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series_by_leastsquare

  

construct the least squares best fit linear subspace of a linear space of series

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

series_by_leastsquare(FS, conditions)

Parameters

FS

-

FPSstruct data structure (see Slode[FPseries])

conditions

-

set of linear conditions for the coefficients of the series

Description

• 

The series_by_leastsquare command determines from the given formal power series a series whose coefficients best satisfy the given linear conditions in the least squares sense and returns the result in form of an FPSstruct data structure.

• 

This command can be used in conjunction with Slode[FPseries] or Slode[FTseries] to construct a least-squares best fit power series solution for a linear ODE with respect to a system of linear constraint equations for some coefficients of the series solution. After constructing a formal series solution FS via Slode[FTseries] or Slode[FPseries], use the series_by_leastsquare command function with the result FS and the linear system as arguments.

Examples

withSlode:

ode1x1yx+x21ⅆⅆxyx=0

ode1:=x1yx+x21ⅆⅆxyx=0

(1)

sys1v3+v84=13,v10v38+v82=64,v850=v10

sys1:=50v8=v10,v3+4v8=13,v108v3+2v8=64

(2)

fps1FPseriesode1,yx,vn:

series_by_leastsquarefps1,sys1

FPSstruct21202133924712120213392471x+2120213392471x22120213392471x3+2120213392471x42120213392471x5+2120213392471x62120213392471x7+2120213392471x82120213392471x9+2120213392471x10+n=11∞vnxn,nvn1+nvn

(3)

ode2x1yx+x21ⅆ3ⅆx3yx=0

ode2:=x1yx+x21ⅆ3ⅆx3yx=0

(4)

fps2FPseriesode2,yx,sn:

sys2s0+s1+s2=3,s4+s6=7720,2s1s4=2,73s0+720s6=80,s1s2=0

sys2:=73s0+720s6=80,s1s2=0,2s1s4=2,s4+s6=7720,s0+s1+s2=3

(5)

series_by_leastsquarefps2,sys2

FPSstruct1+x+x216x3160x5+7720x6+n=7∞snxn,n33n2+2nsn+n36n2+11n6sn1+sn3

(6)

See Also

Slode

Slode[FPseries]

Slode[FTseries]

 


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