Gold Medal Software 2/Gold Medal Software Volume 2 (Gold Medal) (1994).iso/windows/win31/macsyma.arj/MACSDEMO.EXE/BLINALG.OUT

Text File | 1993-09-15 | 25KB | 425 lines

(c1) /* Demonstration of the BLINALG Package */
(showtime:true,remarray(ms,bs))$
Time= 0 msecs

(c2) /* ***********************************************
    LDU DECOMPOSITION OF SQUARE MATRICES
*/
/* 1. A symbolic matrix */
ms2:genmatrix(ms,2);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d2$))$matrix(2,2,$sub(ms,1$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()2),$sub(ms,2$ina($, )$hinge()1),$sub(ms,2$ina($, )$hinge()2))

(c3) ldu_decomp(ms2);
C:\MACGOLD5\share\blinalg.fas being loaded.
Time= 709 msecs
|$label(0,15,Times New Roman,$(d3$))$open([)$matrix(2,2,1,0,$q($sub(ms,2$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1)),1)$ina($, )$hinge()$matrix(2,2,$sub(ms,1$ina($, )$hinge()1),0,0,$sub(ms,2$ina($, )$hinge()2)$in( - )$q($sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1)))$ina($, )$hinge()$matrix(2,2,1,$q($sub(ms,1$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)),0,1)$close(])

(c4) /* 2. A matrix of rational numbers */
(mr[i,j]:=1/(abs(i-j)+1),mr3:genmatrix(mr,3));
Time= 59 msecs
|$label(0,15,Times New Roman,$(d4$))$matrix(3,3,1,$q(1,2),$q(1,3),$q(1,2),1,$q(1,2),$q(1,3),$q(1,2),1)

(c5) ldu_decomp(mr3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d5$))$open([)$matrix(3,3,1,0,0,$q(1,2),1,0,$q(1,3),$q(4,9),1)$ina($, )$hinge()$matrix(3,3,1,0,0,0,$q(3,4),0,0,0,$q(20,27))$ina($, )$hinge()$matrix(3,3,1,$q(1,2),$q(1,3),0,1,$q(4,9),0,0,1)$close(])

(c6) /* 3. A floating point matrix */
mf3:sfloat(mr3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d6$))$matrix(3,3,1.0,0.5,0.33333,0.5,1.0,0.5,0.33333,0.5,1.0)

(c7) ldu_decomp(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d7$))$open([)$matrix(3,3,1,0,0,0.5,1,0,0.33333,0.44444,1)$ina($, )$hinge()$matrix(3,3,1.0,0,0,0,0.75,0,0,0,0.74074)$ina($, )$hinge()$matrix(3,3,1,0.5,0.33333,0,1,0.44444,0,0,1)$close(])

(c8) /* Determinants and inverses */
det_by_ldu(ms2);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d8$))$sub(ms,1$ina($, )$hinge()1)$hinge()$in( )$open($()$sub(ms,2$ina($, )$hinge()2)$hinge()$in( - )$q($sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1))$close($))

(c9) ratsimp(%-newdet(ms2));
Time= 49 msecs
|$label(0,15,Times New Roman,$(d9$))0

(c10) invert_by_ldu(ms2),ratsimp;
C:\MACGOLD5\share\bla_chol.fas being loaded.
Time= 1270 msecs
|$label(0,15,Times New Roman,$(d10$))$matrix(2,2,$q($sub(ms,2$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)),$in( - )$q($sub(ms,1$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)),$in( - )$q($sub(ms,2$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)),$q($sub(ms,1$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)))

(c11) %.ms2,ratsimp;
Time= 59 msecs
|$label(0,15,Times New Roman,$(d11$))$matrix(2,2,1,0,0,1)

(c12) det_by_lduf(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d12$))0.55556

(c13) %-determinant(mf3);
C:\MACGOLD5\share\blinalgl.fas being loaded.
Time= 219 msecs
|$label(0,15,Times New Roman,$(d13$))2.38419e-7

(c14) invert_by_lduf(mf3);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d14$))$matrix(3,3,1.35,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.6,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.35)

(c15) %.mf3;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d15$))$matrix(3,3,1.0,$in( - )1.49012e-7,$in( - )1.78814e-7,$in( - )1.78814e-7,1.0,0.0,1.1921e-7,4.76838e-7,1.0)

(c16) /* ***********************************************
    LU DECOMPOSITION OF SQUARE MATRICES with partial pivoting

*/
/* 1. Symbolic matrices cannot be handled because of pivoting.

   2. A matrix of rational numbers */
lu_decomp(mr3);
C:\MACGOLD5\share\bla_lu.fas being loaded.
Time= 599 msecs
|$label(0,15,Times New Roman,$(d16$))$matrix(3,3,1,$q(1,2),$q(1,3),$q(1,2),$q(3,4),$q(1,3),$q(1,3),$q(4,9),$q(20,27))

(c17) /* 3. A floating point matrix */
lu_decompf(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d17$))$matrix(3,3,1.0,0.5,0.33333,0.5,0.75,0.33333,0.33333,0.44444,0.74074)

(c18) /* Determinants and inverses */
det_by_lu(mr3);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d18$))$q(5,9)

(c19) %-newdet(mr3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d19$))0

(c20) invert_by_lu(mr3);
Time= 59 msecs
|$label(0,15,Times New Roman,$(d20$))$matrix(3,3,$q(27,20),$in( - )$q(3,5),$in( - )$q(3,20),$in( - )$q(3,5),$q(8,5),$in( - )$q(3,5),$in( - )$q(3,20),$in( - )$q(3,5),$q(27,20))

(c21) %.mr3;
Time= 59 msecs
|$label(0,15,Times New Roman,$(d21$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c22) det_by_luf(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d22$))0.55556

(c23) %-determinant(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d23$))2.38419e-7

(c24) invert_by_luf(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d24$))$matrix(3,3,1.35,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.6,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.35)

(c25) %.mf3;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d25$))$matrix(3,3,1.0,1.1921e-7,0.0,$in( - )5.96046e-8,1.0,$in( - )2.38419e-7,0.0,4.76838e-7,1.0)

(c26) /* ***********************************************
    QR DECOMPOSITION OF SQUARE MATRICES
*/

/* 1. A symbolic matrix */
qr:qr_decomp(ms2),ratsimp;
Time= 2309 msecs
|$label(0,15,Times New Roman,$(d26$))$open([)$matrix(2,2,$q($sub(ms,1$ina($, )$hinge()1),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))),$in( - )$q($sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()2)$in( )$sup($sub(ms,2$ina($, )$hinge()1),2),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))$in( )$sqrt($sup($sub(ms,1$ina($, )$hinge()1),2)$in( )$sup($sub(ms,2$ina($, )$hinge()2),2)$in( - )2$in( )$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( + )$sup($sub(ms,1$ina($, )$hinge()2),2)$in( )$sup($sub(ms,2$ina($, )$hinge()1),2))),$q($sub(ms,2$ina($, )$hinge()1),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))),$q($sup($sub(ms,1$ina($, )$hinge()1),2)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))$in( )$sqrt($sup($sub(ms,1$ina($, )$hinge()1),2)$in( )$sup($sub(ms,2$ina($, )$hinge()2),2)$in( - )2$in( )$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( + )$sup($sub(ms,1$ina($, )$hinge()2),2)$in( )$sup($sub(ms,2$ina($, )$hinge()1),2))))$ina($, )$hinge()$matrix(2,2,$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2)),$q($sub(ms,2$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( + )$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,1$ina($, )$hinge()2),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))),0,$q($sqrt($sup($sub(ms,1$ina($, )$hinge()1),2)$in( )$sup($sub(ms,2$ina($, )$hinge()2),2)$in( - )2$in( )$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,1$ina($, )$hinge()2)$in( )$sub(ms,2$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( + )$sup($sub(ms,1$ina($, )$hinge()2),2)$in( )$sup($sub(ms,2$ina($, )$hinge()1),2)),$sqrt($sup($sub(ms,2$ina($, )$hinge()1),2)$in( + )$sup($sub(ms,1$ina($, )$hinge()1),2))))$close(])

(c27) /* Note that R is upper triangular. Check that Q is orthogonal: */
q:part(qr,1)$
Time= 0 msecs

(c28) transpose(q).q,ratsimp;
Time= 279 msecs
|$label(0,15,Times New Roman,$(d28$))$matrix(2,2,1,0,0,1)

(c29) /* Check that q.r=ms2: */
q.part(qr,2)-ms2,ratsimp;
Time= 159 msecs
|$label(0,15,Times New Roman,$(d29$))$matrix(2,2,0,0,0,0)

(c30) /* 2. A matrix of rational numbers */
qr:qr_decomp(mr3);
Time= 59 msecs
|$label(0,15,Times New Roman,$(d30$))$open([)$matrix(3,3,$q(6,7),$in( - )$q(5$in( )$sqrt(2),14),$in( - )$q(1,7$in( )$sqrt(2)),$q(3,7),$q(4$in( )$sqrt(2),7),$in( - )$q(4,7$in( )$sqrt(2)),$q(2,7),$q(3$in( )$sqrt(2),14),$q(9,7$in( )$sqrt(2)))$ina($, )$hinge()$matrix(3,3,$q(7,6),1,$q(11,14),0,$q(1,$sqrt(2)),$q(8$in( )$sqrt(2),21),0,0,$q(10$in( )$sqrt(2),21))$close(])

(c31) /* Note that R is upper triangular. Check that Q is orthogonal: */
q:part(qr,1)$
Time= 0 msecs

(c32) /* Check that q.r=mr3: */
q.part(qr,2)-mr3;
Time= 49 msecs
|$label(0,15,Times New Roman,$(d32$))$matrix(3,3,0,0,0,0,0,0,0,0,0)

(c33) transpose(q).q;
Time= 59 msecs
|$label(0,15,Times New Roman,$(d33$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c34) /* 3. A floating point matrix */
qr:qr_decomp(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d34$))$open([)$matrix(3,3,0.85714,$in( - )0.50508,$in( - )0.10102,0.42857,0.80812,$in( - )0.40406,0.28571,0.30305,0.90914)$ina($, )$hinge()$matrix(3,3,1.16667,1.0,0.78571,0.0,0.70711,0.53875,0.0,0.0,0.67343)$close(])

(c35) /* ***********************************************
    CHOLESKY FACTORIZATION OF POSITIVE DEFINITE SQUARE MATRICES
*/
/* 1. A symbolic matrix (with no check for positive definiteness) */
cholesky_l(ms2);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d35$))$matrix(2,2,$sqrt($sub(ms,1$ina($, )$hinge()1)),0,$q($sub(ms,2$ina($, )$hinge()1),$sqrt($sub(ms,1$ina($, )$hinge()1))),$sqrt($sub(ms,2$ina($, )$hinge()2)$in( - )$q($sup($sub(ms,2$ina($, )$hinge()1),2),$sub(ms,1$ina($, )$hinge()1))))

(c36) /* 2. A matrix of rational numbers */
cholesky_l(mr3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d36$))$matrix(3,3,1,0,0,$q(1,2),$q($sqrt(3),2),0,$q(1,3),$q(2,3$in( )$sqrt(3)),$q(2$in( )$sqrt(5),3$in( )$sqrt(3)))

(c37) /* 3. A floating point matrix */
cholesky_lf(mf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d37$))$matrix(3,3,1.0,0.0,0.0,0.5,0.86602,0.0,0.33333,0.3849,0.86066)

(c38) /* Inversion */
(ms2sym:copymatrix(ms2), ms2sym[2,1]:ms2[1,2],ms2sym);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d38$))$matrix(2,2,$sub(ms,1$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()2),$sub(ms,2$ina($, )$hinge()2))

(c39) invert_sym_by_chol(ms2sym),ratsimp;
Time= 109 msecs
|$label(0,15,Times New Roman,$(d39$))$matrix(2,2,$q($sub(ms,2$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sup($sub(ms,1$ina($, )$hinge()2),2)),$in( - )$q($sub(ms,1$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sup($sub(ms,1$ina($, )$hinge()2),2)),$in( - )$q($sub(ms,1$ina($, )$hinge()2),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sup($sub(ms,1$ina($, )$hinge()2),2)),$q($sub(ms,1$ina($, )$hinge()1),$sub(ms,1$ina($, )$hinge()1)$in( )$sub(ms,2$ina($, )$hinge()2)$in( - )$sup($sub(ms,1$ina($, )$hinge()2),2)))

(c40) %.ms2sym,ratsimp;
Time= 49 msecs
|$label(0,15,Times New Roman,$(d40$))$matrix(2,2,1,0,0,1)

(c41) invert_sym_by_chol(mr3);
Time= 169 msecs
|$label(0,15,Times New Roman,$(d41$))$matrix(3,3,$q(27,20),$in( - )$q(3,5),$in( - )$q(3,20),$in( - )$q(3,5),$q(8,5),$in( - )$q(3,5),$in( - )$q(3,20),$in( - )$q(3,5),$q(27,20))

(c42) %.mr3;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d42$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c43) invert_sym_by_cholf(mf3);
Time= 59 msecs
|$label(0,15,Times New Roman,$(d43$))$matrix(3,3,1.35,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.6,$in( - )0.6,$in( - )0.15,$in( - )0.6,1.35)

(c44) %.mf3;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d44$))$matrix(3,3,1.0,$in( - )2.98024e-7,$in( - )1.1921e-7,0.0,1.0,2.38419e-7,1.1921e-7,2.38419e-7,1.0)

(c45) /* ***********************************************
     INVERSION OF TOEPLITZ MATRICES
*/
load(matfuncs)$
C:\MACGOLD5\share\MATFUNCS.fas being loaded.
Time= 879 msecs

(c46) /* 1. Symbolic Toeplitz Matrices */
toeps2:toeplitz(2);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d46$))$matrix(2,2,t2,t1,t3,t2)

(c47) factor(invert_toeplitz(toeps2));
C:\MACGOLD5\share\bla_toep.fas being loaded.
Time= 1099 msecs
|$label(0,15,Times New Roman,$(d47$))$matrix(2,2,$in( - )$q(t2,t1$in( )t3$in( - )$sup(t2,2)),$q(t1,t1$in( )t3$in( - )$sup(t2,2)),$q(t3,t1$in( )t3$in( - )$sup(t2,2)),$in( - )$q(t2,t1$in( )t3$in( - )$sup(t2,2)))

(c48) factor(toeps2.%);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d48$))$matrix(2,2,1,0,0,1)

(c49) /* 2. Numerical Toeplitz Matrices */
toepn3:subst([t1=1/2,t2=1,t3=2,t4=2,t5=-1/3],toeplitz(3));
Time= 49 msecs
|$label(0,15,Times New Roman,$(d49$))$matrix(3,3,2,1,$q(1,2),2,2,1,$in( - )$q(1,3),2,2)

(c50) invert_toeplitz(toepn3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d50$))$matrix(3,3,1,$in( - )$q(1,2),0,$in( - )$q(13,6),$q(25,12),$in( - )$q(1,2),$q(7,3),$in( - )$q(13,6),1)

(c51) toepn3.%;
Time= 49 msecs
|$label(0,15,Times New Roman,$(d51$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c52) /* 3. Floating Point Toeplitz Matrices */
toepf3:sfloat(toepn3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d52$))$matrix(3,3,2.0,1.0,0.5,2.0,2.0,1.0,$in( - )0.33333,2.0,2.0)

(c53) invert_toeplitzf(toepf3);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d53$))$matrix(3,3,1.0,$in( - )0.5,0.0,$in( - )2.16667,2.08333,$in( - )0.5,2.33333,$in( - )2.16667,1.0)

(c54) toepf3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d54$))$matrix(3,3,1.0,0.0,0.0,9.53675e-7,1.0,0.0,1.90736e-6,0.0,1.0)

(c55) /* ***********************************************
     INVERSION OF SYMMETRIC TOEPLITZ MATRICES
*/
stoepsa[i,j]:=concat(t,abs(i-j));
Time= 0 msecs
|$label(0,15,Times New Roman,$(d55$))$sub(stoepsa,i$ina($, )$hinge()j)$hinge()$in( := )concat$paren(t$ina($, )$hinge()$paren(i$in( - )j,|,|))

(c56) /* 1. Symbolic Symmetric Toeplitz Matrices */
stoeps2:genmatrix(stoepsa,2);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d56$))$matrix(2,2,t0,t1,t1,t0)

(c57) factor(invert_symtoeplitz(stoeps2));
Time= 219 msecs
|$label(0,15,Times New Roman,$(d57$))$matrix(2,2,$in( - )$q(t0,$paren(t1$in( - )t0,$(,$))$in( )$paren(t1$in( + )t0,$(,$))),$q(t1,$paren(t1$in( - )t0,$(,$))$in( )$paren(t1$in( + )t0,$(,$))),$q(t1,$paren(t1$in( - )t0,$(,$))$in( )$paren(t1$in( + )t0,$(,$))),$in( - )$q(t0,$paren(t1$in( - )t0,$(,$))$in( )$paren(t1$in( + )t0,$(,$))))

(c58) factor(stoeps2.%);
Time= 109 msecs
|$label(0,15,Times New Roman,$(d58$))$matrix(2,2,1,0,0,1)

(c59) /* 2. Numerical Symmetric Toeplitz Matrices */
stoepn3:subst([t0=1,t1=1/2,t2=1/5],genmatrix(stoepsa,3));
Time= 49 msecs
|$label(0,15,Times New Roman,$(d59$))$matrix(3,3,1,$q(1,2),$q(1,5),$q(1,2),1,$q(1,2),$q(1,5),$q(1,2),1)

(c60) invert_symtoeplitz(stoepn3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d60$))$matrix(3,3,$q(75,56),$in( - )$q(5,7),$q(5,56),$in( - )$q(5,7),$q(12,7),$in( - )$q(5,7),$q(5,56),$in( - )$q(5,7),$q(75,56))

(c61) stoepn3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d61$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c62) /* 3. Floating Point Symmetric Toeplitz Matrices */
stoepf3:sfloat(stoepn3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d62$))$matrix(3,3,1.0,0.5,0.2,0.5,1.0,0.5,0.2,0.5,1.0)

(c63) invert_symtoeplitz(stoepf3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d63$))$matrix(3,3,1.33929,$in( - )0.71429,0.08929,$in( - )0.71429,1.71428,$in( - )0.71429,0.08929,$in( - )0.71429,1.33929)

(c64) stoepf3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d64$))$matrix(3,3,1.0,5.96046e-8,0.0,1.49012e-7,1.0,2.38419e-7,$in( - )5.96046e-8,0.0,1.0)

(c65) /* ***********************************************
     INVERSION OF HANKEL MATRICES
*/

/* 1. Symbolic Hankel Matrices */
(hanksa[i,j]:=concat(h,i+j-1),hanks2:genmatrix(hanksa,2));
Time= 0 msecs
|$label(0,15,Times New Roman,$(d65$))$matrix(2,2,h1,h2,h2,h3)

(c66) factor(invert_hankel(hanks2));
Time= 169 msecs
|$label(0,15,Times New Roman,$(d66$))$matrix(2,2,$q(h3,h1$in( )h3$in( - )$sup(h2,2)),$in( - )$q(h2,h1$in( )h3$in( - )$sup(h2,2)),$in( - )$q(h2,h1$in( )h3$in( - )$sup(h2,2)),$q(h1,h1$in( )h3$in( - )$sup(h2,2)))

(c67) factor(hanks2.%);
Time= 59 msecs
|$label(0,15,Times New Roman,$(d67$))$matrix(2,2,1,0,0,1)

(c68) /* 2. Numerical Hankel Matrices */
hankn3:subst([h1=1,h2=1/2,h3=1/5,h4=1/6,h5=-1/3],genmatrix(hanksa,3));
Time= 0 msecs
|$label(0,15,Times New Roman,$(d68$))$matrix(3,3,1,$q(1,2),$q(1,5),$q(1,2),$q(1,5),$q(1,6),$q(1,5),$q(1,6),$in( - )$q(1,3))

(c69) invert_hankel(hankn3)$
Time= 0 msecs

(c70) hankn3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d70$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c71) /* 3. Floating Point Hankel Matrices */
hankf3:sfloat(hankn3);
Time= 59 msecs
|$label(0,15,Times New Roman,$(d71$))$matrix(3,3,1.0,0.5,0.2,0.5,0.2,0.16667,0.2,0.16667,$in( - )0.33333)

(c72) invert_hankelf(hankf3)$
Time= 0 msecs

(c73) hankf3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d73$))$matrix(3,3,1.0,3.09944e-6,2.14577e-6,$in( - )1.43052e-6,1.0,3.57629e-6,$in( - )1.43052e-6,1.90736e-6,1.0)

(c74) /* ***********************************************
     INVERSION OF PERSYMMETRIC HANKEL MATRICES
*/
(pshanks2a[i,j]:=concat(h,abs(i+j-3)),
 pshanks3a[i,j]:=concat(h,abs(i+j-4)));
Time= 0 msecs
|$label(0,15,Times New Roman,$(d74$))$sub(pshanks3a,i$ina($, )$hinge()j)$hinge()$in( := )concat$paren(h$ina($, )$hinge()$paren(i$in( + )j$in( - )4,|,|))

(c75) /* 1. Symbolic Symmetric Hankel Matrices */
pshanks3:genmatrix(pshanks3a,3);
Time= 49 msecs
|$label(0,15,Times New Roman,$(d75$))$matrix(3,3,h2,h1,h0,h1,h0,h1,h0,h1,h2)

(c76) invert_psymhankel(pshanks3)$
Time= 49 msecs

(c77) pshanks3.%,factor;
Time= 8180 msecs
|$label(0,15,Times New Roman,$(d77$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c78) /* 2. Numerical Symmetric Hankel Matrices */
pshankn3:subst([h0=1,h1=1/2,h2=1/5],pshanks3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d78$))$matrix(3,3,$q(1,5),$q(1,2),1,$q(1,2),1,$q(1,2),1,$q(1,2),$q(1,5))

(c79) invert_psymhankel(pshankn3)$
Time= 59 msecs

(c80) pshankn3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d80$))$matrix(3,3,1,0,0,0,1,0,0,0,1)

(c81) /* 3. Floating Point Symmetric Hankel Matrices */
pshankf3:sfloat(pshankn3);
Time= 0 msecs
|$label(0,15,Times New Roman,$(d81$))$matrix(3,3,0.2,0.5,1.0,0.5,1.0,0.5,1.0,0.5,0.2)

(c82) invert_psymhankelf(pshankf3)$
Time= 59 msecs

(c83) pshankf3.%;
Time= 0 msecs
|$label(0,15,Times New Roman,$(d83$))$matrix(3,3,1.0,0.0,$in( - )5.96046e-8,2.38419e-7,1.0,1.49012e-7,0.0,5.96046e-8,1.0)

(c84) /* ***********************************************
    INVERSION OF HILBERT MATRICES
*/
(hilberta[i,j]:=1/(i+j-1),hilbert10:genmatrix(hilberta,10));
Time= 219 msecs
|$label(0,15,Times New Roman,$(d84$))$matrix(10,10,1,$q(1,2),$q(1,3),$q(1,4),$q(1,5),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,2),$q(1,3),$q(1,4),$q(1,5),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,3),$q(1,4),$q(1,5),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,4),$q(1,5),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,5),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,6),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,15),$q(1,7),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,15),$q(1,16),$q(1,8),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,15),$q(1,16),$q(1,17),$q(1,9),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,15),$q(1,16),$q(1,17),$q(1,18),$q(1,10),$q(1,11),$q(1,12),$q(1,13),$q(1,14),$q(1,15),$q(1,16),$q(1,17),$q(1,18),$q(1,19))

(c85) /* 1. Gauss Elimination */
invert(hilbert10);
Time= 1210 msecs
|$label(0,15,Times New Roman,$(d85$))$matrix(10,10,100,$in( - )4950,79200,$in( - )600600,2522520,$in( - )6306300,9609600,$in( - )8751600,4375800,$in( - )923780,$in( - )4950,326700,$in( - )5880600,47567520,$in( - )208107900,535134600,$in( - )$num(832431600),$num(770140800),$in( - )389883780,83140200,79200,$in( - )5880600,112907520,$in( - )$num(951350400),$num(4281076800),$in( - )$num(11237826600),$num(17758540800),$in( - )$num(16635041280),$num(8506555200),$in( - )$num(1829084400),$in( - )600600,47567520,$in( - )$num(951350400),$num(8245036800),$in( - )$num(37875637800),$num(101001700800),$in( - )$num(161602721280),$num(152907955200),$in( - )$num(78843164400),$num(17071454400),2522520,$in( - )208107900,$num(4281076800),$in( - )$num(37875637800),$num(176752976400),$in( - )$num(477233036280),$num(771285715200),$in( - )$num(735869534400),$num(382086104400),$in( - )$num(83223340200),$in( - )6306300,535134600,$in( - )$num(11237826600),$num(101001700800),$in( - )$num(477233036280),$num(1301544644400),$in( - )$num(2121035716800),$num(2037792556800),$in( - )$num(1064382719400),$num(233025352560),9609600,$in( - )$num(832431600),$num(17758540800),$in( - )$num(161602721280),$num(771285715200),$in( - )$num(2121035716800),$num(3480673996800),$in( - )$num(3363975014400),$num(1766086882560),$in( - )$num(388375587600),$in( - )8751600,$num(770140800),$in( - )$num(16635041280),$num(152907955200),$in( - )$num(735869534400),$num(2037792556800),$in( - )$num(3363975014400),$num(3267861442560),$in( - )$num(1723286307600),$num(380449555200),4375800,$in( - )389883780,$num(8506555200),$in( - )$num(78843164400),$num(382086104400),$in( - )$num(1064382719400),$num(1766086882560),$in( - )$num(1723286307600),$num(912328045200),$in( - )$num(202113826200),$in( - )923780,83140200,$in( - )$num(1829084400),$num(17071454400),$in( - )$num(83223340200),$num(233025352560),$in( - )$num(388375587600),$num(380449555200),$in( - )$num(202113826200),$num(44914183600))

(c86) hilbert10.%;
Time= 760 msecs
|$label(0,15,Times New Roman,$(d86$))$matrix(10,10,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1)

(c87) /* 2. Invert by Hankel Method (the fastest method) */
invert_hankel(hilbert10);
Time= 269 msecs
|$label(0,15,Times New Roman,$(d87$))$matrix(10,10,100,$in( - )4950,79200,$in( - )600600,2522520,$in( - )6306300,9609600,$in( - )8751600,4375800,$in( - )923780,$in( - )4950,326700,$in( - )5880600,47567520,$in( - )208107900,535134600,$in( - )$num(832431600),$num(770140800),$in( - )389883780,83140200,79200,$in( - )5880600,112907520,$in( - )$num(951350400),$num(4281076800),$in( - )$num(11237826600),$num(17758540800),$in( - )$num(16635041280),$num(8506555200),$in( - )$num(1829084400),$in( - )600600,47567520,$in( - )$num(951350400),$num(8245036800),$in( - )$num(37875637800),$num(101001700800),$in( - )$num(161602721280),$num(152907955200),$in( - )$num(78843164400),$num(17071454400),2522520,$in( - )208107900,$num(4281076800),$in( - )$num(37875637800),$num(176752976400),$in( - )$num(477233036280),$num(771285715200),$in( - )$num(735869534400),$num(382086104400),$in( - )$num(83223340200),$in( - )6306300,535134600,$in( - )$num(11237826600),$num(101001700800),$in( - )$num(477233036280),$num(1301544644400),$in( - )$num(2121035716800),$num(2037792556800),$in( - )$num(1064382719400),$num(233025352560),9609600,$in( - )$num(832431600),$num(17758540800),$in( - )$num(161602721280),$num(771285715200),$in( - )$num(2121035716800),$num(3480673996800),$in( - )$num(3363975014400),$num(1766086882560),$in( - )$num(388375587600),$in( - )8751600,$num(770140800),$in( - )$num(16635041280),$num(152907955200),$in( - )$num(735869534400),$num(2037792556800),$in( - )$num(3363975014400),$num(3267861442560),$in( - )$num(1723286307600),$num(380449555200),4375800,$in( - )389883780,$num(8506555200),$in( - )$num(78843164400),$num(382086104400),$in( - )$num(1064382719400),$num(1766086882560),$in( - )$num(1723286307600),$num(912328045200),$in( - )$num(202113826200),$in( - )923780,83140200,$in( - )$num(1829084400),$num(17071454400),$in( - )$num(83223340200),$num(233025352560),$in( - )$num(388375587600),$num(380449555200),$in( - )$num(202113826200),$num(44914183600))

(c88) %.hilbert10;
Time= 770 msecs
|$label(0,15,Times New Roman,$(d88$))$matrix(10,10,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1)

(c89) /* Clean up */
(remarray(mr,luindex,stoepsa,hanksa,pshanks2a,pshanks3a,hilberta),
 remvalue(ms2,ms2sym,mr3,mf3,q,qr,r,toeps2,toepn3,toepf3,stoeps2,stoepn3,stoepf3),
 remvalue(hanks2,hankn3,hankf3,pshanks3,pshankn3,pshankf3,hilbert10),
 reset(showtime))$

(c90) /* Clean up */
(remarray(n),remvalue(mm,nn,ff))$