x
هدف البحث
بحث في العناوين
بحث في المحتوى
بحث في اسماء الكتب
بحث في اسماء المؤلفين
اختر القسم
موافق
تاريخ الرياضيات
الاعداد و نظريتها
تاريخ التحليل
تار يخ الجبر
الهندسة و التبلوجي
الرياضيات في الحضارات المختلفة
العربية
اليونانية
البابلية
الصينية
المايا
المصرية
الهندية
الرياضيات المتقطعة
المنطق
اسس الرياضيات
فلسفة الرياضيات
مواضيع عامة في المنطق
الجبر
الجبر الخطي
الجبر المجرد
الجبر البولياني
مواضيع عامة في الجبر
الضبابية
نظرية المجموعات
نظرية الزمر
نظرية الحلقات والحقول
نظرية الاعداد
نظرية الفئات
حساب المتجهات
المتتاليات-المتسلسلات
المصفوفات و نظريتها
المثلثات
الهندسة
الهندسة المستوية
الهندسة غير المستوية
مواضيع عامة في الهندسة
التفاضل و التكامل
المعادلات التفاضلية و التكاملية
معادلات تفاضلية
معادلات تكاملية
مواضيع عامة في المعادلات
التحليل
التحليل العددي
التحليل العقدي
التحليل الدالي
مواضيع عامة في التحليل
التحليل الحقيقي
التبلوجيا
نظرية الالعاب
الاحتمالات و الاحصاء
نظرية التحكم
بحوث العمليات
نظرية الكم
الشفرات
الرياضيات التطبيقية
نظريات ومبرهنات
علماء الرياضيات
500AD
500-1499
1000to1499
1500to1599
1600to1649
1650to1699
1700to1749
1750to1779
1780to1799
1800to1819
1820to1829
1830to1839
1840to1849
1850to1859
1860to1864
1865to1869
1870to1874
1875to1879
1880to1884
1885to1889
1890to1894
1895to1899
1900to1904
1905to1909
1910to1914
1915to1919
1920to1924
1925to1929
1930to1939
1940to the present
علماء الرياضيات
الرياضيات في العلوم الاخرى
بحوث و اطاريح جامعية
هل تعلم
طرائق التدريس
الرياضيات العامة
نظرية البيان
Quasi-Minimal Residual Method
المؤلف: Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and van der Vorst, H.
المصدر: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994. http://www.netlib.org/linalg/html_templates/Templates.html.
الجزء والصفحة: ...
1-12-2021
1074
The biconjugate gradient method often displays rather irregular convergence behavior. Moreover, the implicit LU decomposition of the reduced tridiagonal system may not exist, resulting in a breakdown of the algorithm. The quasi-minimal residual method (Freund and Nachtigal 1991) is a related algorithm that attempts to overcome these problems.
The main idea behind the quasi-minimal residual (QMR) method algorithm is to solve the reduced tridiagonal system in a least squares sense, similar to the approach followed in the generalized minimal residual method (GMRES). Since the constructed basis for the Krylov subspace is biorthogonal, rather than orthogonal as in GMRES, the obtained solution is viewed as a quasi-minimal residual solution, which explains the name. Additionally, QMR uses look-ahead techniques to avoid breakdowns in the underlying Lanczos process, which makes it more robust than the biconjugate gradient method.
The convergence behavior of QMR is typically much smoother than for the biconjugate gradient method (BCG). Freund and Nachtigal (1991) present quite general error bounds which show that QMR may be expected to converge about as fast as the generalized minimal residual method. From a relation between the residuals in BCG and QMR (Freund and Nachtigal 1991, relation 5.10) one may deduce that at phases in the iteration process where BCG makes significant progress, QMR has arrived at about the same approximation for . On the other hand, when BCG makes no progress at all, QMR may still show slow convergence.
The look-ahead steps in this version of the QMR method prevent breakdown in all cases except the so-called "incurable breakdown," where no practical number of look-ahead steps would yield a next iterate.
The pseudocode for the preconditioned quasi-minimal residual method with preconditioner is given above. This algorithm follows the two term recurrence version without look-ahead (Freund and Nachtigal 1994, Algorithm 7.1). This version of QMR is simpler to implement than the full QMR method with look-ahead, but it is susceptible to breakdown of the underlying Lanczos process. (Other implementation variations are whether to scale Lanczos vectors or not, or to use three-term recurrences instead of coupled two-term recurrences. Such decisions usually have implications for the stability and the efficiency of the algorithm.)
Computation of the residual is done for the convergence test. If one uses right (or post) preconditioning, that is , then a cheap upper bound for can be computed in each iteration, avoiding the recursions for (Freund and Nachtigal 1991, Proposition 4.1). This upper bound may be pessimistic by a factor of at most .
QMR has roughly the same problems with respect to vector and parallel implementation as the biconjugate gradient method. The scalar overhead per iteration is slightly more than for BCG. In all cases where the slightly cheaper BCG method converges irregularly (but fast enough), QMR may be preferred for stability reasons.
REFERENCES:
Barrett, R.; Berry, M.; Chan, T. F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; and van der Vorst, H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. Philadelphia, PA: SIAM, 1994. http://www.netlib.org/linalg/html_templates/Templates.html.
Freund, R. and Nachtigal, N. "QMR: A Quasi-Minimal Residual Method for Non-Hermitian Linear Systems." Numer. Math. 60, 315-339, 1991.
Freund, R. and Nachtigal, N. "An Implementation of the QMR Method Based on Coupled Two-Term Recurrences." SIAM J. Sci. Statist. Comput. 15, 313-337, 1994.