today's topic is Gaussian integration. So, far when we considered numerical integration rules, we what we did was we fixed the interpolation points. So, we started with interpolation points x 0, x 1, x n. We fitted interpolating polynomial and then integration of that interpolating polynomial that was approximation to integral a to b f x d x. Now, today what we are going to do is we are we will not fix the points interpolation points beforehand. We will try to write a interpolation formula of the form summation w i f x i, where w i's are the bits and x i's are the interpolation points. So, we are going to have i going from 0 to n. So, total we have got 2 n plus 2 constants to be determined. Now, these we will try to determine. So, that there is no error for polynomials or polynomials or polynomials of degree less than or equal to 2 n plus 1. So, that is the idea of Gauss integration. So, when we fix points x 0, x 1, x n in the interval a b and look at p n x to be equal to summation f x i l i x i goes from 0 to n, where l i is the Lagrange polynomial. So, it is product j goes from 0 to n x minus x j divided by x i minus x j j naught equal to i. Now, this is Lagrange polynomial. Then integral a to b f x d x is approximately equal to integral a to b p n x d x, which will be equal to summation i goes from 0 to n f x i integral a to b l i x d x. Now, what you have to note is that l i x the Lagrange polynomial, it depends only on the interpolation points x 0, x 1, x n. So, there is no function f coming into picture there. It is a polynomial of degree n. So, you can integrate. So, you integrate the matrix and then you are going to get the weights w i. So, we have this to be equal to summation i goes from 0 to n f x i w i. So, these are real numbers independent of our function f. Now, this is going to be our starting point. So, let us start with a formula integral a to b f x d x is approximately equal to summation w i f x i i goes from 0 to n. So, our starting point is integral a to b f x d x is approximately equal to summation i goes from 0 to n w i f x i and we want to determine x i's and w i's. Such that integral a to b f x d x is equal to summation w i f x i i goes from 0 to n for polynomials of degree less than or equal to 2 n plus 1. So, we want that there should not be any error if our function f is a polynomial of degree 2 n plus 1. This can be achieved provided there is no error for functions 1 x x square up to x raise to 2 n plus 1 because any polynomial of degree 2 n plus 1 it is going to be combination of these functions 1 x square x raise to 2 n plus 1. So, now, put f x is equal to 1 and then equate integral a to b 1 d x is equal to summation w i f x i i goes from 0 to n. So, f x is 1. So, that will give us 1 equation and like that you will have 2 n plus 2 x raise to 2 n plus 1. So, now, put f x is equal to 1 and then equate a integral a to b 1 d x is equal to summation w i f x i i goes from 0 to n. So, f x is 1. So, that will give us 1 equation and like that we will get 2 n plus 2 equations because there will be f x is equal to 1 then x x square up to x raise to 2 n plus 1 and then our unknowns are also 2 n plus 2 in number. The unknowns are going to be the weights w 0, w 1, w n and x 0, x 1, x n. So, we have got 2 n plus 2 equations into n plus 2 unknowns. So, let us look at some special cases. So, suppose we are taking n is equal to 0. That means we have got x 0 is 1 point and w 0 is the weight. We want integral a to b f x d x is equal to w 0 f x 0 if f is a polynomial of degree 1 which will mean that once take function f x is equal to 1 then take f x is equal to x you will get 2 equations and from that we will try to determine w 0 and x 0. So, let us do that. So, we have integral a to b f x d x is approximately equal to w 0 f x 0, w 0 x 0 these are unknowns f x is equal to 1 constant function. We want that there is no error that means integral a to b d x should be exactly equal to w 0 of f x 0 f of x is 1. So, that means it is equal to w 0. So, this imposes the condition that w 0 should be equal to b minus a. So, which means that if I choose w 0 to be equal to b minus a and x 0 to be any point in the interval a b then the formula w 0 f x 0 is going to be exact for constant polynomials. So, now we want our formula to be exact for linear polynomial. So, we will impose one more condition for f x is equal to x. So, w 0 we have already determined the condition f x is equal to x that will determine our point x 0. So, we have f x is equal to x in this case also no error that will happen provided integral a to b x d x is equal to w 0 x 0 which will imply b square minus a square by 2 which is the integral a to b x d x should be equal to w 0 is b minus a multiplied by x 0. So, this means x 0 should be equal to midpoint a plus b by x 0. So, this means x 0 should be equal to midpoint by 2. So, thus integral a to b f x d x if you approximate it by b minus a into f of a plus b by 2 this is going to be exact for linear polynomials and this is nothing but the midpoint rule. So, thus we have solved the problem for case n is equal to 0. Now, let us look at the case n is equal to 1. So, when you put n is equal to 1 you are trying to approximate integral a to b f x d x by a formula of the type w 0 f x 0 plus w 1 f x 1 unknowns w 0 w 1 x 0 x 1. We will like to determine in such a manner that now the our rule is exact for polynomials of degree less than or equal to 3. This will be achieved provided there is no error for 4 functions which are 1 x x square x cube. Any cubic polynomial is going to be of the form a 0 plus a 1 x plus a 2 x square plus a 3 x cube. So, if there is no error for the 4 functions 1 x x square x cube there will not be any error for a cubic polynomial a general cubic polynomial. So, let us equate and get the 4 equations. So, we have integral a to b f x d x is approximately equal to w 0 f x 0 plus w 1 f x 1. First put f x is equal to 1 then we want b minus a to b minus a should be equal to w 0 plus w 1. So, that is the first condition then f x is equal to x. So, that is going to be b square minus a square by 2 it is the integral a to b x d x and on the right hand side it is going to be w 0 x 0 plus w 1 x 1. Next look at the function f x is equal to x square it is integral is x cube by 3. So, this is going to be b cube minus a cube by 3 on the right hand side it will be w 0 x 0 square plus w 1 x 1 square and f x is equal to x cube that will be b raise to 4 minus a raise to 4 divided by 4 is equal to w 0 x 0 cube plus w 1 x 1 cube. So, let me call these 4 equations 1 2 3 4. So, we have got 4 equations in 4 unknowns which are w 0 w 1 x 0 x 1. Now, these 4 equations we got 4 equations in 4 unknowns, but these are non-linear equations. If they were in linear equations then solving them is easy, but we have got 4 non-linear equations. Now, in this particular case one can do some manipulation and one can try to find w 0 w 1 x 0 x 1. But what we want is we want to look at a general case that when you are considering a formula of the type summation i goes from 0 to n w i f x i. How I should choose the weights w i's and the interpolation points x i's in such a manner that I have no error for as high degree polynomial as possible. So, that is why we will not do manipulation for this, but look at a general case. Now, for the general case the method which we are going to use I am going to explain again for this particular case n is equal to 1 and then we will see how to generalize. So, the first thing we are going to do is we are going to obtain the conditions on x 0 x 1 which will guarantee that if you interpolate if you choose x 0 x 1 in a certain manner fit a linear polynomial integrate then the formula which you are going to get it will be exact for cubic polynomial. So, let us derive the condition which the interpolation points x 0 and x 1 should satisfy. Once we find these conditions there are going to be two conditions then we will see how to choose x 0 and x 1 so that these conditions are satisfied. So, first we are finding the conditions which the interpolation points x 0 and x 1 should satisfy and then find the points x 0 x 1 which satisfy this condition. So, let us look at two points x 0 and x 1 fit a linear polynomial no matter how you choose your two points x 0 and x 1 if you have a linear polynomial then the interpolating polynomial is same as the function if the function is a linear polynomial. So, there will not be any error in our integration formula for linear polynomials, but we want something more we want that the error should not be there or error should be equal to 0 for cubic polynomial. So, we have f x is equal to f x 0 plus divided difference based on x 0 x 1 into x minus x 0 and then you have got error term f of x 0 x 1 x multiplied by x minus x 0 x minus x 1. So, this is our polynomial p 1 x and this is our error and hence integral a to b f x dx will be equal to integral a to b p 1 x dx plus integral a to b f of x 0 x 1 x x minus x 0 x minus x 1 dx. So, this is going to be the error in the integration now we have used this technique before hand if your points x minus x 0 if your points x 0 and x 1 are such that integral a to b x minus x 0 x minus x 1 dx is equal to 0 then we will have to make use of this relation in order to manage our error we had done this for the midpoint rule. In the case of midpoint rule we had integral a to b x minus a plus b by 2 dx is equal to 0. So, our error formula it has got 2 parts it has got a divided difference and then you are multiplying by the function w a. So, this divided difference in the error formula it depends on x. So, we cannot take it out of the integration sign, but if I replace this divided difference which is based on x by a divided difference which is independent of x and then plus extra term then I can take the divided difference out. So, let us look at the error and then f of x 0 x 1 x this divided difference let me write it as f of y 0 x 0 x 1 plus f of y 0 x 1 plus f of y 0 x 1 x 0 x 1 x multiplied by x minus y 0 this is the recurrence relation this you take on the other side divide by x minus y 0 that is the recurrence formula for f of y 0 x 0 x 1 x. So, this I am going to substitute here now here y 0 is a fixed point. So, that will come out of the integration sign and our error will be equal to f of y 0 x 0 x 1 integral a to b w x dx plus integral a to b f of y 0 x 0 x 1 x multiplied by x minus y 0 x 0 x 1 x minus y 0 w x dx where I am calling this as w x. So, if this is equal to 0 then our error is going to have this form. So, our if integral a to b w x dx is equal to 0 then error is equal to integral a to b divided difference based on y 0 x 0 x 1 x w x x minus y 0 dx. So, now, look at the error error has a divided difference of x 0 x 0 x 0 x 0 x 0 x 0 x 0 x x 0 based on 4 points that is f of y 0 x 0 x 1 x. If your function f is a quadratic polynomial then this divided difference will be 0 and that means error will be 0. So, if our points x 0 and x 1 if they are such that integral a to b w x dx is equal to 0 then we get the formula to be exact for quadratic polynomial. So, earlier we had only exactitude for linear polynomials. Now, we got for quadratic polynomial with one condition integral a to b w x dx. Now, we will use the same technique again and then obtain another condition for the equation which will give us the error to be 0 for cubic polynomial.  So, this is the recurrence relation. This we substitute in the error. So, we will get error to be equal to f of y 1 y 0 x 0 x 1.  So, this is the recurrence relation. This we substitute in the error. So, we will get error to be equal to f of y 1 y 0 x 0 x 1 integral a to b w x x minus y 0 dx plus here it should be plus not minus plus to be f of y 1 y 0 x 0 x 1 x x minus y 1. Then, we have got this w x y 0. So, it will be x minus y 0 w x dx. Now, if this is equal to 0, then this will be the expression for the error. If integral a to b w x dx is 0 and integral a to b w x multiplied by x minus y 0 is equal to 0. If both the conditions are satisfied, our error is integral a to b divided difference based on 5 points. The points are the divided difference is based on y 1 y 0 x 0 x 1 and x. So, we have got 5 points. If f is a polynomial of degree 3, then this divided difference will be equal to 0. So, our error is integral a to b a divided difference multiplied by w x into x minus y 0 x minus y 1 dx. If the divided difference term is 0, error is going to be equal to 0. So, thus we have got this. So, we have got this. If the divided difference term is 0, error is going to be equal to 0. So, thus we obtain two conditions which guarantee that there is no error for cubic polynomials and those two conditions are integral a to b w x dx is equal to 0 and integral a to b w x into x minus y 0 dx is equal to 0. We have w x is x minus x 0 x minus x 1. Integral a to b w x dx is equal to 0. Integral a to b w x x minus y 0 dx is equal to 0 implies x is equal to 0 that the error is integral a to b f y 1 y 0 x 0 x 1 x multiplied by w x into x minus y 0 x minus y 1 dx and this term is equal to 0 if f is a cubic polynomial. So, this means error is equal to 0 if f is a polynomial of degree less than or equal to 3. Now, look at the condition integral a to b w x x minus y 0 dx, y 0 is a fixed point and we already have integral a to b w x dx is equal to 0. So, that means this condition will reduce to integral a to b w x x dx is equal to 0. So, the conditions which we get are integral a to b w x dx should be equal to 0 and integral a to b w x multiplied by w x dx is equal to 0. So, we have to choose x 0 and x 1 such that these two conditions are satisfied and then for the error our y 0 and y 1 they can be any points in the interval a b. So, we can choose our y 0 to be equal to x 0 y 1 is equal to x 1 and we will get integral a to b f of x 0 repeated twice x 1 repeated twice x x minus x 0 square x minus x 1 square dx. Now, the reason for writing in this manner will be now we have got a divided difference which is going to be continuous provided your function f is sufficiently differentiable you are multiplying by a function x minus x 0 square x minus x 1 square. So, this function is always bigger than or equal to 0. So, we can apply mean value theorem for integrals and then the divided difference we can take it out as divided difference based on x 0 repeated twice x 1 repeated twice and some point c multiplied by integral a to b x minus x 0 square x minus x 1 square dx. We can integrate this and then obtain a more precise bound as compared to if we had dominated the x minus x 0 and x minus x 1 by b minus a. So, now, our problem is we have reduced our problem to finding x 0 and x 1 such that x minus x 0 x minus x 1 it is integrally 0. So, we can write x 0 x minus x 1 it is 0 and if you multiply x minus x 0 x minus x 1 by x then that integral also should be 0. Now, as I said we want to have a general method that means here we have taken only n is equal to 1. So, we have got two interpolation points x 0 and x 1, but whatever we want to do we want to do it that you should be able to extend it for the more general case like when you have got n plus 1 points then how I should choose my n plus 1 points so that we have got exactitude for polynomials of degree less than or equal to 2 n plus 1. So, in order to do that what we are going to do is we are going to recall what is the inner product. So, on our our space is c a b it is the vector space of continuous real valued functions defined on interval a b. On this we will define a inner product as integral a to b f x into g x d x once we have inner product we can talk of two functions being perpendicular. So, if the inner product is equal to 0 we say that f is perpendicular to g look at our condition which we want it is integral a to b x minus x 0 x minus x 1 d x should be equal to 0 that means we want our function w x to be perpendicular to the constant function 1 and the second condition tells us that our w x should be perpendicular to the function f 1 x is equal to x. So, we will start with three functions function constant 1 then x and then x square to this we will apply Gram-Schmidt orthonormalization process and get a quadratic polynomial which is perpendicular to constant function 1 and function x. This quadratic polynomial look at the roots of this if the roots are x 0 and x 1 those are going to be our desired point. And what we will do is inner product we will defined on c a b, but in order to find the points x 0 and x 1 we will first consider the interval minus 1 to 1 and then we will look at a map from minus 1 to 1 to the interval a b which is 1 to 1 on 2 and a fine. And using this map we will transfer the results to interval a b. So, let us look at the inner product on c a b, so the vector space is c a b f and g these are functions in c a b inner product f comma g is integral a to b f x into g x into d x. This inner product, so this is inner product its properties are inner product of f with itself is going to be bigger than or equal to 0 because we will be looking at integral a to b f square x d a f is a real valued function. Then f comma f is equal to 0 if and only if the function f x is identically 0. If the function is identically 0 then integral a to b f square x d x will be 0. On the other hand if integral a to b f square x d x is 0, so integral a to b f square x d x is equal to 0, then f square is continuous, f square x is bigger than or equal to 0. Then f square is continuous, f square x is bigger than or equal to 0 and which will imply that f x has to be identically 0.  So, these are the properties of inner product, so we have defined the inner product and now we are going to look at the three functions f 0 x is equal to 1, f 1 x is equal to x, f 2 x is equal to x square. Look at g 0 x, this will be equal to we define it as f 0 x divided by inner product of f 0 with itself and raise to half. We have got a induced norm, if you define norm of f to be square root of inner product of f with itself, this is going to be bigger than or equal to 0. So, we can it is going to be greater than or equal to 0, we can take its positive square root and this is generally denoted by norm f 2. So, this has properties that norm f 2 is bigger than or equal to 0, it is equal to 0, if and only if f x is identically 0, second norm of alpha f will be equal to mod alpha times norm f, where alpha belongs to r and the third one is norm of f plus g is less than or equal to norm f plus norm g, this is known as triangle inequality and the triangle inequality is proved using Cauchy-Schwarz inequality, which is modulus of inner product of f with g that is less than or equal to norm f plus norm f into norm g. So, we have look at the three functions 1 x x square, from this we want to construct three other functions, which I will denote by g 0, g 1, g 2, which will have property that if you look at the norm of each function, it is going to be equal to 1 and if you consider any two distinct functions, like if you consider g 0 and g 1, it is inner product will be 0, inner product of g 0 with g 2 will be 0 and inner product of g 1 with g 2 is equal to 0. So, this the way we are going to do that is known as Gram-Schmidt orthonormalization process, we will do it for the three functions, but then one can define it for n functions. So, we have our f 0 x is equal to 1, f 1 x is equal to x, f 2 x is equal to x square, define g 0 x to be equal to f 0 upon norm normalization f 0 to norm. So, which will imply f 0 x, which will imply that norm g 0 to norm is going to be equal to 1.  So, this is my definition, if I look at inner product of r 1 with g 0, it is going to be inner product of f 1 minus inner product of f 1 with g 0, g 0 and then g 0. Now, we use linearity of inner product to write this as inner product of f 1 with g 0 minus, this is going to be a scalar. So, it will come out of the inner product. So, it will be f 1 g 0 and what is left is inner product of g 0 with g 0. Now, our norm g 0 2 is positive square root of inner product of g 0 with itself. So, this is going to be equal to 1 and then this will get cancelled. So, you will get r 1 g 0 to be equal to 0. Now, when you look at r 1, we have got f 1 x is equal to x, this is going to be some scalar, g 0 x is going to be multiple of 1. So, this is going to be a linear polynomial. So, by very construction, our r 1 is perpendicular to g 0 and now, if I want r 1 to have norm 1, define g 1 to be equal to r 1 upon g 0. So, we will have norm g 0 to be equal to 1, norm g 1 to be equal to 1 and inner product of g 0 with g 1 is equal to 0. So, now, we will look at the function x square. We have got g 0 to be a constant polynomial, g 1 to be a polynomial of degree 1, which will be perpendicular to each other. f 2 x is our function x square. So, from this function, we will sort of subtract the component of our f 2 in the direction of g 0 and in the direction of g 1. So, we will construct our r 2 in such a manner that r 2 is perpendicular to both g 0 and g 1. Our r 2 is going to be a quadratic polynomial, which will be perpendicular to g 0 and g 1. Define r 2 x to be equal to f 2 x minus inner product of f 2 with g 0 into g 0 x minus inner product of f 2 with g 1 into g 1 x. When you look at inner product of r 2 with g 0, that will be inner product of f 2 with g 0 minus inner product of g 0 with itself minus inner product of f 2 with g 1 and then inner product of g 1 with g 0 using the linearity of inner product. This is 1, this is 0. So, this gets cancelled and then you get r 2 comma g 0 to be equal to 0. Then you look at inner product of r 2 with g 1. That will be inner product of f 2 with g 1 minus inner product of f 2 with g 0 the coefficient and inner product of g 0 with g 1 minus inner product of f 2 with g 1 and inner product of g 1 with itself. So, this is going to be equal to 1, this is equal to 0. These two will get cancelled and then you get r 2 comma g 1 is equal to 0. Next, define g 2 to be equal to r 2 upon norm r 2. So, this will imply that norm of g 2 with g 1 2 norm is going to be equal to 1. So, this is the procedure for constructing g 0, g 1 g 2, 3 orthonormal functions which are which we have obtained from the functions 1 x x square. This was the general procedure. Now, let us look at the interval to be minus 1 to 1 and find explicit expression for g 0, g 1, g 2. So, when you look at the interval minus 1 to 1, our f 0 x is equal to 1, x belonging to minus 1 to 1, then norm f 0 will be integral minus 1 to 1 d x raise to half. That is going to be equal to root 2 and hence our g 0 x is function 1 by root 2. It is f 0 upon norm f 0.  Let us calculate the inner product f 1 comma g 0 that is integral minus 1 to 1 x, g 0 is 1 by root 2 d x. So, this is already equal to 0. So, we have got our r 1 x to be equal to x. That will give us g 1 x to be equal to x divided by norm r 1. Now, what will be norm r 1? So, norm r 1 to norm will be integral minus 1 to 1 x square d x raise to half. So, this is going to be square root of 2 by 3 and that will give us g 1 x to be root 3 by 2 x. Now, the third one, so let us look at r 2 x. So, r 2 x is going to be equal to x square minus inner product of f 2 with g 0 into g 0 minus inner product of f 2 with g 1, g 1 where our f 2 x is x square. g 0 is 1 by root 2 and g 1 x is equal to root 3 by 2 x. So, inner product of f 2 with g 0, this will be integral minus 1 to 1 x square by root 2 d x. So, this is going to be equal to 2 by 3 root 2 2 by 3. So, this is going to be root 2 by 3. When we look at inner product of f 2 with g 1, this is going to be equal to integral minus 1 to 1 x square into root 3 by 2 x d x. So, this is going to be equal to root 2 by 3 root 2 by 3. So, this is going to be equal to 0. So, our r 2 x becomes x square minus inner product of f 2 with g 0 is root 2 by 3 and g 0 is 1 by root 2 and inner product f 2 comma g 1 is 0. So, that means it is equal to x square minus 1 by 3 and then our g 2 x will be r 2 upon norm r 2. So, it will be equal to some alpha times x square minus 1 by 3. So, we have obtained a polynomial, a quadratic polynomial which is say alpha times x square minus 1 by 3. It is going to be perpendicular to the constant function 1 by root 2 and it is also perpendicular to the function g 1 x which is root 3 by 2 into x. So, this function alpha times x square minus x square minus 1 by 3. So, it has got distinct roots plus or minus 1 by root 3. So, this x plus 1 by root 3 into x minus 1 by root 3 is going to be perpendicular to 1 and perpendicular to x. So, that means we have got x 0 and x 1 in the interval minus 1 to 1 which has got a property that integral minus 1 to 1 x minus x 0 x minus x 1 d x is equal to 0 and integral minus 1 to 1 x minus x 0 x minus x 1 x d x is equal to 0 and that is the property we were looking for that if you have x 0 and x 1 satisfying this property then you fit a linear polynomial, integrate, get a formula for approximate integration and that is going to be exact for our cubic polynomial. Like because we are fitting a polynomial of degree less than or equal to 1, it is natural that the error will be 0 for polynomials of degree less than or equal to 1, but you are getting something more and what happens is when such a thing happens in the error you are going to have a term contain fourth derivative multiplied by the length of the interval and if it is fourth degree then the it will be b minus a raise to 5 and there will be some constant. Now you know that when you go to composite rules then this b minus a becomes important because in the composite rules b minus a will be replaced by h. So, higher the power of h you are going to have faster the conversion. So, for the basic quadrature rule, basic Gauss two point quadrature rule. So, these points which we have obtained those are known as Gauss points and the orthogonal polynomial or rather orthonormal polynomials which we have obtained they are known as the Joindar polynomials. So, for the basic Gauss two point rule our error is going to be of the form fourth derivative of our function evaluated at some point, some constant and then b minus a raise to 5. When we will go to composite rule on each interval we will apply this basic rule. Then if the sub interval length if it is denoted by h that h is going to be b minus a by n and our b minus a in the error for the basic quadrature rule that will be replaced by h we will be summing it over n intervals. So, that is why one power of h will be lost and then you are going to get the error to be less than or equal to constant times h raise to 4. We had obtained the error to be less than or equal to constant times h raise to 4 also in the Simpson's rule. In case of Simpson's rule we had for the composite rules we had to evaluate total function to be total 2 n plus 1 times. Now, for the this particular formula what is going to happen is on each interval you are going to have two points there are n intervals. So, there are going to be two n point. So, this Simpson's rule and Gauss two point rule is going to be on par, but for higher Gaussian rules they will be they are going to be better than the corresponding Newton codes formula. So, in our next lecture we will first derive the error formula for the Gauss two point rule that means we have obtained x square minus one third. So, Gauss two points in the interval minus 1 to 1 they are minus 1 by root 3 and plus 1 by root 3. So, we will look at the general interval a b and then we are going to consider the general Gaussian quadrature and what are the advantages and disadvantages of Gaussian rules. Thank you.


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What is going on guys, today I am back with another Destiny 2 video, now I'm pretty sure I heard Bungie say or mentioned somewhere that Zer will now be located when he comes on a Friday on one of the four new planets. If that is the case I believe I've found where he may be on a certain said planet. If it isn't the case then it's definitely a Zer easter egg for damn sure. But before we get into the video guys if you do enjoy leaving a like really does help me out and if you're new around here and enjoy daily Destiny videos be sure to subscribe. Okay so basically we're gonna open up that director and you want to fly down to Necess and you want to look for Watcher's grave. So yes we want to head down here. Now today is Wednesday, no Tuesday, weekly reset in the morning. The raid comes out Wednesday then this Friday we have trials over the nine and Zer which I can't wait for. I wonder if they're gonna interact somehow. Imagine that, that would be pretty decent. But yes for Zer you will need your legendary shards. I'm not sure how much his exotics will cost but yes I'm guessing probably between 50 and 70 shards maybe. I also think he may sell as well something we ain't seen before within Destiny 2 like heavy ammo packs and special ammo packs or energy ammo packs and power ammo packs so to speak. But yes we've just landed in now on Necess so you want to follow the direction I am taking right here. But yeah I do think Zer will bring something of a shock maybe just maybe maybe bring something like the three coins as well. Imagine that. Would you guys actually welcome that? I'm not sure I think.  I think Exotics are easy enough to get in this game to be honest. If it was the same time within Destiny 1 when Destiny 1 first released I don't think I had a single Exotic within this time. So yeah Exotics are way way easier to get in this game so do we need a three of coins buff? I'm not sure we do to be honest now let me just get rid of these because this is actually area I believe he may come in. Okay that was an epic fail gone out. But we can see this area right here we've got this thing there which I have no idea why it's for. Can't scan nothing there. But if we turn around we've got this. Oh we've got a chest there. We've got this. This triangle. Very reminiscent to that Zer logo or the nine logo. Let me just grab this while we're here. Why not? Okay so yeah if we scan this and listen to this. This reminds me of one of Zer's strange coins a big one come to think of it. Do we even know where strange coins come from? So yeah a clear reference to Zer and strange coins. If you look at that it actually does from above look exactly like a strange coin. So we have a strange coin on a triangle which refers to Zer. I think this possibly could be one of the locations he comes in at if it is true about him spawning in on planets on a Friday. Now isn't actually I'm pretty sure they said that but I just can't remember where so it was it in a weekly update? Was it on a stream? Was it in a trailer video? I don't know but even so guys it still is a easter egg none of the less but I do think this is probably where he will come when he comes to Nessus. Let's hope so anyway. We'll be checking back here this Friday if we can't find him in a social space but yes guys just a quick little video for you guys. Hope you enjoyed it if you did leaving a like really helps me out. Thanks as always for stopping by and if you're new around here and this is the first video you've seen of mine and you like them Bailey Destiny videos be sure to subscribe because that's what you'll get here on my channel. Again thanks for stopping by and hopefully I will see you on that next one.


There is nothing that feels better than when you just like blew out your hair post-shower Wow, okay. Happy Sunday everyone today is the last day of long week.  I'm gonna do a really easy Like soft makeup look maybe some cream blush, you know if you guys are new here Be sure to subscribe. There's been five other vlog or four other vlogs posted this week So be sure to go check those out. I'm still Okay, I'm still trying to figure out like my cream blush game like how does that work like this is too much It's okay, we'll get it together today though. We are going which is why we're in New York. It's for Dom's work trip, right dog We're gonna go see Dom in Nike and times or in store for her Nike campaign in Times Square, so It's a very big day. It's very exciting things happening I put my jewelry on our new rings if you guys saw the vlog yesterday I'm actually trying to post that tick tock as a real right now. I'm using save talk the app To save my tick tock without the watermark, but I'm wearing This white iritsia set makes me feel like a marshmallow. I'm really tired and I'm really hungry So I need to go to brunch. I'm gonna put a coat on. I'm using Dom's new market Jacob bag But I'm using Dom's new market goes bad again because it holds my camera Tell me why I blew out my hair all for it to rain all day long guys all day long Exciting that an ice try it looks so good Literally beautiful. I just left brunch. I met Margot in the East Village I'm actually gonna walk to like Central Park or something at this point Actually, and you got another wife. I'm going to Central Park, but I'm have some work that she's gonna do She's speaking to a university today, so She's doing that. I think I just want to kind of walk around.  I might get cha cha mata to at some point We don't have plans until tonight. We have dinner plans so So beautiful everyone. I think I'm gonna get a cha cha mata It's about a 25-minute walk. I really need my nitro latte I didn't even get it when I was out of cha cha mata like two weeks ago because they were out Watch the New York locations not have the nitro latte. I was literally scream Sorry if they have the nitro latte You can only imagine how happy I am The guy was like not very nice, but that's totally fine Beautiful, but this is my favorite drink in the entire world. Do I spend like eight dollars on it? Yes, I do And it's worth it. I don't even care. It is fully raining. I literally just don't care I've been walking around the city. It's just been so nice listening to Fearless Just really having like a main character moment, if you will I want to come to New York or wherever honestly, I want to start doing solo trips But like they're not really solo trips because it's like I go to cities that I have friends in you know I mean so like I see my friends But I love Traveling and doing things by myself like love traveling and doing things on myself too. Um, so it's really just nice It feels so relaxed so independent. It's really fun I'm just so happy that I have this again. Also the fact that it's actually raining on me right now And I just don't care So I don't know I feel like every time I travel to if I'm in a weird headspace of four Which I was in a weird headspace the past like two weeks if I travel and I get out of my normal routine And I see that the world is bigger. I'm like, oh, I don't even care about like what I was stressing about before So I was really looking forward to this trip because they're honestly just needed it Also, it was obviously like planned because of Dom's work stuff I don't know that's like a bad coping mechanism But I think just in general like travel always inspires me and it always gets me out of my own head Where I was and I'm like, oh, that's like not that big of a dealer like the world is so big or I look at other things And I see people like different like lives and cultures and whatever and I'm like why am I like stressing about that? You know any meme I've also decided that next time Taylor Swift drops an album I'm renting out a karaoke bar in Dallas and it's gonna be like a whole thing I feel like I need to get a cab home. I'm a ways away from what we're staying and it's raining and my hair is getting wet It's later. I've just been working all Sunday Dom just got off her call We're about to go to Times Square, my least ever place in the world, but to see Dom's Nike campaigns. Where's my leather jacket that I don't even like? Oh It's not here. Oh wait, I think I dropped it on the floor. Here it is Like still again nothing to write home about But we're going to dinner after Times Square first Yeah, okay, we're not even in the worst part of Times Square. Oh, is this it the counter? No It's like owned by finish line, which is owned by Nike, right? Okay, the Nike Queen over here. Yeah, I won't be obvious when I just came in there. You're like stop recording Wait, but look here Is it the locker? Wait, that was the other Counter is a burger place that I like have definitely been to by the way. Why did I ask about raining? I'm so mad. I wasn't recording. It just happened on accident.  Hold on Look who did me with my camera Okay, I mean like I Just kind of wait Okay, let's get over here to You want more? Oh, I'm gonna back up a little bit Cuz you know, we've got the whole situation here.   I mean guys like you see We're literally in Times Square. Are you guys understanding that like we're in time for This is so crazy. It literally says my name. Oh, no That's what I'm saying. Dude. This is so crazy. Like also. That's literally the best photo Ever Wow, are you feeling Dominique? Well, Kenzie, thanks for being the best friend ever and coming to New York to see this This is so crazy like this is so crazy literally so crazy Can you believe it? Yeah, also, do you have the green sneakers? Yeah This is like insane. I like cannot get I'll never go over this ever in my entire life. Oh, I'm so clean I know my hair's up. So I'm it. You're like, we'll never get over it. You're in Times Square guys This is crazy. This is so insane and my hair is flat a.m. You're in Times Square. I'm in Times Square It's like insane guys and my boobs look so good in the photos and God. Thanks. Thank you. Thanks, Nike That's it like literally trust my name. Oh My god How are you feeling girl? Hey guys, this is her.  Don fell long story short. We couldn't breathe. We're laughing so hard. It has happened our hair. So we're literally going to dinner Our hair is soaked. We're in Times Square It's just honestly like and I it's really dead right now. I'm holding on to Kenzie Oh, that's why I'm bugging because she's she said if she falls again. I'm going down with her. I'll literally go I'll have to get up but you know Talk about Oh my god, literally, this is foul like no one should ever see you won't see that so I could vlog it Yeah, oh, I just want to be more with the bathroom dinner. We went to Seraphina. It was really good. Um, we This is the diaper. We just But in my diaper we just got oh my god, they're so good Oh Margot hold on.   That's what she's gonna do Call you in five. She's literally gonna die guys. These photos are That's the level. Wait, why do I look snatched?  Like I think that we're wearing no makeup in this. I literally just have my eyebrows done Yeah from kenzie elizabeth llc and dom robert's llc. They should make investments in billboards. Uh, we look so good Not like to be humble and stuff. Yeah, I mean like that in itself See, this is good because if we start posting and then it does well, we'll be able to Holy oh my god. Oh my god the hearts are on it. Yeah, that's why no, but the cover that she made this Like these are so good.  It's been a great five days. I wanted another one soon if you guys want Another let me know we had such a good day Yeah, I'm sure you want me to vlog and edit them for you. No, I'll edit Well, love you guys. I'll see you soon. Bye. Oh my god. I love fiber So I really increased my fiber and take a couple years ago and my life has been completely changed because of fiber and I realized like there was that


Children like to move around. In this game, you can help them learn about positions. You can select a large object, like a stool, and ask the child, can you sit on the stool? Then you might say, now stand beside the stool, or stand in front of the stool. You can continue to ask them to move to different positions, like behind, under, to the left, or to the right. This game helps a child understand their relationship to objects around them.


Hello, and welcome to the STM32G0 training session. The architecture of the STM32G0 is described in this figure. The STM32G0 is a low-power microcontroller whose CPU is an ARM Cortex M0+. Only the GPIO ports are directly accessible from the CPU by using the single-cycle IO bus. Two masters are connected to the main AHB interconnect called bus matrix in the figure, the CPU and also the DMA controller. The AHB slaves are memories SRAM and flash memory, CRC, AES, RCC, RNG, EXTI peripherals, flash controller registers, DMA and DMA MUX registers, the AHB to APB bridge, the APB peripherals are listed in the blue box on the right of the figure. Most peripherals can request a DMA transfer to the DMA MUX and then the DMA controller in order to transfer data to or from SRAM or flash memory. For instance, characters received by USART1 can be transferred to a buffer in SRAM without software intervention by relying on a DMA channel. This slide explains the various fields of the STM32G0 ordering information. Device family is STM32, product type is G. Device sub-family can be 070 for value line 071 for access line and 081 for security line. Four pin counts are supported from 25 to 64 pins with respectively 23 to 60 GPIOs. The flash memory size can be 64 or 128 kilobytes. The following packages are proposed UFPGA, LQFP, UFQFPN and WLCSP. Two temperature ranges are supported, minus 40 to 85°C and minus 40 to 125°C. The last field indicates the options, typically blank or N, depending on whether USB-powered delivery interface number two pins are available in 28 and 32 pin packages. Therefore the STM32G081RBT6 is an STM32G0 security line 64 pin, 128 kilobyte flash memory, LQFP, MCU supporting minus 40 to 85°C temperature range. Most of the pins are GPIOs supporting multiple functions. Dedicated pins are power supply and ground. Power supply pins are VDD, VBAT and VREF+. Note that VBAT and VREF+, pins are not present in 32, 28 and 25 pin packages. The reset signal is multiplexed with GPIO port F number two and boot zero is multiplexed with GPIO port A number 14. So these IO pads have a dedicated function during reset time and become later general purpose IOs. The devices housed in 64 and 48 pin packages provide two USB-C power delivery ports. The devices housed in 28 or 32 pin packages come in two variants, GP with a single port limited USB-C powered livery and PD with two port USB-C powered livery. This slide highlights the differences between the packages supporting and not supporting two port USB-C powered livery. On the left, the STM32G071KXT supports a unique powered livery interface, PD1. On the right, the STM32G071KXTXN supports two power delivery interfaces, PD1 and PD2. The STM32G0 has three power supply pins, VDD or VDDA. VBAT and VREF+, VDD or VDDA is the power supply of the IO pads and integrated flash memory and also supplies voltage regulators which provide the V-Core supply. V-Core voltage is 1.2V for a system clock frequency of 64 MHz. The VBAT power domain consuming very little energy includes RTC and LSE oscillator, temper detection and backup registers. The VREF+, is the input reference voltage for ADCs and DACs. It is also the output of the internal voltage reference buffer when enabled. On packages where VREF+, and VBAT are not available, these power supplies are internally connected to the VDD pin. The STM32G0 offers a new security feature called Securable Memory Area. The main purpose of the Securable Memory Area is to protect a specific part of flash memory against undesired access. This allows implementing software security services such as Secure Key Storage or Safe Boot. The general purpose input outputs or GPIOs can sync or source up to plus or minus 8 mA and sync or source up to plus or minus 20 mA. The STM32G0 supports flex power control which enables efficient running 7 low power modes and several sub modes. In run mode, the consumption is 100 mA per MHz. Stop 0 and Stop 1 are low power modes in which the state of the cortex M0+, is maintained. Stop 1 is equivalent to Stop 0 with main regulator off, resulting in a smaller current consumption but longer wake-up time. In standby mode, by default, there is neither SRAM nor register as retention because voltage regulators are in power down state. However, there is an option to retain the contents of the entire SRAM. Shutdown mode is similar to standby but without power monitoring and the unique clock source is the low-speed external oscillator. The STM32G0 integrates the following subsystems. System modules PWR, RCC, RTC and Tampers, SysTick, GPIO, CRC and SysCFG. Processor modules Cortex M0+, Core, MPU, NVIC and Software Debug. DMA controller supporting 7 channels. Interconnect based on an AHB light matrix and an AHB to APB bridge. Embedded memories 128 kilobyte flash memory plus 36 kilobyte SRAM. Control modules 132-bit timer, 116-bit timer that can be used to control a motor, 516-bit timers with PWM capability, 2 low-power timers. Analog modules temperature sensor, 12-bit ADC, 12-bit DAC and 2 comparators. USB power delivery modules, 2 modules including BMC and PHY. Connectivity modules, 2 SPI modules also supporting I2S, 4 USARTS, 1 LPU ART, 2 I2C controllers and HDMI CEC. Encryption modules AES and TrueRNG. This session is organized to provide you with the most important information to ensure that you can develop your application as easily as possible. You will find a technical description of all the STM32G0 modules including peripherals and development tools organized into specific sections, system, ARM, Cortex, M0+, CPU, interconnect and DMA controller, memories, control, analog, USB power delivery, connectivity, encryption and development ecosystem. You can browse each section separately and learn about each module in the order of your choice and at your convenience. This session also allows you to search directly for a keyword and you will have a direct access to sections covering this information. Now let's get started with the training. Do not hesitate to follow the events and news about this product on our website at www.st.com.stm32g0. Enjoy!


From the Cosmopolitan Hotel in Las Vegas, Nevada, it's theCUBE, covering Coupa Inspire 2019, brought to you by Coupa. Welcome to theCUBE, Lisa Martin on the ground at Coupa, Inspire 19 from Las Vegas. Very excited to welcome one of Coupa's spend-setters from the USO, Rick Quaintan's Senior Director of Procurement and Contract Management. Hey, Rick, welcome. Thank you, welcome. Glad to be here. Yeah, so this is one of the things that I really appreciate with all of the tech conferences that we go to on theCUBE, which is many, many a year, is when vendors like Coupa really share their successes through the voices and the stories of their successful customers. You got called out yesterday during general session today. There's a big cardboard color W behind us there. But one of the things also that I find intriguing is looking at older organizations. USO is 77 years young. We think of older organizations challenging to maneuver that in this digital era and really be able to transform the business so that you can do what the mission of the USO is, which is to help men and women in our US armed forces from the time that they enter to the time that they transition back to civilian life. Talk to us a little bit about the USO, what your role is in procurement, and then we'll talk about how you're achieving these great things with Coupa. Well, I've been with the USO for four years, almost four years, when I first interviewed for this position with my boss, the VP controller, I asked her if they had a procure to pay solution. She said, no. And so again, when I was hired for this position, my goal was to get the organization automated. They were processing everything by paper. All the requisitioning was being processed by paper. It would take four, seven, 10 days for a requisition to be approved because it would literally be something printed out and move from desk to desk to desk on approvals. And on the back end for invoicing would occur the same, filling out a cover sheet, everything was printed out, processed manually. So that was kind of my first project when I started and my position was new. Procurement had been under the director of accounting operations. So procurement was just a small piece of it. So they made a decision after he left to create my position. And so again, that was my goal initially when I started. So it was going through an RFP process, looking at our requirements and then selecting the vendor that's the best value to the USO, which was Coupa. And Coupa is what I think we all love about it is it's so customizable. And the USO has a lot of different requirements in our procurements from, we have entertainment tours to our programs, care packages we send out to the military, our operations, our USO centers, construction projects, our development campaigns for online and direct mail. So there are a lot of different requirements. So I had to really work with each department in kind of setting up those requirements. And Coupa was able to do that for us. We were able to customize a lot of it. But for us, the innovation part is really thinking outside the box because- Tough to do for a 77 year old organization, right? Especially one that has paper everywhere. You guys are now 90.4% paperless with Coupa. That's a massive, that's a cultural change, right? It's huge. And it took, again, another thing when I interviewed was, I interviewed with the CFO as well. And I said, if you don't support me, I will not be successful. So they have been very supportive. My supervisor, the CFO, the entire organization, the CEO has been extreme. He loves Coupa. So he loves the app in approving invoices and requisitions. So it was really that communication, the socialization, the training, because it was a huge cultural shift. And some were embraced, it was a little tougher for others moving, but eventually you move them on because that's the new process for us as an organization. So it's become very successful. We're moving towards new modules, contracts, CLM, expense sourcing. So we're really expanding the Coupa picture at USO. So what would you say before you came on board when there was so much paper floating around everywhere? Can you imagine the security risk of all these, you know, personal information or what have you lying around on someone's desk? But what was the percent, if you could guess, visibility into where the USO was spending money prior to bringing on Coupa versus what is it today? Extremely small. The percentage would have been very small.  Not, you know, it's just, it's hard to see the visibility. Now it's 100% visibility. We see all of the requisitioning occurring overseas. You know, we have centers all over the world and they all have access to Coupa now because they have to submit requisitions through Coupa. And so we now have 100% visibility. And for our reporting, you know, we've been able to pull all that information and we've got controls in place. Coupa gave us the ability to put some controls in place in our approval workflows and making sure that contracts are reviewed before budgets are approved, et cetera. A lot of those things, we were able to set those controls in place in Coupa.  We've been talking about that for the last couple of days and it's the same when we were talking with Suzy Orman earlier who was one of the keynotes and when she talks about, you know, personal finance, it's sort of the same thing. We all as individuals, whether we're consumers, you know, in our personal lives, buying whenever we want from anything.com to being buyers or managers of even lines of business within whatever company we work for, we need to have that picture, that control. And control is really that kind of accountability and that awareness. Are we managing everything appropriately? Are there other parts of the business that are doing the same thing that they're maybe getting the same services at a better price than we are? We should know that. But without having that visibility and being able to control all those processes, it's an inhibitor to any business being able to transform digitally and be competitive and to really get back to your core mission.  I mean, we are a 501C3, so we need to have that visibility and make sure that our donor dollars are being spent wisely and this enables us to do that, enable us to have that total visibility and making sure those controls are in place. Has that actually, speaking of donor dollars, has this actually been a facilitator of actually being able to increase donations because the donors now have this a much easier transaction process? Can you imagine that would be a positive impact there? Well, I mean, this is more for our procurements. I mean, COOP is kind of more for our actual procurements, but what it does do is it does, creates process savings and avoidant savings, which we can reinvest in our programs. So that's where we're seeing it. That's where CFO is seeing it. We've communicated that to him. And then we're also able to provide our CFO with reporting tools. So we pull all this information from Cuba through reports and then we're able to create a spreadsheet and he can see how we spend as an organization, how we spend in commodities, where are our unbudgeted, kind of get a total of how much unbudgeted we have for a specific period of time. So we're able to see all this kind of information, he can see all this kind of information on one spreadsheet that we've created through all the reports that we can create in Cuba. So I want to get your perspectives on the changing role of the chief procurement officer and the chief financial officer. Now they have the opportunity to leverage technology, emerging technologies like artificial intelligence and machine learning to be able to get that visibility and that control, but also be farmer strategic and really drive top line or bottom line for their business. Give me your perspective on the last few years alone and how were you able to help a 77 year old organization like the U.S. so embrace the opportunities that these emerging technologies can deliver? Well I think one key is because our organization is all over the world and then there are centers that can be rule and it's the whole vendor presence and the amount of vendors that we as an organization do bring on and some of them, it's totally understandable where some of them they do need to bring on based on their availability, but what I'm trying to do and what Coupé has helped me try to do with Coupé Advantage is to try to leverage our volume, our organizational volume. That was not occurring previously. I think people were just, when they needed a vendor they brought it on because we have a lot of events, supplies for the centers, et cetera. So really trying to strategically as an organization to be able to work with the regions on where can we find synergies to kind of consolidate and leverage our volume for vendors and with Coupé we've been able to do that because we can see the span where it's occurring, kind of all the duplications that are occurring. So that's where I'm seeing a big opportunity and trying to work on. One of the coolest things about what you guys are doing in procurement with Coupé is this is affecting human lives. Give us a little bit of an overview of what you guys were able to facilitate with Hurricane Harvey, which struck Houston just about two years ago. I loved that story that kind of shared this matter. And those are kind of those spur-of-the-moment emergency type requisitions that we get and we're able to, those can get processed a lot quicker when we have Coupé as opposed to previously the way they had processed it was very labor-intensive or manually, verbally instead of being able to see it in, what's great about the requisitioning piece of it is the comments kind of audit that people can see in all the conversations. So those types of requests that are considered emergencies, they can go, get processed a lot sooner and so we can get those services or the goods out to that particular project. So that's what we were able to do with that particular one as well, being able to support the National Guard during the Hurricane Harvey disaster. And accelerate the things that really, based on the data that you can see, really need to have acceleration and all the actions that the USO needs to deliver. I mean, just in just to our programs team, they support the care packages that we send to the military. Now that we have Coupé in place, we use a third party fulfillment center. When they receive the product, the receipts are automatically fed into Coupé and applied against the purchase orders and then they are approved a lot quicker so then they can get received, kicked the product and ship it out overseas because these are based on requests that the military bases have requested to have this particular product being sent over. So this turns, the process is cut in half to get the care packages out to the military. That's awesome. Getting care packages to the troops 50% faster is outstanding. Last question for you, Rick. Some of the things that Coupé has announced in the last day and a half, what excites you about the direction that this company is going in? For me, the constant changing. I mean, and I was not in the military. So I mean, I just, we moved around a lot when I was growing up. I adapt to change very quickly but understand some people don't as quickly but it's bettering themselves, finding the opportunity, listening to the customer and really making those enhancements based on customer feedback. And I think it helps with the community intelligence that we talk with, you know, with the communities and find out what are you doing? How are you doing this? Because a lot of companies will say, well, I have specific requirements and a lot of them are pretty similar if people talk, you know, the community talks.  And so it's, yeah, it's pretty exciting. I like how Chandar this morning, you know, showed the word community and said, really it's communication and unity. And you just articulated that beautifully. It's listen to the customers, get the synergies from them. That's why we should, any software business should be developing software. So Rick, thank you so much for joining me on theCUBE today, sharing the big impact that you guys are making at the USO charity near and dear to my heart. We appreciate your time. All right, thank you very much. For Rick Quaintance, I'm Lisa Martin. You're watching theCUBE from Coupé Inspire 19. Thanks for watching.