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Recursive collocation for the numerical solution of stiff ordinary differential equations
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| Zeitschriftentitel: | Mathematics of Computation |
|---|---|
| Personen und Körperschaften: | |
| In: | Mathematics of Computation, 28, 1974, 126, S. 475-481 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
American Mathematical Society (AMS)
|
| Schlagwörter: |
| author_facet |
Brunner, H. Brunner, H. |
|---|---|
| author |
Brunner, H. |
| spellingShingle |
Brunner, H. Mathematics of Computation Recursive collocation for the numerical solution of stiff ordinary differential equations Applied Mathematics Computational Mathematics Algebra and Number Theory |
| author_sort |
brunner, h. |
| spelling |
Brunner, H. 0025-5718 1088-6842 American Mathematical Society (AMS) Applied Mathematics Computational Mathematics Algebra and Number Theory http://dx.doi.org/10.1090/s0025-5718-1974-0347089-9 <p>The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval <italic>I</italic> is approximated, on each subinterval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to a partition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>I</italic>, by a linear combination <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of exponential functions. The function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The unknown vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are computed recursively by requiring that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the given system at certain suitable points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (collocation), with the additional condition that the collection of these functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper U Subscript k Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {U_k}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> represent a continuous function on <italic>I</italic> satisfying the given initial conditions.</p> Recursive collocation for the numerical solution of stiff ordinary differential equations Mathematics of Computation |
| doi_str_mv |
10.1090/s0025-5718-1974-0347089-9 |
| facet_avail |
Online Free |
| finc_class_facet |
Mathematik |
| format |
ElectronicArticle |
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blob:ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAyNS01NzE4LTE5NzQtMDM0NzA4OS05 |
| id |
ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAyNS01NzE4LTE5NzQtMDM0NzA4OS05 |
| institution |
DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 DE-Ch1 DE-L229 DE-D275 DE-Bn3 DE-Brt1 DE-Zwi2 DE-D161 |
| imprint |
American Mathematical Society (AMS), 1974 |
| imprint_str_mv |
American Mathematical Society (AMS), 1974 |
| issn |
0025-5718 1088-6842 |
| issn_str_mv |
0025-5718 1088-6842 |
| language |
English |
| mega_collection |
American Mathematical Society (AMS) (CrossRef) |
| match_str |
brunner1974recursivecollocationforthenumericalsolutionofstiffordinarydifferentialequations |
| publishDateSort |
1974 |
| publisher |
American Mathematical Society (AMS) |
| recordtype |
ai |
| record_format |
ai |
| series |
Mathematics of Computation |
| source_id |
49 |
| title |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_unstemmed |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_full |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_fullStr |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_full_unstemmed |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_short |
Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_sort |
recursive collocation for the numerical solution of stiff ordinary differential equations |
| topic |
Applied Mathematics Computational Mathematics Algebra and Number Theory |
| url |
http://dx.doi.org/10.1090/s0025-5718-1974-0347089-9 |
| publishDate |
1974 |
| physical |
475-481 |
| description |
<p>The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval <italic>I</italic> is approximated, on each subinterval <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>σ<!-- σ --></mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> corresponding to a partition <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper N">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>π<!-- π --></mml:mi>
<mml:mi>N</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\pi _N}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of <italic>I</italic>, by a linear combination <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>U</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of exponential functions. The function <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>U</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>σ<!-- σ --></mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. The unknown vectors in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>U</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> are computed recursively by requiring that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>U</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> satisfy the given system at certain suitable points in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>σ<!-- σ --></mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> (collocation), with the additional condition that the collection of these functions <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper U Subscript k Baseline right-brace">
<mml:semantics>
<mml:mrow>
<mml:mo fence="false" stretchy="false">{</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msub>
<mml:mi>U</mml:mi>
<mml:mi>k</mml:mi>
</mml:msub>
</mml:mrow>
<mml:mo fence="false" stretchy="false">}</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\{ {U_k}\}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> represent a continuous function on <italic>I</italic> satisfying the given initial conditions.</p> |
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| description | <p>The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval <italic>I</italic> is approximated, on each subinterval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to a partition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>I</italic>, by a linear combination <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of exponential functions. The function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The unknown vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are computed recursively by requiring that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the given system at certain suitable points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (collocation), with the additional condition that the collection of these functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper U Subscript k Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {U_k}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> represent a continuous function on <italic>I</italic> satisfying the given initial conditions.</p> |
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| geogr_code | not assigned |
| geogr_code_person | not assigned |
| id | ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAyNS01NzE4LTE5NzQtMDM0NzA4OS05 |
| imprint | American Mathematical Society (AMS), 1974 |
| imprint_str_mv | American Mathematical Society (AMS), 1974 |
| institution | DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14, DE-Ch1, DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-Zwi2, DE-D161 |
| issn | 0025-5718, 1088-6842 |
| issn_str_mv | 0025-5718, 1088-6842 |
| language | English |
| last_indexed | 2024-03-01T14:19:05.319Z |
| match_str | brunner1974recursivecollocationforthenumericalsolutionofstiffordinarydifferentialequations |
| mega_collection | American Mathematical Society (AMS) (CrossRef) |
| physical | 475-481 |
| publishDate | 1974 |
| publishDateSort | 1974 |
| publisher | American Mathematical Society (AMS) |
| record_format | ai |
| recordtype | ai |
| series | Mathematics of Computation |
| source_id | 49 |
| spelling | Brunner, H. 0025-5718 1088-6842 American Mathematical Society (AMS) Applied Mathematics Computational Mathematics Algebra and Number Theory http://dx.doi.org/10.1090/s0025-5718-1974-0347089-9 <p>The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval <italic>I</italic> is approximated, on each subinterval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to a partition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\pi _N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>I</italic>, by a linear combination <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of exponential functions. The function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The unknown vectors in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are computed recursively by requiring that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript k Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_k}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the given system at certain suitable points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\sigma _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (collocation), with the additional condition that the collection of these functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper U Subscript k Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {U_k}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> represent a continuous function on <italic>I</italic> satisfying the given initial conditions.</p> Recursive collocation for the numerical solution of stiff ordinary differential equations Mathematics of Computation |
| spellingShingle | Brunner, H., Mathematics of Computation, Recursive collocation for the numerical solution of stiff ordinary differential equations, Applied Mathematics, Computational Mathematics, Algebra and Number Theory |
| title | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_full | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_fullStr | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_full_unstemmed | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_short | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_sort | recursive collocation for the numerical solution of stiff ordinary differential equations |
| title_unstemmed | Recursive collocation for the numerical solution of stiff ordinary differential equations |
| topic | Applied Mathematics, Computational Mathematics, Algebra and Number Theory |
| url | http://dx.doi.org/10.1090/s0025-5718-1974-0347089-9 |