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function [ynew,a,b,c,d] = NatSpline(x,y,xnew)
% ynew = NatSpline(x,y)
% Natural spline interpolation.
% x,y are column n-vectors.
% Assume that n >= 4 and x(1) < ... x(n).
% xnew -- is a set of points in the interval x(1) to x(n) for which
% the natural spline needs to be evaluated.
% On the i-th interval [x(i),x(i+1)] the spline has the form:
% S(z) = a(i) + b(i)(z-x(i)) + c(i)(z-x(i))^2 + d(i)(z-x(i))^2(z-x(i+1).
%
% Outputs:
% ynew -- the values of the natural spline evaluated at the points xnew.
% a,b,c,d,-- the vector of coefficients of the splines
% Usage:   
%   [ynew,a,b,c,d] = NatSpline(x,y,xnew)

      
% First, set up all but the first and last equations that
% define the vector of interior slopes s(2:n-1).

n = length(x);
Dx = diff(x);
yp = diff(y) ./ Dx;
T = zeros(n-2,n-2);
r = zeros(n-2,1);
for i=2:n-3
   T(i,i)   = 2*(Dx(i) + Dx(i+1));
   T(i,i-1) = Dx(i+1);
   T(i,i+1) = Dx(i);
   r(i)    = 3*(Dx(i+1)*yp(i) + Dx(i)*yp(i+1));
end

T(1,1) = 2*Dx(1) + 1.5*Dx(2);
T(1,2) = Dx(1);
r(1) = 1.5*Dx(2)*yp(1) + 3*Dx(1)*yp(2);
T(n-2,n-2) = 1.5*Dx(n-2)+2*Dx(n-1);
T(n-2,n-3) = Dx(n-1);
r(n-2) = 3*Dx(n-1)*yp(n-2) + 1.5*Dx(n-2)*yp(n-1);
%we now need to solve
%stilde = T\r;
A=diag(T);
B=zeros(size(A));
for i=2:n-2
    B(i)=T(i,i-1);
end
C=zeros(size(A));
for i=1:n-3
    C(i)=T(i,i+1);
end

%call the tri diagonal solver: (below)
stilde=tridiag( A, B, C, r );

s1 = (3*yp(1) - stilde(1));
sn = (3*yp(n-1) - stilde(n-2));
s = [s1;stilde;sn];


% Compute the a,b,c,d vector of coefficients for the splines.
   
a = y(1:n-1);
b = s(1:n-1);
c = (yp - s(1:n-1)) ./ Dx;
d = (s(2:n) + s(1:n-1) - 2*yp) ./ (Dx.* Dx);

%next evaluate at the points xnew to obtain the values ynew


for j=1:length(xnew)
    x_val=xnew(j);
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      • Illuminate\Cache\Events\CacheHit (173μs)
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      • Illuminate\Cache\Events\CacheHit (133μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
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      • Illuminate\Cache\Events\CacheHit (127μs)
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      • Illuminate\Cache\Events\CacheHit (143μs)
      • Illuminate\Cache\Events\CacheHit (142μs)
      • creating: site.layouts.app (386μs)
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      • creating: components.open-graph (150μs)
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      • creating: site.headers.header (224μs)
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      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (258μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (140μs)
      • creating: components.forms.contact-us (417μs)
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      • creating: components.forms.get-started (127μs)
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      • 313 x Illuminate\Cache\Events\CacheHit (14.93%)
        67.84ms
        18 x Illuminate\Database\Events\StatementPrepared (9.57%)
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        127μs
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        106μs
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        50μs
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        49μs
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      uri
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        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        1.46mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (29853)
        3.52mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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        6.06mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
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        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (47)
        1.59mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
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        1.57mstwenty4_siteHomeworkLibrary.php#260
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
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        Backtrace
        • 18. app/Models/Subject.php:100
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        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
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        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.08mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.16mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        Backtrace
        • 18. app/Models/Subject.php:100
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        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
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      App\Models\HomeworkLibrary\HomeworkLibraryFile
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