MATLAB Lecture 6 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr MATLAB Lecture 6-Polynomial 多项式 Ref: MATLAB-Mathematics-Polynomials and Interpolation Vocabulary: polynomial多项式 root根 arithmetic operation算术运算 multiply乘法 divide除法 derivative导数 differentiation微分法 求值 action 部分分式 展开 convolution卷积 product乘积 deconvolution去卷积 quotient商 remainder余项 multiple roots重根 direct直接的 term transfer function转换函数,传递函数 ● Some functions conv decon poly polyder polyval polyvalm roots *residue polyfit Y Representing Poly nomials MATLAB represents polynomials as row vectors containing coefficients ordered descending powers >>p=[10-2-5] %represents x'-2x-5 >> sym p= poly 2sym(p) represents a polynomial in sym form sym p- X^3-2*x-5 ☆ Create polynomials >>p=[10-2-5 % represents a polynomail x'-2x-5 0 >>r=[0, 1, -1]; poly(r) %generate a polynomial x(x-D(x+1), whose roots are 0, 1,-1 >>a=[12: 3 4; poly(a) generate the characteristic polynomials of matrix Lec6-I
MATLAB Lecture 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec61 MATLAB Lecture 6 – Polynomial 多项式 Ref: MATLAB→Mathematics→Polynomials and Interpolation l Vocabulary: polynomial 多项式 root 根 arithmetic operation 算术运算 multiply 乘法 divide 除法 derivative 导数 differentiation 微分法 evaluation 求值 partialfraction 部分分式 expansion 展开 convolution 卷积 product 乘积 deconvolution 去卷积 quotient 商 remainder 余项 multiple roots 重根 direct 直接的 term 项 transfer function 转换函数，传递函数 l Some functions conv deconv poly polyder polyval polyvalm roots * residue * polyfit l Polynomials ² Representing Polynomials MATLAB represents polynomials as row vectors containing coefficients ordered by descending powers. >> p = [1 0 2 5]; % represents 3 x - 2x -5 p = 1 0 2 5 >> sym_p = poly2sym(p) % represents a polynomial in sym form sym_p = x^32*x5 ² Create Polynomials >> p = [1 0 2 5] % represents a polynomail 3 x - 2x -5 p = 1 0 2 5 >> r = [0, 1, 1]; poly(r) %generate a polynomial x(x -1)(x +1) ,whose roots are 0,1,1 ans = 1 0 1 0 >> a = [1 2; 3 4]; poly(a) % generate the characteristic polynomials of matrix… 1 2 3 4 l l Ê - - ˆ Á ˜ Ë - - ¯ , i.e. 2 1 2 5 2 3 4 l l l l - - = - - - -
ATLAB Lecture 6 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr 1.0000-5.0000-2.0000 t Polynomial Evaluation > polyval(p, 5) evaluates a polynomial at a specified value, say 5 > subs(sym p, 5) %o substitute the sym variable x in sym p with 5 110 X=[245;-103;71 >>Y=polyvalm(p, X)% create a square matrix X and evaluate the polynomial p at X Y=X3-2X-5 377179439 490253639 ☆ Polynomial Roots >>r=roots(p) calculates the roots of a polynomial p 2.0946 -1.0473+ 1.13591 1.0473 1.13591 Y Polynomial Arithmetic operation Addition >>p2=02-13: add p=p+p2 %calculates sum of two polynomials p and p2 Here the matrix dimensions must agree add p= >>p3= poly2sym(p2); add p sym= sym p+ p3 %sym p pulses p3 and display the result in sym form dd p sym x^3-3*x-2+2*x^2 > sym2poly(add p sym) returns a row vector containing the coefficients of the symbolic polynomial P Subtraction(Omit. It is similar to addition) Multiplication( Correspond to the operations convolution >a=[12; b=[20-1; c=conv(a, b); poly2sym(c)
MATLAB Lecture 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec62 ans = 1.0000 5.0000 2.0000 ² Polynomial Evaluation >> polyval(p,5) % evaluates a polynomial at a specified value, say 5. ans = 110 >> subs(sym_p,5) % substitute the sym variable x in sym_p with 5 ans = 110 >> X = [2 4 5; 1 0 3; 7 1 5]; >> Y = polyvalm(p, X) % create a square matrix X and evaluate the polynomial p at X… 3 Y = X - 2X -5I Y = 377 179 439 111 81 136 490 253 639 ² Polynomial Roots >> r = roots(p) % calculates the roots of a polynomial p r = 2.0946 1.0473 + 1.1359i 1.0473 1.1359i ² Polynomial Arithmetic operation Addition >>p2 = [0 2 1 3]; add_p = p+p2 %calculates sum of two polynomials p and p2. … Here the matrix dimensions must agree. add_p = 1 2 3 2 >> p3 = poly2sym(p2); add_p_sym = sym_p + p3 %sym_p pulses p3 and display the … result in sym form. add_p_sym = x^33*x2+2*x^2 >> sym2poly(add_p_sym) % returns a row vector containing the coefficients … of the symbolic polynomial P ans = 1 2 3 2 Subtraction (Omit. It is similar to addition) Multiplication (Correspond to the operations convolution) >> a = [1 2]; b = [2 0 1]; c = conv(a, b); poly2sym(c) …
ATLAB Lecture 6 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr compute the product of (x+2)(2x-1) 2*x^3+4*x^2-X Division( Correspond to the operations convolution and deconvolution >>a, r=decon(c, a) %dividing c by a is quotient q and remainder r Y Polynomial Derivatives q= polder(p) %computes the derivative of polynomial (x'-2x-5) >>a=[135: b=[246;c=polyder(a, b) %computes the derivative of the Ict of two polynomials [(x+3x+5)(2x+4x+6] 8305638 >>[q, d]= polyder(a, b) %computes the derivative of the quotient of two polynomial (x2+3x+5)_q(x) (2x2+4x+6)」d(x) 4164 t *Partial Fraction Expansion residue finds the partial fraction expansion of the ratio of two polynomials. This is particularly useful for applications that represent systems in transfer function form. For polynomials b and a, if there are no multiple roots b(x) P1 x- p2 P where r is a column vector of residues, p is a column vector of pole locations, and k is a row vector of direct terms >>b=[-48: a=[168[, p, k]=residue(b, a)% x2+6x+8x+4x+2
MATLAB Lecture 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec63 % compute the product of 2 (x + 2)(2x -1) ans = 2*x^3+4*x^2x2 Division (Correspond to the operations convolution and deconvolution) >> [q, r] = deconv(c, a) %dividing c by a is quotient q and remainder r q = 2 0 1 r = 0 0 0 0 ² Polynomial Derivatives >> q = polyder(p) %computes the derivative of polynomial 3 (x - 2x - 5)¢ q = 3 0 2 >> a = [1 3 5]; b = [2 4 6]; c = polyder(a,b) %computes the derivative of the … product of two polynomials 2 2 [(x + 3x + 5)(2x + 4x + 6)]¢ c = 8 30 56 38 >> [q,d] = polyder(a,b) %computes the derivative of the quotient of two polynomials… 2 2 ( 3 5) ( ) (2 4 6) ( ) x x q x x x d x ¢ È + + ˘ = Í ˙ + + Î ˚ q = 2 8 2 d = 4 16 40 48 36 ² *Partial Fraction Expansion residue finds the partial fraction expansion of the ratio of two polynomials. This is particularly useful for applications that represent systems in transfer function form. For polynomials b and a, if there are no multiple roots, 1 2 1 2 ( ) ... ( ) n s n b x r r r k a x x p x p x p = + + + + - - - where r is a column vector of residues, p is a column vector of pole locations, and k is a row vector of direct terms. >> b = [4 8]; a = [1 6 8]; [r, p, k] = residue(b, a) % 2 4 8 12 8 6 8 4 2 x x x x x - - - = + + + + + r = 12 8
ATLAB Lecture 6 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr k Given three input arguments(r, p, and k), residue converts back to polynomial form due(r, p, k)% x+4x+2x2+6x+8 Polynomial Function Summary Function Description Multiply polynomials decoy Divide polynomials Poly fied roots I Po olynomial derivative orval Polynomial evaluation olyvalm Matrix polynomial evaluation oots i Find polynomial roots. polyfit Polynomial curve fitting
MATLAB Lecture 6 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu.cn Lec64 p = 4 2 k = [ ] Given three input arguments (r, p, and k), residue converts back to polynomial form. >> [b2, a2] = residue(r, p, k) % 2 12 8 4 8 4 2 6 8 x x x x x - - - + = + + + + b2 = 4 8 a2 = 1 6 8 ² Polynomial Function Summary Function Description conv Multiply polynomials. deconv Divide polynomials. poly Polynomial with specified roots. polyder Polynomial derivative. polyval Polynomial evaluation. polyvalm Matrix polynomial evaluation. roots Find polynomial roots. residue Partialfraction expansion (residues). polyfit Polynomial curve fitting