<p> tridiag(a,d,b,r) ... Stem(data) </p>

37db2745-3ae9-496e-81ac-dcc021b7bccc 21 3 5 decimal

&[DATE] &[TIME] - &[FILENAME]

1. Solving a tridiagonal matrix

Create a user tridiagonal matrix [adapted Valery Ochov] a b c n Tridiagonal i 1 n range j 1 n range F i j el a i j ≡ b i j 1 + ≡ c i j 1 - ≡ 0 cases : for for F 21 line :

t0 1 time :

The tridiagonal 'vminor' a b c n Axx U a b c n Tridiagonal : x 1 n range ω x el U x x alg : idx x el x : 21 line for idx ω augment 31 line :

a 1 : b 2 : c 3 : n 8 : 14 mat Ω a b c n Axx : 21 sys

<= vminor # # # alg

U a b c n Tridiagonal :

ω Ω 2 col :

i 1 n range x i el i : u i el x i el sqrt : 21 line for

sol U 1 -^u * : 9.288 2.763 - 4.8 - 4.019 2.527 2.776 - 0.057 2.714 81 mat

U 130000002130000002130000002130000002130000002130000002130000002188 mat

2. Solving a tridiagonal system of vectors

coded from "Numerical Recipes", Steven Finch a b c u tridiag n b rows : β 1 el b 1 el : T 1 el u 1 el : j 2 n range β j el b j el a j el β j 1 - el / c j 1 - el * - : T j el u j el a j el β j el / T j 1 - el * - : 21 line for u n el T n el β n el / : j n 1 - (1 n 2 - range u j el T j el c j el u j 1 + el * - β j el / : for u 71 line :

a b c u tridiag

a 0.1 - 0.1 - 0.1 - 0.1 - 0.1 - 0.1 - 61 mat :

b 11111161 mat :

c 0.1 0.1 0.1 0.1 0.1 0.1 61 mat :

u 0.6 1.1 0.8 01 1.5 61 mat :

Sol_Tridiag a b c u tridiag :

Sol_Tridiag 0.4942 1.0583 0.9052 0.0063 0.8422 1.5842 61 mat

3. Solving advection

<= Euler PDE @ Advection.sm

<p> tridiag(a,d,b,r) ... Stem(data) </p>

Solve a user tridiagonal vector system [adapted]. 
If you don't need more matrix manipulation, you
can solve the system directly. [Mathsoft Staff] a b c u tridiag n b rows : β 1 el b 1 el : T 1 el u 1 el : j 2 n range β j el b j el a j el β j 1 - el / c j 1 - el * - : T j el u j el a j el β j el / T j 1 - el * - : 21 line for u n el T n el β n el / : j n 1 - (1 n 2 - range u j el T j el c j el u j 1 + el * - β j el / : for u 71 line :

Spectrum stem plot for QuickPlot unicolor data Stem stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

ν 1 :

length of Advection p i 1100 range p i el i : for p 21 line :

i 1100 range j 1 a i j el ν - 2 / : a 11 el 0 : 21 line for for

i 1100 range j 1 d i j el 1 : for for

i 1100 range j 1 b i j el ν 2 / : b 1001 el 0 : 21 line for for

i 1100 range j 1 r i j el 100 i 5 ≤ 0 cases : for for

init p a d b r tridiag augment :

a 0 0.5 - 0.5 - 0.5 - 0.5 - 0.5 - 0.5 - 71 mat

d 111111171 mat

b 0.5 0.5 0.5 0.5 0.5 0.5 0.5 71 mat

r 63.085 73.83 95.425 87.647 119.22 49.37 20.451 71 mat

init Stem

number of collected solutions K 200 :

r soln s r : count 1 r count el r : count count 1 + : 21 line for count 2 K range r count el a d b s tridiag : s r count el : count count 1 + : 31 line for r 41 line :

CWD C:\SmathFile CurrentDirectory :

r soln CWD Advection wfile 1

A CWD Advection rfile :

data_1 p A 50 el augment : data_2 p A 125 el augment : 21 line

where Advection reflects return 1003010030 - 22 mat :

t anim p A t el augment : 11 line

τ 120010 range :

1 time t0 - min 1.8

data_1 Stem data_2 Stem return 3 1 sys t anim Stem return 2 1 sys

4. The L_P_C splines tridiagonal systems

Essentially, the coefficients of the cubic spline consists in solving the basis tridiagonal matrix wrt the 'br' vector of derivatives . Independently of the type of cubic spline [type: 0, 1, 2 = l, p, c] two steps of preparation are needed: 1. Install the constant vector 'br' 2. Install the blank tridiagonal matrix.

XY 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

X XY 1 col :

Y XY 2 col :

N Y rows 1 - :

N 5

r 2 N range :

Prep.0 Install the constant vector 'br'

i r br i el Y i 1 + el Y i el - X i 1 + el X i el - / (Y i el Y i 1 - el - X i el X i 1 - el - / (- : br N 1 + el 0 : 21 line for

br 0 0.6782 1.44 - 0.74 0.052 - 061 mat

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

M 10000000000000000000000000000000000166 mat

Observe the population of the rows upper left row ---------------------- ----------------------lower right row respectively to linear/parabolic/cubic. The basis tridiagonal is the "linear", further modified to suit user l_p_c .

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0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

s M 1 -^br * :

M 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat

s 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

==============================================================

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

s M 1 -^br * :

M 11 - 0000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00001 - 166 mat

s 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

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2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

s M 1 -^br * :

M 1 1.52 - 0.52 000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00012 - 166 mat

s 7.1661 1.0623 2.8867 - 1.8447 0.052 - 1.9487 - 61 mat