7.1 Roots of a Polynomial

Transcription

1 7.1 Roots of a Polynomial A. Purpose Given the oeffiients a i of a polynomial of degree n = NDEG > 0, a 1 z n + a 2 z n a n z + a n+1 with a 1 0, this subroutine omputes the NDEG roots of the polynomial. Sine the roots may be omplex, they will always be returned as omplex numbers, even when they are real. Subroutines are provided for four ases: SPOLZ DPOLZ CPOLZ ZPOLZ for single preision real oeffiients. for double preision real oeffiients. for single preision omplex oeffiients. for double preision omplex oeffiients. See Setion E for remarks on the method of storing double preision omplex numbers. B. Usage B.1 Program Prototype, Single Preision Real Coeffiients REAL A( NDEG + 1), H( NDEG**2) COMPLEX CALL SPOLZ(A, NDEG, Z, H, IERR) B.2 Program Prototype, Double Preision Real Coeffiients A( NDEG + 1), H( NDEG**2) COMPLEX*16 Z(2, NDEG) or CALL DPOLZ(A, NDEG, Z, H, IERR) B.3 Program Prototype, Single Preision Complex Coeffiients 1997 Calif. Inst. of Tehnology, 2015 Math à la Carte, In. COMPLEX A( NDEG + 1) REAL COMPLEX H( 2 * NDEG**2) CALL CPOLZ(A, NDEG, Z, H, IERR) B.4 Program Prototype, Double Preision Complex Coeffiients A(2, NDEG + 1) or COMPLEX*16 A( NDEG + 1) COMPLEX*16 H ( 2 * NDEG**2) Z(2, NDEG) or CALL ZPOLZ(A, NDEG, Z, H, IERR) B.5 Argument Definitions A() [in] Contains the oeffiients of a polynomial, high order oeffiient first, with A(1) 0. The ontents of this array will not be modified by the subroutine. In the ase of ZPOLZ, if the delaration DOUBLE PRECISION A(2, NDEG+1) is used, the user must store the real part of a i in A(1, i) and the imaginary part in A(2, i). NDEG [in] Degree of the polynomial. H() [srath] Work spae for the subroutine. Z() [out] Contains the polynomial roots stored as omplex numbers. In the ases of DPOLZ and ZPOLZ, if the delaration Z(2, NDEG) is used, the real part of z i will be returned in Z(1, i) and the imaginary part in Z(2, i). IERR [out] Error flag. Set by the subroutine to 0 on normal termination. Set to 1 if A(1) = 0. Set to 2 if NDEG < 1. Set to J > 0 if the iteration ount limit has been exeeded and roots 1 through J have not been determined. July 11, 2015 Roots of a Polynomial 7.1 1

2 C. Examples and Remarks The program, DRSPOLZ, uses SPOLZ to ompute the roots of the two polynomials x 3 4x 2 + x 4, and x 5 15x x 3 225x x 120 The output is shown in ODSPOLZ. The Fortran 77 standard does not support a double preision omplex data type, although suh a type is supported in many ompilers using the delaration, COM- PLEX*16. The subroutines desribed here onform to the standard and thus do not use COMPLEX*16. For ases in whih double preision omplex quantities are ommuniated using the arrays A() or Z(), these subroutines store a omplex number as an adjaent pair of double preision numbers, representing respetively the real and imaginary parts of the omplex number. This is ompatible with the Fortran 90 storage onvention for double preision omplex. If the user has the COM- PLEX*16 delaration available and wishes to use it, it will generally be ompatible with this storage onvention. D. Funtional Desription Method The degree, NDEG, and oeffiients, a 1,..., a NDEG+1, are given. An error ondition is reported if NDEG < 1 or if a 1 = 0. Let k + 1 be the index of the last nonzero oeffiient. If k < NDEG, the polynomial has a root at zero repeated NDEG k times and these roots are reorded. The remaining roots will be roots of the polynomial of degree k whose oeffiients are the first k + 1 given oeffiients. If k = 1, the root a 2 /a 1 is reorded. If k > 1, the subroutine forms the k k ompanion matrix H for this k th degree polynomial. The matrix H is zero exept for the first row, h 1,j = a j+1 /a 1, j = 1,..., k and the elements just below the diagonal, h i,i 1 = 1, i = 2,..., k A saling algorithm is applied to H to balane the sizes of the nonzero elements. Testing showed that auray ould be very unsatisfatory if no saling were done. The saling method used is a modifiation of the subroutine BALANC from the EISPACK olletion of matrix eigensystem subroutines [1], [2]. The modifiation avoids treating the elements that are known to be zero due to the form of the ompanion matrix. The searh for useful row or olumn permutations done in BALANC is also deleted sine, with a k+1 known to be nonzero, none of these permutations are possible. The eigenvalues of the balaned H matrix are then omputed using appropriate ode for the QR algorithm from EISPACK. For SPOLZ and DPOLZ the ode for the EIS- PACK subroutine, HQR, has been inorporated in-line. For CPOLZ or ZPOLZ, a all is made to SCOMQR or DCOMQR, respetively. These latter two subroutines are, respetively, single preision and double preision versions of the EISPACK subroutine, COMQR. Referene [3] reports that ompared with other widely used methods, the onversion of the root finding problem to an eigenvalue problem is superior in reliability, generally superior in auray, and omparable in speed. Auray tests A root of a polynomial is alled ill-onditioned if a small relative hange in the oeffiients makes a signifiantly larger relative hange in the root. Roots that are bunhed losely together relative to their magnitude, or relative to their distane from other roots, tend to be illonditioned. An extreme ase of ill-onditioning ours with multiple roots. If the oeffiients of a polynomial are only known to some relative preision, say p, then a double root will typially have a relative unertainty of about p 1/2, and a triple root a relative unertainty of about p 1/3. The auray of a omputed root is limited by the inherent onditioning of the root. Rather than diretly testing the auray of the omputed roots, we have hosen to test the auray with whih polynomial oeffiients ould be reonstituted from the omputed roots. To test these subroutines we seleted eight sets of roots to test SPOLZ and DPOLZ and a different group of eight sets of roots to test CPOLZ and ZPOLZ. These sets inluded double and triple roots, roots that formed a small luster relative to other roots, roots that differed by a fator of 10 8, and both real and omplex roots. The number of roots in eah set varied from 1 to 7. From eah set of roots we reated 8 test sets by multiplying all roots in the set by one of the fators, 10 3, 10 2,..., 10 3, or This produed a total of 64 test sets for eah subroutine. To exeute the test for one subroutine, say xpolz, and one set of roots, say r 1,..., r n, we did the following: 1. Compute, in double preision, the oeffiients, say a i, of the moni polynomial having the roots, r i. Subroutine ZCOEF of Chapter 15.3 is used for this. 2. Use subroutine xpolz to ompute the roots, say s i, of this polynomial Roots of a Polynomial July 11, 2015

3 3. Compute, in double preision, the oeffiients, say b i, of the moni polynomial having the roots, s i. 4. Compute δ = the maximum of the relative differenes between orresponding oeffiients, a i and b i. 5. Compute ε = δ /ρ, where ρ is the relative preision of the arithmeti being used by xpolz. The value for ρ was obtained as R1MACH(4) for single preision and D1MACH(4) for double preision. (See Chapter 19.1.) Ideally, Step 3 should be done using signifiantly higher preision than is used by the subroutine xpolz being tested, so the error measures, δ and ε, will be attributable only to xpolz with no inflation from Step 3. Sine we used double preision in Step 3 for all ases, we onlude that the reported values of ε are solely attributable to SPOLZ and CPOLZ in the tests of those subroutines, while the values of ε reported for DPOLZ and ZPOLZ are somewhat inflated due to errors from Step 3. These tests were run on an IBM PC/AT equipped with the Intel math oproessor. The preision was ρ for single preision and for double preision. The results may be summarized as follows: Max. ε over Number of Cases Subroutine 64 test ases with ɛ > 10 SPOLZ DPOLZ CPOLZ 8 0 ZPOLZ 43 8 Referenes 1. B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines EISPACK Guide, Leture Notes in Computer Siene 6, Springer Verlag, Berlin (1974) 387 pages. 2. B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler, Matrix Eigensystem Routines EISPACK Guide Extension, Leture Notes in Computer Siene 51, Springer Verlag, Berlin (1977) 343 pages. 3. S. Goedeker, Remark on algorithms to find roots of polynomials, SIAM J. on Sientifi Computing 15, 5 (Sept. 1994) E. Error Proedures and Restritions The error indiator, IERR, will be set as follows: = 0 when no errors are deteted. = 1 if A(1) is zero, or A(1, 1) and A(2, 1) are both zero. = 2 if NDEG < 1. = J > 0 if the iteration ount limit has been exeeded and roots 1 through J have not been determined. The matrix eigenvalue odes allow a maximum of 30 iterations for eah root. This limit is ample. When IERR is set nonzero, the subroutine will also issue an error message using ERMSG, of Chapter 19.2, with an error level of 0. F. Supporting Information Entry Required Files CPOLZ AMACH, CPOLZ, ERFIN, ERMSG, SCOMQR DPOLZ AMACH, DPOLZ, ERFIN, ERMSG SPOLZ AMACH, ERFIN, ERMSG, SPOLZ ZPOLZ AMACH, DCOMQR, DZABS, ERFIN, ERMSG, ZPOLZ, ZQUO, ZSQRT Designed by C. L. Lawson, JPL, May Programmed by C. L. Lawson and S. Y. Chiu, JPL, May 1986, Feb July 11, 2015 Roots of a Polynomial 7.1 3

4 DRSPOLZ Program DRSPOLZ >> DRZPOLZ Krogh Mods to g e t omples used in s i n g l e pre. >> DRZPOLZ Krogh Set f o r d e r i v i n g s i n g l e p r e i s i o n C v e r s. >> DRSPOLZ WVS Remove 0 i n format >> DRSPOLZ CLL >> DRSPOLZ Lawson I n i t i a l Code. Conversion s h o u l d only be done from D to S f o r p r o e s s i n g to C. S r e p l a e s? : DR?POLZ,?POLZ Demonstration d r i v e r f o r SPOLZ. real A1 ( 4 ), A2 ( 6 ), H( 2 5 ) ++ CODE f o r.d..c. i s i n a t i v e C r e a l Z1 ( 2, 3 ), Z2 ( 2, 5) omplex Z1 ( 3 ), Z2 ( 5 ) integer N1, N2, IERR integer I, k %% l o n g i n t i, k ; data A1 / 1. E0, 4.E0, 1. E0, 4.E0 / data A2 / 1. E0, 15.E0, 8 5. E0, 225.E0, E0, 120.E0 / N1 = 3 N2 = format (, Degree =, I2 /, C o e f f i i e n t s =, ( T20, 4 ( F10. 4, 1X) ) ) 200 format (, Roots = /(2(1X, (,1X, F8. 5,,,1X, F8. 5, 2X, ) : 2X) ) ) 300 format (// ) print 100, N1, (A1( I ), I =1,4) %% p r i n t f ( Degree =%2l d \n C o e f f i i e n t s =, n1 ) ; %% f o r ( i = 0 ; i < 4 ; i++) p r i n t f ( %10.4 f, a1 [ i ] ) ; ++ End a l l SPOLZ(A1, N1, Z1,H, IERR) ++ CODE f o r.d. &.C. i s i n a t i v e C p r i n t 200, ( ( Z1 (K, I ),K=1,2), I =1,3) print 200, ( Z1 ( I ), I =1,3) print 300 print 100, N2, (A2( I ), I =1,6) %% p r i n t f ( \n Roots =\n ) ; %% f o r ( i = 0 ; i < 3 ; i +=2) { %% p r i n t f ( ( %8.5 f, %8.5 f ), z1 [ i ] [ 0 ], z1 [ i ] [ 1 ] ) ; %% i f ( i <2) p r i n t f ( ( %8.5 f, %8.5 f ), z1 [ i + 1 ] [ 0 ], z1 [ i + 1 ] [ 1 ] ) ; %% p r i n t f ( \n ) ; } %% p r i n t f ( \n\n \n Degree =%2l d \n C o e f f i i e n t s =, n2 ) ; %% f o r ( i = 0 ; i < 6 ; i +=4) { %% f o r ( k = i ; k < ( i < 2? i + 4 : 6 ) ; k++) Roots of a Polynomial July 11, 2015

5 %% p r i n t f ( %10.4 f, a2 [ k ] ) ; %% i f ( i < 4) p r i n t f ( \n ) ; } ++ End a l l SPOLZ(A2, N2, Z2,H, IERR) ++ CODE f o r.d. &.C. i s i n a t i v e C p r i n t 200,(( Z2(K, I ),K=1,2), I =1,5) print 200,( Z2 ( I ), I =1,5) %% p r i n t f ( \n Roots =\n ) ; %% f o r ( i = 0 ; i < 5 ; i +=2) { %% p r i n t f ( ( %8.5 f, %8.5 f ), z2 [ i ] [ 0 ], z2 [ i ] [ 1 ] ) ; %% i f ( i <4) p r i n t f ( ( %8.5 f, %8.5 f ), z2 [ i + 1 ] [ 0 ], z2 [ i + 1 ] [ 1 ] ) ; %% p r i n t f ( \n ) ; } %% p r i n t f ( \n ) ; end ODSPOLZ Degree = 3 C o e f f i i e n t s = Roots = ( , ) ( , ) ( , ) Degree = 5 C o e f f i i e n t s = Roots = ( , ) ( , ) ( , ) ( , ) ( , ) July 11, 2015 Roots of a Polynomial 7.1 5