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CS4413 Numeric Algorithms (These materials are used in the classroom only). Goals. Evaluate polynomials. More efficient matrix multiplication. Solve linear equations. Describe growth rates and order. Why numeric algorithms?.
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CS4413 Numeric Algorithms (These materials are used in the classroom only)
Goals • Evaluate polynomials. • More efficient matrix multiplication. • Solve linear equations. • Describe growth rates and order.
Why numeric algorithms? • Mathematical calculation forms the basis for a wide range of programs. • Computer Graphics and Computer Vision both require a large number of calculations involving polynomials and matrices. • Because these are typically done for each location in an image, small improvements can have a great impact.
Why numeric algorithms? • For example: a typical image can be created with 1024 pixels per row and 1024 pixels per column. • Improving the calculation for each of these locations by even one multiplication would reduce the creation of this image by 1,048,576 multiplications overall. • Some software will repeatedly evaluate complex polynomial equations. • Another application is trigonometric functions such as sine and cosine having power series expansions that take the form of polynomial equations.
Calculate polynomials • The standard evaluation algorithm is very straightforward. Evaluate (x) x the value to use for evaluation of the polynomial result = a[0] + a[1] * x xPower = x for i = 2 to n do xPower = xPower * x result = result + a[i] * xPower end for return result
Horner’s method • Horner’s method gives a better way to do evaluation without making the process very complex. p(x) = ({…[(anx + an-1) * x + an-2] * x + … + a2} * x + a1) * x + a0 HornersMethod(x) x the value to use for evaluation of the polynomial result = a[n] for i = n – 1 down to 0 do result = result * x result = result + a[i] end for return result
Complexity • Standard: 2N – 1 Multiplications and N Additions. • Horner’s:N Multiplications and N Additions.
Preprocessed Coefficients • For preprocessed coefficients to work, we need our polynomial to be monic (an = 1) and to have its largest degree equal to 1 less than a power of 2 (n = 2k -1 for some k = 1).
Matrix multiplication • A matrix is a mathematical structure of numbers arranged in rows and columns that is equivalent to a two-dimensional array. • Two matrices can be added or subtracted element by element if they are the same size. • Two matrices can be multiplied if the number of columns in the first is equal to the number of rows in the second.
Matrix multiplication … • If we multiply a 3 x 4 matrix by a 4 x 7 matrix, we will get a 3 x 7 matrix as our answer. • Matrix multiplication is not commutative. • The standard matrix multiplication algorithm will do a * b * c multiplications and a * (b -1) * c additions for two matrices of size a x b and b x c.
Winograd multiplication • If you look at each element of the result of a matrix multiplication, you will see that it is nothing more than the dot product of the corresponding row and column of the original matrices. • Consider two of these vectors: V=(v1,v2,v3,v4) and W=(w1,w2,w3,w4). Their dot product is given by V·W.
Winograd Multiplication • V=(v1,v2,v3,v4) • W=(w1,w2,w3,w4) • V·W=v1w1+v2w2+v3w3+v4w4 • V·W=(v1+w2)(v2+w1) + (v3+w4)(v4+w3) -v1v2-v3v4-w1w2-w3w4
Generalization Compute Once, Use many times
Winograd algorithm • Multiplying G a x b and H b x c to get result R a x c. d = b / 2 //calculate rowFactors for G for i = 1 to a do rowFactor[i] = Gi,1 * Gi,2 for j = 2 to d do rowFactor[i] = rowFactor[i] + Gi,2j-1 * Gi,2j end for j end for i //calculate columnFactors for H for i = 1 to c do columnFactor[i] = H1,i * H2,i
Winograd algorithm… for j = 2 to d do columnFactor[i] = columnFactor[i] + H2j-1,i * H2j,i end for j end for i //calculate R for i = 1 to a do for j = 1 to c do Ri,j = -rowFactor[i] – columnFactor[j] for k = 1 to d do Ri,j = Ri,j + (Gi,2k-1 + H2k,j) * (Gi,2k + H2k-1,j) end for k end for j end for i
Winograd algorithm… //add in terms for odd shared dimension if (2 * (b/2) ≠ b) then for i = 1 to a do for j = 1 to c do Ri,j = Ri,j + Gi,b * Hb,j end for j end for i end if
Analysis • Ordinary Matrix Multiply, n3 • Winograd’s Matrix Multiply, (n3/2)+n2 • Additions, n3-n2, v.s. (3/2)n3+2n2-2n • Lower Bound, Best known is n2 • Best known Upper Bound and Best Known Lower Bound are not the same
Two By Two Multiplication • c1,1=a1,1b1,1+a1,2b2,1 • c1,2=a1,1b1,2+a1,2b2,2 • c2,1=a2,1b1,1+a2,2b2,1 • c2,2=a2,1b1,2+a2,2b2,2
2x2 Works for Matrices • C1,1=A1,1B1,1+A1,2B2,1 • C1,2=A1,1B1,2+A1,2B2,2 • C2,1=A2,1B1,1+A2,2B2,1 • C2,2=A2,1B1,2+A2,2B2,2 A1,1 A1,2 B1,1 B1,2 C1,1 C1,2 ´ = A2,1 A2,2 B2,1 B2,2 C2,1 C2,2
Divide and Conquer? • Assume matrices of size 2n´2n • Multiplications: M(n) = 8M(n/2), M(1)=8 • Additions: A(n) = 8A(n/2)+n2 • M(n) = 8lg n = nlg 8 = n3 • Additions are also Q(n3), but point is moot • Can we reduce the 8 multiplications in the base equations
Strassen’s Equations • x1=(a1,1+a2,2)(b1,1+b2,2) • x2=(a2,1+a2,2) b1,1 • x3=a1,1(b1,2-b2,2) • x4=a2,2(b2,1-b1,1) • x5=(a1,1+a1,2) b2,2 • x6=(a2,1-a1,1)(b1,1+b1,2) • x7=(a1,2-a2,2)(b2,1+b2,2)
Using The Equations • c1,1 = x1 + x4 - x5 + x7 • c1,2 = x3 + x5 • c2,1 = x2 + x4 • c2,2 = x1 + x3 - x2 + x6
Analysis of Strassen • Only 7 multiplications are used • Will work with matrices, because associative property is not used • Multiplications: M(n) = 7M(n/2), M(1) = 7 • Additions: A(n) = 7A(n/2)+18(n2/4) • M(n) = 7lg n = nlg 7 = n2.81 • Additions are also Q(n2.81) • First algorithm to break the n3 “barrier”
Gauss-Jordan Method • Any system of linear equations must satisfy one of the following: (1) has no solution. (2) has an unique solution. (3) has an infinite number of solutions.
Elementary Row Operations (1)TypeⅠoperation:A’ is obtained by multiplying any row of A by a nonzero scalar c. e.g.
Elementary Row Operations (2)Type Ⅱ operation:For some j≠i, let row i of A’=c(row i of A )+row j of A. And let the other rows of A’ be the same as the row of A. e.g.
Elementary Row Operations (3)Type Ⅲ operation:Interchange any tow rows of A. e.g. -2(1)+(2) (2)'
Continued (1)-(2)” is the unique solution of such problem. • Note: the above 4 systems of linear equation are equivalent.
Example [A∣b] = want
Example + + +
Example + + +
Usage of type Ⅲ operations ex [A│b]=
Special Cases:No Solution ex one row no solution
Special Cases: Infinite Solutions ex Homework: one row infinite solutions