invertible from the existence and uniqueness of the polynomial coefficients the invertibility will be established. But keep also in mind this fact that is even though this matrix is invertible, but typically it is ill conditioned that is the solution process will quite often have numerical errors because of the ill conditioned. Now, we try to see the uniqueness of this polynomial coefficient set suppose there are two polynomials that means two sets of values for these coefficients which match the function values at the given data point and those two polynomials are p 1 and p 2.  So, their difference will be at most an n s degree polynomial. Now, at n plus 1 data points p 1 matches it exactly that means at those n plus 1 points the p 1 polynomial p 1 will have exactly these values p 2 also. So, the difference of these two polynomials will have zero value at n plus 1 points that will mean that delta p x is an n s degree polynomial of x, but at n plus 1 values of x it has got zero value. Now, you know from the fundamental theorem of algebra that an n s degree polynomial will have exactly n roots it cannot have n plus 1 roots and that means that if delta p satisfies such a requirement that will mean that it is not only zero at those n plus 1 values, but it is the zero polynomial that means the entire polynomial is zero that means all the coefficients are zero which will mean that delta p is zero as a polynomial will mean that p 1 and p 2 are actually the same polynomial that shows that p x is unique and p x is p x exist that we can argue based on the properties of linear system of equations, but let us go another round and existence also we establish in a little short cut argument. How for that we do something which we would have done otherwise also because we know that solution of this will require as to solve an ill conditioned system. So, typically we may not like to do that for large value of n that is for high degree polynomials. So, what we can do the alternative is Lagrangian deposition for that we consider the basis functions not as 1 x s square x cube the way we did last time in this function representation the basis functions were 1 x s square x cube etcetera and coefficients were a 0 a 1 a 2 etcetera. So, we have noted that with this kind of a polynomial expression it will be very easy to differentiate and integrate etcetera, but then evaluation through the solution of this kind of a system is going to be unnecessarily time consuming. So, what we can do is that we can use basis functions not as 1 x s square etcetera, but this kind of basis functions examine carefully in this expression we have got in the denominator product of x k minus x j j running from 0 to n except j equal to k of course, because for j equal to k this will become 0. So, that term we do not have all other terms we have. Now, in the numerator in place of x k we have x. So, the denominator is a constant number. So, that means for the second basis function k equal to 2 we have got here x 2 minus x 0 x 2 minus x 1 x 2 minus x 3 x 2 minus x 4 and so on up to x 2 minus x n and that is a number. In the numerator we will have the corresponding x minus x 0 x minus x 1 x minus x 3 x minus x 4 etcetera up to x minus x n omitting x minus x 2. So, that means that this numerator is an nth degree polynomial in x denominator is a number. So, l 2 x is a is an nth degree polynomial. Similarly, l 0, l 1, l 2, l 3, l 4 all of them will be nth degree polynomials. So, we have got n plus 1 nth degree polynomials. So, any linear combination of them also will be an nth degree polynomial and this is the linear combination.  Now, these functions l 0, l 1, etcetera basis functions at x 0 then note that l 0 will be found to be 1 at x equal to x 0 because in that expression for the expression of l 0 this x minus x 0 will be missing all others will be there and here x 0, x 0, x 0 will be there. So, when we put x equal to x 0 this and this will cancel next, next will cancel and so on all all of them will cancel and we will get the value 1. At that same point x 0 if we try to evaluate l 1 then in the expression of l 1 this term will be missing this term will be there. So, at x 0 when we try to evaluate then we will get x 0 minus x 0 which is 0 and that means that the expression and here it will be x 1 minus x 0. So, that means that l 1, l 2, l 3, l 4 all of them will be 0 at x equal to x 0. So, that means that suppose this is x 0, this is x 1, this is x 2, this is x 3 and this is x 4. So, that means l 0 will be 1 here, 0 here, 0 here, 0 here, 0 here, l 1 will be 0 1, 0 0 0, l 2 will be 0 0, 1 0 0 at other places which are not the given data points all of these expressions will have some other values, but at the jth data point l j will have value 1 all others will have value 0. And therefore, in this when we try to equate the function values at x 0, x 1 etcetera then we will find that at x 0 when we try to put the value then we will have alpha 0 into 1 plus alpha 1 into 0 plus alpha 2 into 0 and so on. So, we will get alpha 0 into 1 is equal to f at x 0. So, immediately we will get alpha 0 as the function value itself at x 0. Similarly, alpha 1 will be the function value at x 1 and so on. So, at the data points we have this that is at the at as k is equal to y we have 1 and k not equal to y we have 0. So, that means that basically we are trying to solve a system of equations in which the coefficient matrix is identity this and that means alpha i is going to be just f of x y x i.  So, the Lagrang interpolation formula we get like this. Now, note that in this development from here to here we have done nothing which will not be possible for arbitrary data arbitrary points and arbitrary function values. That means for any set of points and for any function values this polynomial will always exist and this is a nanothetical polynomial and earlier we have seen that whatever energy degree polynomial we can find to exactly satisfy n plus 1 data point that has to be unique. So, that means we basically get this polynomial which is the same as the polynomial that we would have got otherwise through the earlier process except that here we get the polynomial very easily without any difficulty, but then differentiation integration etcetera will be more cumbersome from this kind of expression. So, here you will find that we have got 2 interpolation formulae one costly to determine with the basis functions 1 x x square etcetera, but easy to process afterwards. The other trivial to determine the Lagrange interpolation, but costly to process later differentiation integration etcetera depending upon your situation depending upon your need you will be in a position to frame the polynomial coefficients and the basis functions either this way or that way. If you want a formula a methodology by which it will not be so costly to determine on the other side it will not be so costly to process you have an intermediate case also that is the Newton interpolation which is an intermediate data there you have got the formula in this manner. So, in this case if you try to put x equal to x 0 you will find all these are 0. So, you will get C 0 is equal to f of x 0 then when you put x equal to x 1 you will have this and this and all others having a factor x minus x 1 will become 0. So, you will get another equation in C 0 and C 1 and so on. So, this will have a triangular coefficient matrix. So, it is easier to develop compared to the first case and more difficult to develop compared to the second case Lagrange interpolation. On the other hand the processing that is differentiation integration etcetera also will have intermediate computational cost. Now, another kind of interpolation is there which uses not only the function value, but also some of the derivative such an interpolation is called Hermite interpolation it has a lot of applications because in many situations many problems you need to satisfy not only the function values, but the rates as well they are part of the problem that is you cannot have an have a polynomial which changes in a rate which is different from what is already known to be the rate of change of that function which you are trying to model this is one important point. The second important point is that sometimes with the help of interpolation which uses derivative information also at certain points or in the locality in the local neighborhood of certain points we can establish we can develop function approximations which have the required trends. So, therefore, Hermite interpolation is quite well used kind of interpolation. So, in this what kind of data you will talk about you will still have a few points say x 0 x 1 x 2 up to x m a handful of points at which you will use the data. And at every point the data may be function and some of its derivatives it is not even necessary that at every point you have the same number of derivatives to be used it is possible that at x 0 the function value and two derivatives are given at x 1 the function value and one derivative is given at x 2 only function value is given at x 3 the function value as and one derivative is given that is possible now whatever be the number of data items function and its derivative values at every point. So, if you add up those number of thus those conditions the total number of conditions that is given in terms of function values and their derivatives that will equal one more than the degree of the polynomial that you can model why because these many conditions will give you the number of that many equations in the unknown coefficients of the polynomial and if you have got n plus 1 coefficients in the polynomial that means you can go up to n at the degree n plus 1 equations n plus 1 unknowns. So, the number of the conditions will be one larger than the degree of the polynomial by up to which you can correctly model the function. Now that way you can think of a function in which at this point three pieces of data are given at this point two at this point four at this point again two and at this point one. So, what is the total number of conditions that you have got three two four two one that means two s conditions that means through these points satisfying all these function values and derivative values that are given you can make an eleventh degree polynomial which will fit this data exactly, but then there are very strong objections to high degree polynomials as function representation. One is of course the computational cost and numerical imprecision that is you will have to solve a 12 by 12 system of equations and that may be in condition. So, that may lead to computational cost and numerical imprecision, but this is a minor objection. The major objection is that the high degree polynomials sometimes fail in their prime duty to represent the function well. Let us see how as you know a linear function of x either goes all the way up or goes all the way down or stays constant in the particular exceptional case that is if you remove this then you will say that typically a linear function goes all the way down or goes all the way up. A typical quadratic function has this kind of a loop it goes down takes one turn and then goes up never takes another turn or it goes up takes a turn then goes down this will be the case if the coefficient of x square is positive this will be the case if the coefficient of x square is negative. So, a quadratic function of degree two has two trends once going down and once going up a cubic equation a cubic expression a cubic function that is degree three will have at most three such trends going down then going up and then possibly going down again. I said at most because a cubic could also go like this that is also possible, but then a cubic has the possibility of showing this kind of a trend. A four degree polynomial similarly will have the possibility of showing a trend of this kind or of this kind this is the situation is a four degree function. Now if you talk of an eleventh degree function that may have scope to one two three four five six seven eight nine ten eleven this kind of a profile it might show that is up to that much it can oscillate. That means that if your data are like this it is possible for an eleventh degree polynomial to give you this representation which is a very bad representation because through these data points the function that you were possibly approximating was this it is a nice function, but because you allowed an eleventh degree polynomial the polynomial representation turned out to be this which is matching the data exactly, but other than that it is doing nothing sensible. So, this is the prime objection to high degree polynomial representation of functions that it may be failing in the prime duty that to represent the function faithfully that itself may be failed in a problem of this sort. So, that is why single high degree polynomial interpolation is something which we usually do not do then what do we do we make piecewise polynomial interpolations. Simplest piecewise interpolation is piecewise linear interpolation. So, that means that if we have a large number of data points like this then one straight forward sensible way to find a piecewise interpolation of this could be to join these with straight lines and say that is a function representation which is good enough for many purposes. So, for dense data where function values are available at a large number of close by points there quite often we conduct piecewise linear interpolation, but we have to keep in mind that this representation of the function will not be differentiable because at the corners from this side there will be some slope and from this side there will be some other slope. So, this will not be differentiable. So, piecewise linear interpolation is not differentiable you can see that this formula gives you the piecewise linear interpolation between x i minus 1 and x i you will get this formula which matches x i and x i minus 1 exactly and it is a linear expression. Next higher interpolation typically used and very popular representation is piecewise cubic interpolation. Now, this is typically hermit segments what you do in this piecewise cubic interpolation is that at the data points you have got the function values and the derivative values and then for every pair of points you try to frame a cubic with the help of two values here function and derivative and two values here. You see function and derivative value at the two end points of every segment give you 2 plus 2 4 conditions and with those 4 conditions you can determine the 4 coefficients of a cubic exactly and that you go on determining for every segment. So, for that what you will get? So, for every segment you will get here you have got the function value and the derivative here you have got the function value and the derivative. So, the cubic that you will be framing will perhaps be this cubic, but then the part of it which is outside this sub interval that will not be used and you will have only this much and note that since the adjoining segment between these two points will also use this function value and this derivative value and match that data exactly the trend of the function value here will match this segment up to the first order derivative. And this way you will get a curve a function which is differentiable up to first order. So, you will have an exact interpolation with first order derivative. So, the data for the jth segment will be function value at point j minus 1 at point j and derivative values at the same two points with the help of these 4 pieces of data you will get the 4 coefficients from here because as you force these data values on this polynomial you will get 4 linear equations in a 0 a 1 a 2 a 3 and then as you solve them you will find that the coefficients will turn out to be linear combinations of these 4 items of data these. And if the coefficients are linear interpolations of these then you will get a composite function which is c 1 continuous at not points that means at the junctions of the intervals they will have first order continuity that is the derivative is also first order derivative also continuous. In between the not points that is during the segment of course it will have infinite order differentiability because they are single polynomials it is the differentiability at the junctions which we are trying to establish. The differentiability in the interior of a segment is of course up to whatever order you want. Now, this same formulation can be made in a general setup if we reparameterize in a normalized interval say 0 to 1. So, that is quite often done and the result is something quite useful say the variable x for that you have the interval say x 3 to x 4 j equal to 4 say x 3 to x 4. So, now between x 3 and x 4 if you put another variable t in terms of which you reparameterize the scale then at t equal to 0 you will have x 3 x j minus 1 at t equal to 1 you will have this and this will get cancelled you will have x j. So, the interval 0 to 1 will scale up to the x interval that is x j minus 1 to x j and then the function f of x can be talked of as f of x of t which you can call as g of t and then the g prime from this expression will be derivative of f with respect to x and then into derivative of x with respect to t which you get from here x j minus x j minus 1. So, whatever is the derivative given in terms of f prime that can be mapped in terms of g prime. So, with g mapped like this and g prime mapped like this both functions of t you will have these as the representative data. So, the with these 4 pieces of data you determine the cubic polynomial for the j th segment. Once you do that you can note something interesting this q j t you can write as the row vector alpha 0 alpha 1 alpha 2 alpha 3 into 1 t t square t cube like this and we have already seen that alpha 0 alpha 1 alpha 2 alpha 3 will be linear combinations of these 4 pieces of data and with this fixed interval 0 to 1 these coefficients of these linear interpolations these linear combinations are going to be constant. And therefore, you can have alpha 0 alpha 1 alpha 2 alpha 3 as constant in terms of numbers known linear interpolations of these 4 pieces of data and that those linear interpolations those coefficients will turn out to be a matrix will constitute 4 by 4 matrix W. In the textbook there is a small exercise in which the steps to evaluate this matrix W is given and I suggest that you attempt that problem. Now, with this representation we can write for the j th segment g j into W into t. Now, this way we have packaged the data interpolation type and variable terms separately that means g j this vector has nothing other than the items of data this W has just constant numbers with are which reflect the type of interpolation that we have decided upon that is in this case piece wise cubic. So, that cubic hermit taking function values and derivative values at both the ends of the sub interval based on that decision of the interpolation type we get this matrix W. If we had taken some other interpolation type we would get some other matrix W there and in the third segment t we have the variable terms which we will get their values at the time of evaluation of the function at a particular point. So, this modularity of representation we get if we convert if we rescale every sub interval to the normalized interval 0 to 1 and the same expression the same formula we can use for every segment of one composite curve and for every segment of every other composite curve for that matter everywhere we use the same matrix W of fixed numbers and the data changes and at the time of evaluation we have to give these values of t. Now, the question is arises that is there are many situations where we have got the function values at several points through which we want a smooth curve smooth or say continuous curve differentiable function passing through all these points. Now, at these points if we do not know the derivative values then how do we supply it to a hermit interpolation program hermit interpolation algorithm the requirement of the first derivative becomes a problem that is if the problem as it is does not come with the derivative data then how do we supply them and also the question arises why should we supply can we not say that we do not care what derivative is here as the curve goes from this segment to this segment we do not care in which direction it goes whether it goes like this or this all that we care is that whichever way it terminates at the end of this segment exactly in that direction it should commence at the beginning of the next segment and so on only this much we want to say. So, in that case we do not need to give the derivative values, but we can demand first order continuity and if we do not want to specify the derivative values there then we can demand not only first order continuity, but we can demand second order continuity as well note that when we were earlier giving the value of the derivative here at this point then through that value actually we were supplying two conditions one is that as part of this segment the function should have this slope here and as part of this segment the function should have this slope here. So, at this point through the derivative we were actually giving two conditions now that we want to omit that piece of data all that we say is that we are going to demand first order continuity of derivative at this point that is whatever is the derivative here we do not care similarly whatever is the derivative on this side we do not care all that we need is that the derivative at this point from this segment and the derivative at this point from this segment should be equal. That means that the derivative value we are not concerned about, but derivative continuity we are concerned about. So, when we ask for the derivative continuity at this point then we are actually specifying only one condition now we have got a slot for another condition. So, from where to get that condition so we can say that at this point we want the continuity of the second derivative also. So, that way with the help of less data that we supply from our hand that is derivative value we are not supplying, but we are demanding two conditions first derivative continuity and second derivative continuity. If we do that that is we can do that when the requirement does not specify the derivative value. So, when we do that we can develop a function representation which is not only first order continuous, but second order continuous across the junctions and that kind of an approximation is called spline interpolation. Spline basically classically it is a drafting tool to draw a smooth curve through key points which are already marked on the drawing board. Now, from that particular drawing drafting tool the spline interpolation has got its name how do we do this say there are n plus 1 points at which the data values are given. Now, if the derivative value at the junction points are k j unknown to us currently then p j the j th segment say let us take j equal to 0 that means the first segment. So, this segment this segment can be planned or determined in terms of f j minus 1 f j k j minus 1 k j that means function values at this point at this point and the rate here the rate here. So, in terms of these four quantities we can determine this segment. Similarly, for this segment we will require the function values here k j and k j plus 1. So, this segment is determined based on function values at these two points which are known and k j minus 1 and k j. Similarly, the segment here this one can be determined based on function values at this point k j and k j plus 1. If so then the derivative at this point can be determined based on the same four pieces of data and similarly on this side. So, apart from the known quantities the slope here can be determined in terms of k j minus 1 and k j from this direction and k j and k j plus 1 from this direction all that we want to do is to equate these two. So, the second derivative first derivative is going to be same because k j is used on this side as well as on this side. Now, the second derivative also we can evaluate and equate on both sides that is whatever is the second derivative here that is in terms of k j minus 1 and k j and the second derivative from this side will be in terms of k j and k j plus 1. So, these two quantities we determine and equate. So, p j is determined based on these four quantities p j plus 1 is determined in terms of these four quantities. So, the derivative first derivative continuity is ensured because k j is used here and the same k j is used here. Now, if we develop the second derivative from this expression and the second derivative from this expression and evaluate both of them at the same point at the junction point that is at the end point of this interval and at the beginning point of this interval and equate these two that will give us one equation in k j minus 1 k j and k j plus 1 in these three unknowns. So, that means that at the junction 1 2 3 4 we will get equations in k 0 k 1 k 2 k 1 k 2 k 3 k 2 k 3 k 4 and so on. So, we will get n minus 1 such junction points and at that we will get n minus 1 such equations in k 0 to k n plus 1 and this equation system will be a tri diagonal system and as it happens it gives us a tri diagonal diagonally dominant system of equations. So, from the n minus 1 interior junctions we will get n minus 1 linear equations in the derivative values these are n plus 1 values. So, to determine them completely we will need two values two of these n plus 1 unknowns we will need and then the rest of the n minus 1 can be determined typically the derivative value at the beginning and at the end are supplied and then the rest of them can be determined from these n minus 1 linear equations together framing a diagonally dominant tri diagonal system. So, this way we get a smooth curve like this which matches the second derivatives at the junction points and in between the junctions in a sub interval of course it is differentiable up to infinite order that is of course known. So, this is called spline interpolation with C 2 continuity. Now, in this same manner we can talk of multivariate functions also and interpolate them that is if we have a function of two variables. So, for that also we can have bilinear interpolations by cubic interpolations by quadratic interpolations and piecewise bilinear interpolation whatever we did for one variable we can have for multiple variable the spline also can be set in that same manner. Now, the we will take one example that is piecewise bilinear interpolation for piecewise bilinear interpolation we have the domain structured in the form of a rectangular grid. So, we typically have x 0, x 1, x 2, x 3, x 4 and so on. Similarly, on this side y 0, y 1, y 2, y 3 and so on. So, suppose our domain is this over which we want the function representation for this region this rectangular region. So, for that we can have a rectangular grid like this and typically for bilinear interpolation we typically want the grid to be quite dense and then the function expression will look like this constant term plus something into x plus something into y plus something into x y which can be represented in this manner 1 x here 1 y here and the coefficients here. Now, the coefficients form a matrix. So, with the data at 4 corner points function values only with 4 pieces of data we can determine these 4 coefficients. So, as we put these 4 pieces of data with the corresponding values x values and y values then we will get this will be a known matrix this will be a known matrix this will be a known matrix and we need to determine this matrix. So, pre multiplication of inverse of this and post multiplication of inverse of this will determine this matrix for us which means the coefficients here. So, like this for every small rectangular region we will get the function representation and this will be continuous representation only no differentiability. So, differentiability can be ensured if we have a piecewise bi cubic interpolation and so on. So, we will omit all these details because in the same manner you as the representation of single variable functions you can develop these and everywhere we will have coefficients in the form of matrices data also in the form of matrices this is piecewise bi cubic interpolation. Here you will need the derivatives these derivatives which again if you normalize then you will scale in this manner as we did in the case of single variable function. Now, what is important for us at this stage is to have a quick look of what we have been doing all this while. Now, typically a common strategy of function approximation is to express a function as a linear combination of several known functions that is a set of known functions which are called the basis functions. Now, the question arises which basis functions we use. Similarly, we determine the coefficients based on some criteria what is the criteria that is the second question. Now, if we decide that our criterion is exact agreement with sample data then we get one kind of approximation which is interpolatory approximation which we have been doing in this lecture. There is another kind of criterion that is least square approximation that is we do not mind the some errors in the at the data points, but then what we ask for is that the errors all over should be limited that typically we do when we have super abundance of data that is more data than necessary to fix the coefficients. In that case we expect error and then we say that error square sum should be minimized that is the least square approximation. Earlier in the linear problems in the linear segment and then in the optimization segment we have seen how we conduct least square approximation in the modeling of functions. Least square approximation is based on minimizing the sum of squares of errors. Now, when these sums are sums of infinite terms organized arranged very close to each other then that sum is replaced by integral. So, even that is a kind of least square approximation which we will study after our study of differential equations. And around the same time we will study the third criterion of function representation also that is minimax approximation which also has a certain advantage which also has a certain application. There we do not try to make exact agreements with certain data points. We do not want the overall least square of approximation what we want is to keep the largest error within a limit that means we try to minimize the maximum error and that is why that kind of an approximation is called minimax approximation. So, after the study of ordinary differential equations when we take up our topics on approximation theory there we will consider this kind of an approximation also. Now, this is the kind of approximations that we can talk by using different criteria. Now, which set of basis functions we use one possible choice which we have been doing in this particular lecture in this particular lesson are polynomials. So, in this lecture we have done single polynomial which with basis functions as 1 x x square etcetera then Lagrange polynomials then Newton approximation or piecewise polynomials. So, polynomials is one large class of basis functions that can be used sinusoids, sines and cos cos sines and cosines are another set of popular basis functions which give us Fourier series. Orthogonal Eigen functions which will be encountered here are another vertical area of choices of basis functions and then sometimes field specific heuristic choices are also possible. The kind of choices we have been talking about when we were talking about least square approximation in the context of optimization. Now, all these are sensible all these are proper choices of basis functions and we choose the set of basis functions and we choose the criterion depending upon what is our field of application. Now, one important issue here is that we have been talking about the approximation of functions of this kind f of x and then f of x y. Now, consider this independent variable we will just change the names and then see what kind of interpretations we can make in place of this independent variable t x suppose we put t and for this function in place of f we put x, x of t will be a function of parameter t. Now, similarly y of t can be another function of parameter t and z of t can be another function of parameter t. Now, all these we can do exactly the way we have been doing for function f x and now these three functions can be taken as x coordinate, y coordinate and z coordinate of a point or a curve as t varies. So, enclosing it like this we get this. So, this is a vector function of a scalar variable t. Now, with this kind of a setup with the same interpolation tools we can actually model curves in 3D space. Similarly, the bivariate functions that we have been talking about in terms of such bivariate functions we can model triplets of such functions that is we can model vector functions of two parameter two parameters u and v and that way with a lot of data over the surface we can make a representation of the surface like this surface as functions of two parameters u and v and with the help of that we can model surfaces. So, typically curve and surface modeling is one large area which stems from this kind of interpolatory or other function approximations. For example, suppose you have got a large number of data points in 3D space and you want to find out the analytical representation of a surface on which all these data points lie. Then what you can do? You can say that suppose I have got a bi cubic expression of u and v and a bi cubic expression not just one expression, but three expressions one for x coordinate, one for y coordinate, one for z coordinate. So, you can say that we have got a bi cubic vector function r x y z that means three scalar functions and each of them bi cubic with the with two variables u and v in hand that means you have got 4 by 4 matrix like this. So, this will give you a bi cubic expression. Now, you have got three such expressions for x of u v, y of u v and z of u v and then you want to determine the three 4 by 4 matrices sitting here for the x coordinate y coordinate and z coordinate and with all the points that are given to you as points on the surface you can use that those points as the set of data and then you can frame equations for x coordinate, y coordinate and z coordinate and from that you can determine these three 4 by 4 matrices. Once you determine these 4 by 4 matrices you have got x coordinate y coordinate and z coordinate as functions of u v with known coefficients here and that means that you have got a complete analytical representation for the surface for which the data was just a lot of three points. So, this way you can do a lot of curve and surface modeling with the help of these kinds of interpolatory formulas. So, for this particular problem you can also do a bi cubic approximation based on the function values and the derivatives at the end points. So, in that case you will have the data points, the values of the function and then their u derivatives, their v derivatives and the cross derivatives that is del 2 f by del x del y, this kind of data you can use to model bivariate functions and when you have vector functions like that with three scalar bivariate components you have got a model of a surface. So, such curve and surface modeling are often done in the case of computer aided engineering design or computer aided geometric design. So, some of these examples such some of such examples are there in the exercises in the chapter of the textbook and I will suggest you strongly to attempt some of these exercises to develop an ability to work with such approximations. Thank you.


and welcome to the show it's me JP it's time for another episode of JP's product pick of the week I am so happy to be here and I am glad you're joining me this week's product pick is going like hotcakes so the first thing I want to do is send you over to the product page so go there to that QR code or to that URL head to the product page this show is being displayed right inside of there and you're going to see we've got a 50% off discount on this week's product pick they're going like hotcakes I saw it jump from about a hundred down to 50 something so go there right now if you think you want to get them I don't want you to delay you don't need a coupon code or anything like that you just head there and you'll find the discounted price uh so first thing I'm going to do is have lady aida introduce us to this product pick so take it away lady aida the esp32 s2 tft feather um so we basically took a 1.14 inch color tft display and slapped it onto our feather esp32 s2 to give you a wi-fi microcontroller board based on the esp32 s2 with native usb two uh megabytes of ps ram four megabytes of flash just got a lot of capability a lot of speed and it's got this color tft display and um you can see I also have a little neopixel up top and a stem acutee port so you can plug and play i-squared c reset and boot button um it's went through a couple different redesigns due to part shortages and etc etc um but you know if you want something that's just ready to go it has a display um you can even use the you know the boot button as an input so you might be able to do even some simple projects that just need like one button um and a display and and i-squared c you know plug for external sensors go to the other hand expanders you have got usbc battery input and you can see here's the stem acutee port you can plug in vertically so you can have i-squared c um sensors and expanders reset and boot buttons color tft display it's all feather compatible of course you wouldn't want to plug a feather um on top of it like a feather wing on top of it but you could use a doubler uh on the bottom we've got um an esp 32 um s2 mini with four megabytes of flash two megabytes of ps ram um all the passives for monitoring the battery there's an lc 709 203 um and that's a i-squared c ultra low power battery monitoring that does like um voltage and not cool i'm counting but it like kind of tracks the voltage you just tell it the capacity of the battery and it'll automatically um it'll automatically tell you the percentage and voltage of the battery so you can like track percentage and not have to do the the math to convert like 3.7 volts to whatever and there's also a second regulator um the second regulator is used to um uh uh disable power um from the i-squared c or tft display so you can do like ultra low power uses because i think that could be you know more people are trying to use our boards for low power and i thought um i had time with this one to because i had to redesign it doing due to the component shortage so you can turn off the tft power and the power to the i-squared c port and get down to like 8200 micro ampere which is about as low as you can get in deep sleep when using an esp 32 s2 and still have like you know the regulator and then of course in a lighter sleep like a sleep where you want to resume execution i think it's about one milliamp um so overall pretty um full featured boards it sounds like the most stuff it's uh double sided um so again you know you you don't want to put a feather on top i recommend like a doubler or something with this if you want to add um more hardware but this is also great for a breadboard because you can use on a breadboard and have the tft display um and um you know plug it in and it'll the the height of all the components on the bottom isn't higher than um the header spacer so that's kind of what i recommend it to be used for um but otherwise it's basically just like a feather esp 32 s2 with a high speed tft display and for circuit python it's great because you'll have um the grapple show up on the tft when you plug it in all right yeah let me go ahead and uh grab one i've got one right here let's take a look at oh yes there it is fits so nicely in that box let's talk about this so product pick of the week this week is the feather esp 32 s2 tft this is a fantastic board it has what two megs of uh ps ram and four megs of flash on it you can use it for es for usb things such as h id stuff and midi stuff um it has wi-fi this is not a chip that has bluetooth so some of the esp's do this one is a wi-fi chip and lemore was talking about plugging it into things but honestly it's got everything you need on there for a lot of projects and then it also has that stem of qt breakout to add a sensor or a chain of things to it so if we take a look in fact i'm gonna jump over to our product page for this if you take a look uh first of all jump on this because it is almost out of stock i know we had a hundred of them to start and we're down to nine of them so uh so grab one if you want to get it it's half price which is terrific you just can't beat that uh that price for the amount of microcontroller you're getting right there uh it's terrific for all in one projects because it's got the display right on there if you scroll down here you'll see some stats about including this is a 240 megahertz tensilica processor on here uh some info about the uh tft display this is 240 by 135 yeah that's right 240 by 135 it's a ips display so it's got some some good angles on it uh i'll show you i'll show you a couple demos running on here in a second if you scroll down in here you will get to not only our video happening right inside of here but also some guides so we have a main learn guide right here uh from catney which will tell you all about it how to get it set up you can use this with circuit python you can use this with arduino uh there's even some whippers whippersnapper setup info there for doing uh very easy internet of things projects and then we have a couple of uh cool projects here one of them is adding a case to it so there's a a little 3d printable case from the reese brothers i did a modified version of that i'll show you in a second but this case they did allows you to hang it to things add sensors uh even put a battery inside there's also this esp whole ad blocker guide that i just came out with as well as a uh personal portable web server and that's the one i have uh my wordle game on so let me show you some of these let's uh let's jump into a couple of these little demos so first of all here's this little case i built for it you can just uh snap fit this in this is just a uh a model i derived based on the reese brothers model just uh removed some features really to make it a little more streamlined for my needs and i'll go ahead and just plug in usb power to this one so this one when it boots up this is the esp whole and if i use the ip address on there 1.133 as my dns server then i can uh have ad blocking happen on my phone on any laptops or any devices using wi-fi if we pop that out of there and take a look at this one this is a different project this is that wordle server but this can be any kind of web server a little mobile web server so you can connect directly to this so you don't need a wi-fi access necessarily right now it's going through my local network but you can also if this is not connected to a network it'll serve up something from the from the board itself and you can see i add a little space in this one so that we can put in this big switch as well as a battery that's there uh the board has charging so if i plug that in you'll see the little yellow led light up there and it's now charging that board um and this is a really great all-around um board you know the esp boards are the ones the esp chips rather the ones that we can get a hold of now so you'll see we're doing a lot of uh different esp 32 based uh projects and boards that that l'amour is is designing and this one's really terrific it's it's kind of one of the most modern feathers that we have since it has usb c on it which i love it has a stem of qt port on it so you can just plug a sensor or something into that just plug that right into the uh the top there and uh we've got the couple buttons on there so you can use the let's see if i can get this to go to the the boot there uh i can't remember if this one wants a double click i think this one wants the boot and the reset one of those um nope that's just reset it uh but if you double click these you can uh you can get to the reset put circuit python on there or if you're if you're boot loading from arduino you can use that as well uh on the back side you see we've got the big esp 32 s2 package on there as well as most of the rest of the charging circuit and passives and voltage regulator and so on uh this can plug into a breadboard this can have things plugged into it it can go into a doubler into a tripler you can solder things directly to it so i've also got mounting holes on there so you can really use this like you use any feather but the fact that you've got this great looking uh sharp colorful display on there allows you to get a lot of projects done uh really really easily uh let's see if we let me uh jump back to a main view here put that one up in the corner there zoom down in on that uh yeah unfortunately i think we have sold out so that is the double-edged sword of having this uh great pricing is that they go quick i think we are now officially out of stock uh if you want to get one of course you're not going to get it at the half price but you can sign up uh there's a little out of stock on the on the webpage you can fill out the little uh your email notify me you'll find out when these come back it being an esp 32 baseboard we will be making lots and lots more of them it is one of the chips like i said that we've got uh we've got coming in so you will see this one again it's not going away anytime soon uh let's see any questions over in the chat um let me know we got a couple of if you take a look at the discord there we've got a couple of links to some of these cases yeah rufus says esp hole then i could reclaim my pi zero w exactly the uh reason one of the reasons that lamar wanted to do this esp whole project was that we we don't have pi zeros and and uh and pies in general they're really hard to get so if you're using a pie for some basic ad blocking this is not as sophisticated as piehole but it works really well it will block a ton of domains uh and you can download a blacklist onto it to to update i think right now it says we're blocking or checking 7,061 domains um so yeah use this it's a a great way to get um ad blocking going on your devices and then you can free up a pie zero w that you had in a in a piehole project uh let's see i think that's it yeah we have a few people who got some and they definitely went like hot cakes uh if you are uh interested there are other feather esp 32 s2 boards that we have uh again the the status of stock will will come and go but uh in general those are the chips that we're able to get so thank you so much for stopping by that's my product pick of the week it is the feather esp 32 s2 tft and i'll go ahead and put a hanger on there and we'll put that up on the board i say that like it's easy there we go let's hang that up on the board here and that is going to do it for another episode of jp's product pick of the week thank you everyone for stopping by and don't forget we have a bunch of shows coming up there's going to be three hangouts tomorrow morning there's a show and tell that liz and i'll be co-hosting as well as ask an engineer uh and then i'll be back with a workshop show on thursday and i believe we have a deep dive with tim on friday so uh if i missed anything head on over to our discord it's adafruit.it slash discord and you can type in showtimes or go to the live stream uh channels and check out what's coming up all right thanks everyone i will see you next time bye-bye


Great, let me just ask that news to get top stories and crypto right amount of bite-sized pieces today Just the thumbnail the title suggest we're going to take a look at a whistleblower or multiple whistleblowers talking about how certain specific government agency is Suppressing the banks working with crypto related companies That'll be our main story for today But of course we'll look at the markets take a look at some charts from looking at Bitcoin Which shows a pretty good accumulation They're going to take a look at some macro events as there's a major Housing market issue not here in the United States, but but elsewhere and also take a look at more layoffs caused by Three-arrow's capital take a look at how Coinbase is going to profit big time on the Ethereum merge and when that's going to happen And also some misconceptions about the Ethereum merge And lastly we'll take a look at a sweat coin as they roll out more public Information and I'm going to tell you exactly How much I got into sweat coin as I paid them to be a part of the project So we'll go over all those things but first take a look at what's going on into the market So today what is it 17th full 45, Texas time and Yeah, kind of just chopping sideways. Nothing too fantastic Bitcoin's down a little bit. There aims down a little bit I think a theorem will rally going into the merge But I don't expect it to last big time after that But who knows it could drop precipitously and do the exact opposite way say that's why I just dollar-cost average And that's pretty much across the board nothing's up too big XRPs up percent, you know something to write home about Congratulations, that's pretty much it and then of course, we're down Which means of course, you know traditional markets are down a little bit S&Ps down NASDAQ's down a little bit and so so far We've been pretty correlated with bits and bouts of here and not being so much Also, what I like to do is just take a look at some charts My favorite ones are looking at Bitcoin because they are high quality charts and it's a website It's a hundred percent free and I even put it in the the link in the description But it's very simple look into Bitcoin.com you can find it right there And there's gonna be three we're gonna take a look at one is the two-year MA multiplier It's very simple. All it does is it just takes a two-year moving average Which is in the green right here and then it takes a two-year moving average times five and that's the red line And also just say it states is look once it goes above the red That's a good time to think about selling and once it go below the green That's a good time to probably accumulate and you can see there's not that many chances when you've had a chance to really accumulate Right here in 2019 right here back in 2015 and oh look Exactly right now is not a bad idea to accumulate. Does that mean that we can't go lower? Of course. It's crypto Who knows they could go down to 10k tomorrow? I have no idea But I think in the next two three five years you might be happy if you accumulate enough financial advice It's not a financial advisor, but also take a look a logarithmic regression the Bitcoin price chart indicator I like this one because it's colorful and it's simple to look at the big thing I also want to make mention of is that is of course as It scales down we can see just tells us the very bottom is just called the fire sale Territory and we've been in that sense around June on June 13th or so and we just upgraded barely From fire sale to buy the next one is accumulate and then after that still cheap hodl Is this a bubble FOMO intensifies? Da da da and of course this is where you want to sell off So right now still a good time looks like as far as what the charts will tell us and then also the well multiple which is basically just Bitcoin miners selling off and It's calculated by dividing the daily issuance value of bitcoins by the 365 day moving average And it just takes a look at the supply try to the supply side of the Bitcoin economy And we can still see right here that it's I love these charts are so simple Down here was a good time to accumulate we've we've gone above that but still not a bad time But we're still in that territory where it looks pretty good. So that's what's going on in the markets and the charts Let me just think about that in the comment section. Let's move into the day story. This one's interesting. I This came across my desk and it's a it's a pretty lengthy article From market watch whistleblowers accused FDIC of stifling crypto companies access to banks But I'm just gonna skip that and just go right to the letter. So what this is is the whole article from this is Based on senator Pat Toomey's letter to director Gruenberg Who is the head of the FDIC and this was August 16th? Yeah, so just yesterday. Wow pretty fast and Here's what he says and it's it's interesting because it talks about whistleblowers and what they are doing or excuse me Lack of doing and I found this just to be par for the course of what's going on and in Crypto world and banks and traditional finance working together. It's not so Dear director Gruenberg, this is again from senator Pat Toomey It's come to my attention that the federal the deposit insurance corporation FDIC may be improperly Taking action to deter banks from doing business with lawful Crypto related companies my office has received information from effective parties and protect a whistleblower communications Alleging the FDIC is taking such actions Even though the FDIC has not determined that these companies are providing services that are unlawful or impermissible Given the FDIC is involvement under your leadership in the Obama administration's notorious operation choke point Which sought to coerce banks into denying services to legally yet politically disfavored businesses It's important to better understand the actions the FDIC is now taking in a legal basis for them Wow to me getting into the point FDIC this is what's been going on the FDIC original offices have been sending letters to multiple banks requesting that they refrain From expanding relationships with crypto related companies without providing any legal basis Whistleblowers reports have also alleged that the FDIC may be abusing its supervisory powers to deter banks From extending credit to crypto related companies FDIC headquarter employees have contacted FDIC original office bank examination staff To question their review of a loan made by a bank To a crypto related company and to urge them to downgrade Their classification alone. So not only are they telling the banks don't do business But the business that they have done with these crypto related companies are like Let me take a look at what you're doing what you're loaning and maybe you should downgrade that because We don't like what that's what's going on And then of course it goes into some other stuff and they ask some more questions But it just goes to show you that sometimes the legal process does work whistleblowers come in they say this is what's going on Senate takes action talks to the director of the FDIC Grunberg and says what are you doing? You can't do that. Tell me why you're doing that. We can move forward So again, is this the end all be all no, but I think it's the step in the right direction Just to let you know exactly what's going on as far as suppression of the crypto market So let me actually think about that in the comments. Sure. That'll be good later. I'm going to do the q&a. Let's jump into some macro and We know that the that the housing market is slowing down here in the United States I've got friends who are still in the real estate business and they say, yeah, you know, things are slowing down But we still have our tricks up our sleeve We can bring people in and say that's uh, you know buyers will help pay for the closing costs We can do certain upgrades and of course different contracts have fallen through because The interest rates have increased because of the the Fed but we're still moving houses and the buyers are still building houses just not as big as houses The most expensive but there's a bigger issue coming on not the United States And I hate to say this but i'm kind of happy that this is happening It's in china and I hate to say it like that because I don't wish this ill upon people but it's nice to see sometimes that That us isn't the only one that's screwing up So this is a tweet done some pretty good information from graham steven fellow youtuber and If if look if graham knows anything it's about it's about real estate now as far as the the macro stuff I can't uh Confirm most of this and of course you can you can help me out But the but the data that he shows looks pretty compelling. This is what he states Chinese Are obsessed and that's a that's a keyword obsessed With real estate and 70 percent of china's wealth is tied to real estate twice as high as the us reports are now emerging about real estate Ponzi schemes Mortgage boycotts and an unfolding banking crisis. I like I love this uh this this meme 2008 veterans In the housing crisis that happened here in the united states. That's exactly what happened real estate bubble defaults banking crisis You know the story. You know what's going to happen next And this is something I didn't know the shanghai composite which is similar to our s&p 500 has not yet recovered from its 2008 peak Despite china's gdp Growing by nearly three times since then and we can see right here If this is the the shanghai composite It peaked in 2008 and it hasn't come back since now, of course nasdaq and s&p over here is As uh has dramatically rallied since then and just very interesting though that the stocks over there are not as big as What's uh, I guess some they claim to be and of course the increase of of gdp Give that a second He says for some context and how crazy the real estate market is in china the median home price in new york is around 10 times The median annual annual income, but in beijing this can be as high as 25 times The annual income look at this Chinese and us real estate shenzhen beijing This is Wow sale price per square meter is seven thousand dollars sweet joseph phoenix arizona 1400 st. Louis missouri $1400 seven thousand dollars Man, and that's look in california San Francisco california is five thousand wow And it's crazy ridiculous prices over there so This demand has led to ponzi schemes. I think we can appreciate that here in uh, the crypto verse as well Ponzi schemes were developers pre-sale homes, which are not even built use the pre-sale money to start more projects and then collect more pre-sales from new projects buyers started a mortgage buy boycott as the properties have not been completed And that's I think is some of the times when you see like those run the bank videos where people are throwing, uh And they can't at the banks and there's big riots in in china not everywhere, but in certain provinces The crisis spreads the banks as they had loaned out depositors money to developers who were now Defaulting on payments as frozen depositors or deposits mounted and protests erupted all over china investigations revealed critical systemic issues If anybody has been around since 2008 for the housing crisis, this is par for the course Reports suggest over four thousand customers have fallen victim to such activities That they know of now public trust in the banking system plummeted and the government stepped in to avoid bank runs and liquidity crunches it's like it's playing out again and The government has agreed to pay up to fifteen thousand dollars for lost deposits. I think that's a lot of money But whatever But only if the banks weren't involved in non-compliant transactions government strategy for over leveraged properties is to lend more money To finish them pretend that everything is fine And that's pretty much it's in a nutshell. So follow graham steven on twitter. I think I linked in the description, but uh this is just one more issue of like, um See I I have a flaw my flaws. I look too much at the united states and I don't look globally as much as I should when we see stories like this coming about and then There was another issue that came up in uh twitter early this morning where people Especially in italy and france and the uk Are commenting just how crazy ridiculous high Their their gas prices are not gas prices electricity prices are for their homes And I've seen it's like 4x 5x sometimes 10x in certain areas And if you want to find out more about that just follow me on twitter and uh, it wasn't just one post It was I asking people like you actually live in those areas like can you collaborate that corroborate? Yes This is exactly what's going on. Some people say it's like a 4x. Some people say it's an 8x It just depends but uh, I see some problems on the horizon Which would lead me to the next story Oh, I forgot about this goldman sacks Uh cuts china's 2022 gdp forecast at 3 from 3.3. That's shocker, but this one uh crypto broker genesis Cutting 20 of workforce as ceo michael morris and to me before yesterday I wouldn't really have cared about this story, but in plays into a video. We just did last night. This what's going on Genesis ceo michael morrow is stepping down as the company slashes One fifth 20 percent of its 260 person workforce following major losses tied To the collapse of three arrows capital genesis owned by digital currency group dcg is also the parent of coin desk Look for those to collapse. Maybe filed a 1.2 billion dollar claim against failed crypto hedge fund three arrows capital in july dcg assumed the claim associated with genesis um, even though they have 1.2 billion we did a video about this I I put this out last night. It's kind of late But uh, I made this video Just that hope that tells the whole three arrows capital story from start to finish Where zu and jeffreys came from I mean even back to their early days and college and the difference Jobs that they held and and how they got into the industry and and arbitrage and and uh some Some trading but really what they were doing is just borrowing from everybody and I know I've complained a lot about The different platforms, but I understand now Where they fell victim because and I'm not going to make excuses for anybody. It was a boneheaded move. Let's be honest But watch that video. It's about 30 minutes long But it goes from a to z everything that happened with three arrows capital and why these things happened And uh, it really just just come down to they were Barrowing from peter to pay paul and instead of putting up collateral what they were doing was promises was promising outrageous Amount of interest and they were paying it back From the next one and the next one the next one and we go were all that stuff. So Uh, that genesis capital we just talked about They said it was 1.2 billion dollars that they were owed They may be more because in the video that we covered genesis capital gave them over 2 billion dollars And a lot of different companies that had come forward. They were doing so anonymously in the article Because they didn't want to be attached to three arrows capital because if they do there's a stigma against that And of course their creditors will look at them and say when are you guys going to fail? So I know we talk about contagion, but there might be more on the horizon. Sorry So that's what's going on there and then uh, lastly to finish up Coinbase is going to profit. I think they're gonna profit big time from the the eath merger. Uh, this what's happening And I I found this article odd because jp morgan, which is also I think investors Into ethereum and uh metamask and so on and so forth. It's an entanglement But anyhow coinbase will be meaningful beneficiary of ethereum says jp morgan And this is why jp morgan estimates that coinbase has a 15 market share at eath assets, which makes sense Uh, they have to own a lot of them to be to be market makers Which trumps its seven percent share of the overall crypto ecosystem. So they own a 50 percent of eath, but you know Assets but seven percent of the overall Coinbase's market share is probably tilted towards institutions, which we know because that's who mostly trades Not mostly trades, but uh, there's a lot of traders there. Unfortunately, you know who pays all the fees We do retail does it's crazy Coinbase's market share is tilted institutions, which are more likely to own eath and bitcoin The bank estimates that coinbase can generate an incremental annual staking revenue of 650 million from the merge With eath at $2,000 when it hits that and a five percent yield The exchange began offering ethereum staking for institutional clients earlier this month Coinbase is bigger and ether then was intuitive to us. That's leading directly to a bigger revenue opportunity. So You know, maybe kathy wood and arc investment. Maybe they're like, maybe we should buy some more Coinbase back and just to be clear, they didn't sell all of it. It's they just sell a little bit of it but there's there's this part but when they were saying about The staking and the revenue which is is pretty good It got me thinking about the different questions that I've been asked as far as the merge the merge Which is supposed to be happening. I think in 27 days And if you want to do a find a countdown, there's a great website called when the merge w en m e r g e dot com nice little countdown with updates But there's a couple of this was put out by the ethereum foundation some misconceptions and Just so everybody knows the merge isn't going to be anything like the greatest thing of all time I'm just going to go from proof of work to proof of stake. That's it. So there's there's a couple ones I want to make sure everybody knows Misconception running a node requires staking 32 eath. It's false Anyone is free to sync their own self-verified copy of eath now I think for requires a node running a node requires taking is taking 32 eath. That's not true Another misconception the merge will reduce gas fees. I want to make this very clear The merge is a change of consensus mechanism not an expansion on network capacity and will not result in lower gas fees So gas fees are kind of low right now because people aren't using it as much Especially as people get it get out of the nft game and other different transactions so Even though they're low right now Just because of lack of use it's not like they're going to drop drastically in the pennies They're still going to be in the dollars three five dollars. Maybe even 10 when things go up also Misconception transactions will be faster. No though some slight changes exist transaction speed will mostly remain the same as on layer one Misconception you can withdraw staked eath once the merge occurs. No, you can't Staking withdrawals are not yet enabled with the merge the following shanghai upgrade Will enable staking withdrawals and as far as the shanghai upgrade as far as I know correct me the comment section There's not a hard date on that yet. I could be wrong. Let me know and Next misconception when withdrawals are unable stakers will all exit at once. No, they stay validator exits are Are limited for security reasons. So they're going to only allow so much Misconceptions taking our apr is going to triple. That's not true It's maybe up to 50 percent increase but not a 200 percent increase And then the merge will result in downtime of the chain and that is I thought this was true But it's false the merge upgrade is designed to transition to proof of stake with absolutely zero downtime Now that's just the best case scenario. We'll see if it all plays out and actually works I'm rooting for it. I would like to see this actually happen in my lifetime So, uh, hopefully it goes through. So yeah, let me know which thing about that and then let's finish up With a little sweat coin And if you haven't been on my channel that much, you know, I love me some sweat coins And uh, I like it because it's free And they don't steal your data and you can get it right now And you can download on your phone and you can compete against me and it gives you these sweat coins for free And there's nothing else you need to do. It's very simple. And then uh, and let me see actually in 20 26 days they're going to airdrop everybody who has accumulated these sweat tokens And you're going to be able to get these tokens for every So much that you've walked actually, let me show you right here. Let me scroll down So for every thousand steps you get a sweat coin like I walk about 7 000 steps a day Not too bad, not the best But I get uh, roughly seven sweat tokens And then if you add that up over, you know, weeks and months that I've been doing it adds up pretty well And then of course as time goes on In 2023 1 000 steps, you only get a third of a sweat coin a thousand steps. You only get 20 of a sweat coin 1 000 steps you get point zero six and so on and so forth. So like I say like if you want to try it out It's not like like step in where you have to buy an expensive nft and walk it all up and and use these things to To repair your your nft shoes. It's free Anyhow I love this part because it talks about this is how we generate revenue and before we go on There's a link in the description I'm not shy I'll tell you about it. It's right here Sweat coin walk and earn sweat coin. There's a deep dive bid Uh, you can be done on the sweat coin and then here's the link to download everything, right? It's an affiliate link. So if you can't send an affiliate link, so it's okay Just go to what is this website sweat economy.com then you can totally bypass me. That's fine. I don't care But here's how we here's how they generate revenue user engagement businesses pay to access our vast network of users Just as you know, I'm sure when you got here for youtube You probably saw an ad and that's what generates revenue for youtube ads. It's the same thing over here on on on this On their platform transaction fees on the sweat wallet app features. It's going to include a fiat on ramp Sorry, united states. You're not getting it Fiat on ramp off ramp crypto to crypto exchange empty purchases and more And then here's how the revenue returns the token. They're going to buy and burn They're going to invest in rewards. You can buy different things on here with your sweat tokens Ecosystem development and a bunch of other cool stuff And then the tge the token generation event it's in like I said 26 days somebody that Only 10 percent gets unlocked 90 percent remains locked over 24 months And let me just show you this So you see right here. It's kind of hard to see it's really hard to see So it's like there's an airdrop team advisors private round public sales sweatco. See this private round right here That's what I got into and uh, I've got a lock up My lock up for this On the tge in 26 days. That's when you get your tokens. I don't get my tokens You know when I get my tokens I get mine the first Is a month or no, sorry 12 months after this token generation event and then over the next 12 months past that So over two years Then I'll get the tokens that I bought into how much did I buy great question? Glad you asked So in the links in the description uh Right here under strategies That says uh 5% degen plays there's a A spreadsheet looks like this and these are all the the degenerate cryptos that I got into and uh We do I talk about these over on dan degen my second channel. I don't talk about them this channel This is the the very basic medium potato stuff. But over there is just straight up gambling And uh, I can I only put you know, two to three percent of my portfolio into these projects Nothing big, but they paid off pretty well and we did ginsokishi right now. It's at 20 cents, but Over time it's been like a dollar 45 and I've sold along the way. I've taken profits like I always tell you to do Everdome we bought it at point zero zero five and then it went up to like 20 cents again We took profits fame we bought it at a penny a hundred thousand tokens and I went to a dollar 50 took some big profits and uh And then for sweat coin You have to remember for all these different Projects I don't get all the tokens. I still have to wait but when I get them I take profits along the way and for sweat coin We did the deep divots right here the purchase price was point zero one four two nine so like a penny and a half and I bought one million seven hundred forty nine thousand four hundred seventy five tokens and uh That's what's up. So When people are always saying like how much they pay how much they pay they didn't pay me squat man I I bought into this and I have to wait a long time I mean up to two years But I think it's a great project. It's free. I like where where things are going the deep dive explains it all And uh, that's what we got so I pay them to be a part of it That's what's going on. So look that's it for today.  Uh, thanks for coming by a lot of you today. Appreciate it So thanks so much for that. Let's get into a little q and a