### Related

##### linear

linear(vds,vgsvth) :=...

Calculate

? linear ;

Calculate

? linear;

Calculate

##### linear

linear(vds,vgsvth) :=...

Calculate

? linear ;

Calculate

? 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
```sphere_error:(x[i]-a)^2 + (y[i]-b)^2 + (z[i]-c)^2 - r2;
2           2           2
(%o1)              - r2 + (z  - c)  + (y  - b)  + (x  - a)
i           i           i
(%i2) sphere_error_square:sphere_error^2;
2           2           2 2
(%o2)             (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
i           i           i
(%i3) declare(sum,linear);
(%o3)                                done
(%i4) sphere_error_sum:sum(sphere_error_square, i, 1, n);
n
====
\                     2           2           2 2
(%o4)           >    (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/               i           i           i
====
i = 1
(%i5) sphere_da: diff(sphere_error_sum, a);
n
====
\                              2           2           2
(%o5)    - 4  >    (x  - a) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i6) sphere_db: diff(sphere_error_sum, b);
n
====
\                              2           2           2
(%o6)    - 4  >    (y  - b) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i7) sphere_dc: diff(sphere_error_sum, c);
n
====
\                              2           2           2
(%o7)    - 4  >    (z  - c) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i8) sphere_dr2: diff(sphere_error_sum, r2);
n                 n                 n
====              ====              ====
\             2   \             2   \             2
(%o8) - 2 (- n r2 +  >    (z  - c)  +  >    (y  - b)  +  >    (x  - a) )
/       i         /       i         /       i
====              ====              ====
i = 1             i = 1             i = 1
(%i9) sphere_da_exp: expand(sphere_da=0);
n
====
\                   2          2        3
(%o9) - 4 a n r2 + 4 ( >    x ) r2 + 4 a c  n + 4 a b  n + 4 a  n
/      i
====
i = 1
n                 n              n                   n
====              ====           ====                ====
\         2       \      2       \                   \
- 4  >    x  z  + 4 a  >    z  + 8 c  >    x  z  - 8 a c  >    z
/      i  i       /      i       /      i  i         /      i
====              ====           ====                ====
i = 1             i = 1          i = 1               i = 1
n                 n              n                   n            n
====              ====           ====                ====         ====
\         2       \      2       \                   \            \      3
- 4  >    x  y  + 4 a  >    y  + 8 b  >    x  y  - 8 a b  >    y  - 4  >    x
/      i  i       /      i       /      i  i         /      i     /      i
====              ====           ====                ====         ====
i = 1             i = 1          i = 1               i = 1        i = 1
n               n               n                n
====            ====            ====             ====
\      2      2 \             2 \              2 \
+ 12 a  >    x  - 4 c   >    x  - 4 b   >    x  - 12 a   >    x  = 0
/      i        /      i        /      i         /      i
====            ====            ====             ====
i = 1           i = 1           i = 1            i = 1
(%i10) sphere_db_exp: expand(sphere_db=0);
n
====
\                   2        3        2
(%o10) - 4 b n r2 + 4 ( >    y ) r2 + 4 b c  n + 4 b  n + 4 a  b n
/      i
====
i = 1
n                 n              n                   n            n
====              ====           ====                ====         ====
\         2       \      2       \                   \            \      3
- 4  >    y  z  + 4 b  >    z  + 8 c  >    y  z  - 8 b c  >    z  - 4  >    y
/      i  i       /      i       /      i  i         /      i     /      i
====              ====           ====                ====         ====
i = 1             i = 1          i = 1               i = 1        i = 1
n            n                 n                  n
====         ====              ====               ====
\      2     \      2          \                2 \
+ 12 b  >    y  - 4  >    x  y  + 8 a  >    x  y  - 4 c   >    y
/      i     /      i  i       /      i  i        /      i
====         ====              ====               ====
i = 1        i = 1             i = 1              i = 1
n               n              n                n
====            ====           ====             ====
2 \             2 \              \      2         \
- 12 b   >    y  - 4 a   >    y  + 4 b  >    x  - 8 a b  >    x  = 0
/      i        /      i       /      i         /      i
====            ====           ====             ====
i = 1           i = 1          i = 1            i = 1
(%i11) sphere_dc_exp: expand(sphere_dc=0);
n                                               n
====                                            ====
\                 3        2          2         \      3
(%o11) - 4 c n r2 + 4 ( >    z ) r2 + 4 c  n + 4 b  c n + 4 a  c n - 4  >    z
/      i                                        /      i
====                                            ====
i = 1                                           i = 1
n            n                 n               n
====         ====              ====            ====
\      2     \      2          \               \      2
+ 12 c  >    z  - 4  >    y  z  + 8 b  >    y  z  - 4  >    x  z
/      i     /      i  i       /      i  i     /      i  i
====         ====              ====            ====
i = 1        i = 1             i = 1           i = 1
n                   n               n               n
====                ====            ====            ====
\                 2 \             2 \             2 \
+ 8 a  >    x  z  - 12 c   >    z  - 4 b   >    z  - 4 a   >    z
/      i  i         /      i        /      i        /      i
====                ====            ====            ====
i = 1               i = 1           i = 1           i = 1
n                n              n                n
====             ====           ====             ====
\      2         \              \      2         \
+ 4 c  >    y  - 8 b c  >    y  + 4 c  >    x  - 8 a c  >    x  = 0
/      i         /      i       /      i         /      i
====             ====           ====             ====
i = 1            i = 1          i = 1            i = 1
(%i12) augcoefmatrix([sphere_da_subst, sphere_db_subst, sphere_dc_subst], [a, b, c]);
[ 0  0  0  sphere_da_subst ]
[                          ]
(%o12)                   [ 0  0  0  sphere_db_subst ]
[                          ]
[ 0  0  0  sphere_dc_subst ]
(%i13) ```
Run Example
```sphere_error:(x[i]-a)^2 + (y[i]-b)^2 + (z[i]-c)^2 - r2;
2           2           2
(%o1)              - r2 + (z  - c)  + (y  - b)  + (x  - a)
i           i           i
(%i2) sphere_error_square:sphere_error^2;
2           2           2 2
(%o2)             (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
i           i           i
(%i3) declare(sum,linear);
(%o3)                                done
(%i4) sphere_error_sum:sum(sphere_error_square, i, 1, n);
n
====
\                     2           2           2 2
(%o4)           >    (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/               i           i           i
====
i = 1
(%i5) sphere_da: diff(sphere_error_sum, a);
n
====
\                              2           2           2
(%o5)    - 4  >    (x  - a) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i6) sphere_db: diff(sphere_error_sum, b);
n
====
\                              2           2           2
(%o6)    - 4  >    (y  - b) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i7) sphere_dc: diff(sphere_error_sum, c);
n
====
\                              2           2           2
(%o7)    - 4  >    (z  - c) (- r2 + (z  - c)  + (y  - b)  + (x  - a) )
/       i                i           i           i
====
i = 1
(%i8) sphere_dr2: diff(sphere_error_sum, r2);
n                 n                 n
====              ====              ====
\             2   \             2   \             2
(%o8) - 2 (- n r2 +  >    (z  - c)  +  >    (y  - b)  +  >    (x  - a) )
/       i         /       i         /       i
====              ====              ====
i = 1             i = 1             i = 1
(%i9) sphere_da_exp: expand(sphere_da=0);
n
====
\                   2          2        3
(%o9) - 4 a n r2 + 4 ( >    x ) r2 + 4 a c  n + 4 a b  n + 4 a  n
/      i
====
i = 1
n                 n              n                   n
====              ====           ====                ====
\         2       \      2       \                   \
- 4  >    x  z  + 4 a  >    z  + 8 c  >    x  z  - 8 a c  >    z
/      i  i       /      i       /      i  i         /      i
====              ====           ====                ====
i = 1             i = 1          i = 1               i = 1
n                 n              n                   n            n
====              ====           ====                ====         ====
\         2       \      2       \                   \            \      3
- 4  >    x  y  + 4 a  >    y  + 8 b  >    x  y  - 8 a b  >    y  - 4  >    x
/      i  i       /      i       /      i  i         /      i     /      i
====              ====           ====                ====         ====
i = 1             i = 1          i = 1               i = 1        i = 1
n               n               n                n
====            ====            ====             ====
\      2      2 \             2 \              2 \
+ 12 a  >    x  - 4 c   >    x  - 4 b   >    x  - 12 a   >    x  = 0
/      i        /      i        /      i         /      i
====            ====            ====             ====
i = 1           i = 1           i = 1            i = 1
(%i10) sphere_db_exp: expand(sphere_db=0);
n
====
\                   2        3        2
(%o10) - 4 b n r2 + 4 ( >    y ) r2 + 4 b c  n + 4 b  n + 4 a  b n
/      i
====
i = 1
n                 n              n                   n            n
====              ====           ====                ====         ====
\         2       \      2       \                   \            \      3
- 4  >    y  z  + 4 b  >    z  + 8 c  >    y  z  - 8 b c  >    z  - 4  >    y
/      i  i       /      i       /      i  i         /      i     /      i
====              ====           ====                ====         ====
i = 1             i = 1          i = 1               i = 1        i = 1
n            n                 n                  n
====         ====              ====               ====
\      2     \      2          \                2 \
+ 12 b  >    y  - 4  >    x  y  + 8 a  >    x  y  - 4 c   >    y
/      i     /      i  i       /      i  i        /      i
====         ====              ====               ====
i = 1        i = 1             i = 1              i = 1
n               n              n                n
====            ====           ====             ====
2 \             2 \              \      2         \
- 12 b   >    y  - 4 a   >    y  + 4 b  >    x  - 8 a b  >    x  = 0
/      i        /      i       /      i         /      i
====            ====           ====             ====
i = 1           i = 1          i = 1            i = 1
(%i11) sphere_dc_exp: expand(sphere_dc=0);
n                                               n
====                                            ====
\                 3        2          2         \      3
(%o11) - 4 c n r2 + 4 ( >    z ) r2 + 4 c  n + 4 b  c n + 4 a  c n - 4  >    z
/      i                                        /      i
====                                            ====
i = 1                                           i = 1
n            n                 n               n
====         ====              ====            ====
\      2     \      2          \               \      2
+ 12 c  >    z  - 4  >    y  z  + 8 b  >    y  z  - 4  >    x  z
/      i     /      i  i       /      i  i     /      i  i
====         ====              ====            ====
i = 1        i = 1             i = 1           i = 1
n                   n               n               n
====                ====            ====            ====
\                 2 \             2 \             2 \
+ 8 a  >    x  z  - 12 c   >    z  - 4 b   >    z  - 4 a   >    z
/      i  i         /      i        /      i        /      i
====                ====            ====            ====
i = 1               i = 1           i = 1           i = 1
n                n              n                n
====             ====           ====             ====
\      2         \              \      2         \
+ 4 c  >    y  - 8 b c  >    y  + 4 c  >    x  - 8 a c  >    x  = 0
/      i         /      i       /      i         /      i
====             ====           ====             ====
i = 1            i = 1          i = 1            i = 1
(%i12) sphere_dr2_exp: expand(sphere_dr2=0);
n              n
====           ====
2        2        2       \      2       \
(%o12) 2 n r2 - 2 c  n - 2 b  n - 2 a  n - 2  >    z  + 4 c  >    z
/      i       /      i
====           ====
i = 1          i = 1
n              n            n              n
====           ====         ====           ====
\      2       \            \      2       \
- 2  >    y  + 4 b  >    y  - 2  >    x  + 4 a  >    x  = 0
/      i       /      i     /      i       /      i
====           ====         ====           ====
i = 1          i = 1        i = 1          i = 1
(%i13) sphere_dr2_solve: solve(sphere_dr2_exp, r2);
n              n          n
====           ====       ====
2    2    2      \      2       \          \      2
(%o13) [r2 = ((c  + b  + a ) n +  >    z  - 2 c  >    z  +  >    y
/      i       /      i   /      i
====           ====       ====
i = 1          i = 1      i = 1
n          n              n
====       ====           ====
\          \      2       \
- 2 b  >    y  +  >    x  - 2 a  >    x )/n]
/      i   /      i       /      i
====       ====           ====
i = 1      i = 1          i = 1
(%i14) sphere_da_subst: subst(sphere_dr2_solve, sphere_da_exp);
n                             n              n          n
====                          ====           ====       ====
\            2    2    2      \      2       \          \      2
(%o14) (4 ( >    x ) ((c  + b  + a ) n +  >    z  - 2 c  >    z  +  >    y
/      i                      /      i       /      i   /      i
====                          ====           ====       ====
i = 1                         i = 1          i = 1      i = 1
n          n              n
====       ====           ====
\          \      2       \
- 2 b  >    y  +  >    x  - 2 a  >    x ))/n
/      i   /      i       /      i
====       ====           ====
i = 1      i = 1          i = 1
n              n          n              n
====           ====       ====           ====
2    2    2      \      2       \          \      2       \
- 4 a ((c  + b  + a ) n +  >    z  - 2 c  >    z  +  >    y  - 2 b  >    y
/      i       /      i   /      i       /      i
====           ====       ====           ====
i = 1          i = 1      i = 1          i = 1
n              n                                            n
====           ====                                         ====
\      2       \                2          2        3       \         2
+  >    x  - 2 a  >    x ) + 4 a c  n + 4 a b  n + 4 a  n - 4  >    x  z
/      i       /      i                                     /      i  i
====           ====                                         ====
i = 1          i = 1                                        i = 1
n              n                   n            n
====           ====                ====         ====
\      2       \                   \            \         2
+ 4 a  >    z  + 8 c  >    x  z  - 8 a c  >    z  - 4  >    x  y
/      i       /      i  i         /      i     /      i  i
====           ====                ====         ====
i = 1          i = 1               i = 1        i = 1
n              n                   n            n               n
====           ====                ====         ====            ====
\      2       \                   \            \      3        \      2
+ 4 a  >    y  + 8 b  >    x  y  - 8 a b  >    y  - 4  >    x  + 12 a  >    x
/      i       /      i  i         /      i     /      i        /      i
====           ====                ====         ====            ====
i = 1          i = 1               i = 1        i = 1           i = 1
n               n                n
====            ====             ====
2 \             2 \              2 \
- 4 c   >    x  - 4 b   >    x  - 12 a   >    x  = 0
/      i        /      i         /      i
====            ====             ====
i = 1           i = 1            i = 1
(%i15) ```

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