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linear

linear(vds,vgsvth) :=...

Calculate

linear

? linear ;

Calculate

linear

linear(vds,vgsvth) :=...

Calculate

linear

? linear ;

Calculate

linear

Run Example
(%i1)remvalue(n,z);
(%o1)                           [false, false]
(%i2) matchdeclare(a,lambda([x],freeof(x,n)));
(%o2)                                done
(%i3) matchdeclare(b,lambda([x],freeof(x,n) and integerp(x) and x>
0));
(%o3)                                done
(%i4) matchdeclare(f,all);
(%o4)                                done
(%i5) matchdeclare(g,lambda([x],freeof(n-1,x)));
(%o5)                                done
(%i6) defrule(r1,zt(1,n,z),z/(z-1));
                                                 z
(%o6)                      r1 : zt(1, n, z) -> -----
                                               z - 1
(%i7) defrule(r2,zt(n,n,z),z/(z-1)^2);
                                                z
(%o7)                    r2 : zt(n, n, z) -> --------
                                                    2
                                             (z - 1)
(%i8) defrule(r3,zt(1/n,n,z),log(z/(z-1)));
                                1                 z
(%o8)                   r3 : zt(-, n, z) -> log(-----)
                                n               z - 1
(%i9) defrule(r4,zt(a^n,n,z),z/(z-a));
                                   n             z
(%o9)                     r4 : zt(a , n, z) -> -----
                                               z - a
(%i10) defrule(r5,zt(a^(n-1),n,z),1/(z-a));
                                 n - 1             1
(%o10)                  r5 : zt(a     , n, z) -> -----
                                                 z - a
(%i11) defrule(r6,zt(binomial(n,b),n,z),z/(z-1)^(b+1));
                                                       - b - 1
(%o11)         r6 : zt(binomial(n, b), n, z) -> (z - 1)        z
(%i12) defrule(r7,zt(n^b,n,z),block([tmp:z/(z-1)],  for i:1 thru b do tmp:-z*diff(tmp,z),  factor(tmp)));
                b                          z
(%o12) r7 : zt(n , n, z) -> block([tmp : -----], 
                                         z - 1
                           for i thru b do tmp : - diff(tmp, z) z, factor(tmp))
(%i13) defrule(r8,zt(sin(a*n),n,z),z*'sin(a)/((z^2-2*z*'cos(a)+1)));
                                                sin(a) z
(%o13)          r8 : zt(sin(a n), n, z) -> -------------------
                                            2
                                           z  - 2 cos(a) z + 1
(%i14) defrule(r9,zt(cos(a*n),n,z),z*(z-'cos(a))/(z^2-2*z*'cos(a)+1));
                                             z (z - cos(a))
(%o14)          r9 : zt(cos(a n), n, z) -> -------------------
                                            2
                                           z  - 2 cos(a) z + 1
(%i15) defrule(r10,zt(kron_delta(b,n),n,z),1/z^b);
                                                        1
(%o15)              r10 : zt(kron_delta(b, n), n, z) -> --
                                                         b
                                                        z
(%i16) defrule(r11,zt(a^n*f,n,z),'subst(z/a,z,ztrans(f,n,z)));
                     n                        z
(%o16)     r11 : zt(a  f, n, z) -> substitute(-, z, ztrans(f, n, z))
                                              a
(%i17) defrule(r12,zt(n^b*g,n,z),block([tmp:ztrans(g,n,z)],  for i:1 thru b do tmp:-z*'diff(tmp,z),  return(tmp)));
                   b
(%o17) r12 : zt(g n , n, z) -> block([tmp : ztrans(g, n, z)], 
                                                           dtmp
                                   for i thru b do tmp : - ---- z, return(tmp))
                                                            dz
(%i18) defrule(r13,zt(exp(-a*n),n,z),z/(z-exp(-a)));
                                - a n               z
(%o18)               r13 : zt(%e     , n, z) -> ---------
                                                      - a
                                                z - %e
(%i19) defrule(r14,zt(f(n-m),n,z),z^(-m)*ztrans(f(n),n,z));
                                            ztrans(f(n), n, z)
(%o19)          r14 : zt(f(n - m), n, z) -> ------------------
                                                     m
                                                    z
(%i20) declare(zt,linear);
(%o20)                               done
(%i21) ztrans1(e):=apply1(e,r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12,r13,r14);
(%o21) ztrans1(e) := apply1(e, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, 
                                                                 r12, r13, r14)
(%i22) ztrans(f,n,z):=ev(ztrans1(zt(f,n,z)),diff,subst);
(%o22)     ztrans(f, n, z) := ev(ztrans1(zt(f, n, z)), diff, subst)
(%i23) defrule(ir1,izt(z/(z-a),z,n),a^n);
                                     z              n
(%o23)                   ir1 : izt(-----, z, n) -> a
                                   z - a
(%i24) defrule(ir2,izt(1/(z-a),z,n),a^(n-1));
                                   1              n - 1
(%o24)                 ir2 : izt(-----, z, n) -> a
                                 z - a
(%i25) declare(izt,linear);
(%o25)                               done
(%i26) invztrans1(e):=apply1(e,ir1,ir2);
(%o26)               invztrans1(e) := apply1(e, ir1, ir2)
(%i27) invztrans(f,z,n):=invztrans1(izt(f,z,n));
(%o27)          invztrans(f, z, n) := invztrans1(izt(f, z, n))
(%i28) ztrans(1,n,z);
                                       z
(%o28)                               -----
                                     z - 1
(%i29) ztrans(a,n,z);
                                      a z
(%o29)                               -----
                                     z - 1
(%i30) expand(ztrans((-1)^n,n,z));
                                       z
(%o30)                               -----
                                     z + 1
(%i31) ztrans(n,n,z);
                                      z
(%o31)                             --------
                                          2
                                   (z - 1)
(%i32) ztrans(n^2,n,z);
                                   z (z + 1)
(%o32)                             ---------
                                          3
                                   (z - 1)
(%i33) ztrans(a^(n-1),n,z);
                                       1
(%o33)                               -----
                                     z - a
(%i34) factor(ztrans(a^(n-1)+n^2,n,z));
                         3      2      2
                      2 z  - a z  - 2 z  - a z + 3 z - 1
(%o34)                ----------------------------------
                                      3
                               (z - 1)  (z - a)
(%i35) 
Run Example
f(a,b):=sum((y[i]-a*x[i]-b)^2,i,1,n);
                                                 2
(%o1)              f(a, b) := sum((y  - a x  - b) , i, 1, n)
                                    i      i
(%i2) f1:diff(f(a,b),a)=0,expand;
                     n
                    ====
                    \                       2
(%o2)                >    (- 2 x  y  + 2 a x  + 2 b x ) = 0
                    /           i  i        i        i
                    ====
                    i = 1
(%i3) f2:diff(f(a,b),b)=0,expand;
                        n
                       ====
                       \
(%o3)                   >    (- 2 y  + 2 a x  + 2 b) = 0
                       /           i        i
                       ====
                       i = 1
(%i4) declare(sum, linear);
(%o4)                                done
(%i5) ff:solve([f1,f2],[a,b]);
              n                 n         n
             ====              ====      ====
             \                 \         \
            ( >    x  y ) n - ( >    x )  >    y
             /      i  i       /      i  /      i
             ====              ====      ====
             i = 1             i = 1     i = 1
(%o5) [[a = -------------------------------------, 
                   n              n
                  ====           ====
                  \      2       \        2
                 ( >    x ) n - ( >    x )
                  /      i       /      i
                  ====           ====
                  i = 1          i = 1
                                   n         n              n         n
                                  ====      ====           ====      ====
                                  \         \              \      2  \
                                 ( >    x )  >    x  y  - ( >    x )  >    y
                                  /      i  /      i  i    /      i  /      i
                                  ====      ====           ====      ====
                                  i = 1     i = 1          i = 1     i = 1
                           b = - --------------------------------------------]]
                                            n              n
                                           ====           ====
                                           \      2       \        2
                                          ( >    x ) n - ( >    x )
                                           /      i       /      i
                                           ====           ====
                                           i = 1          i = 1
(%i6) rhs(ff[1][2]);
                    n         n              n         n
                   ====      ====           ====      ====
                   \         \              \      2  \
                  ( >    x )  >    x  y  - ( >    x )  >    y
                   /      i  /      i  i    /      i  /      i
                   ====      ====           ====      ====
                   i = 1     i = 1          i = 1     i = 1
(%o6)           - --------------------------------------------
                             n              n
                            ====           ====
                            \      2       \        2
                           ( >    x ) n - ( >    x )
                            /      i       /      i
                            ====           ====
                            i = 1          i = 1
(%i7) 
Run Example
f(a,b):=sum((y[i]-a*x[i]-b)^2,i,1,n);
                                                 2
(%o1)              f(a, b) := sum((y  - a x  - b) , i, 1, n)
                                    i      i
(%i2) f1:diff(f(a,b),a)=0,expand;
                     n
                    ====
                    \                       2
(%o2)                >    (- 2 x  y  + 2 a x  + 2 b x ) = 0
                    /           i  i        i        i
                    ====
                    i = 1
(%i3) f2:diff(f(a,b),b)=0,expand;
                        n
                       ====
                       \
(%o3)                   >    (- 2 y  + 2 a x  + 2 b) = 0
                       /           i        i
                       ====
                       i = 1
(%i4) declare(sum, linear);
(%o4)                                done
(%i5) ff:solve([f1,f2],[a,b]);
              n                 n         n
             ====              ====      ====
             \                 \         \
            ( >    x  y ) n - ( >    x )  >    y
             /      i  i       /      i  /      i
             ====              ====      ====
             i = 1             i = 1     i = 1
(%o5) [[a = -------------------------------------, 
                   n              n
                  ====           ====
                  \      2       \        2
                 ( >    x ) n - ( >    x )
                  /      i       /      i
                  ====           ====
                  i = 1          i = 1
                                   n         n              n         n
                                  ====      ====           ====      ====
                                  \         \              \      2  \
                                 ( >    x )  >    x  y  - ( >    x )  >    y
                                  /      i  /      i  i    /      i  /      i
                                  ====      ====           ====      ====
                                  i = 1     i = 1          i = 1     i = 1
                           b = - --------------------------------------------]]
                                            n              n
                                           ====           ====
                                           \      2       \        2
                                          ( >    x ) n - ( >    x )
                                           /      i       /      i
                                           ====           ====
                                           i = 1          i = 1
(%i6) rhs(ff[1][1]);
                       n                 n         n
                      ====              ====      ====
                      \                 \         \
                     ( >    x  y ) n - ( >    x )  >    y
                      /      i  i       /      i  /      i
                      ====              ====      ====
                      i = 1             i = 1     i = 1
(%o6)                -------------------------------------
                            n              n
                           ====           ====
                           \      2       \        2
                          ( >    x ) n - ( >    x )
                           /      i       /      i
                           ====           ====
                           i = 1          i = 1
(%i7) 

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