NUMERICAL ANALYSIS: LINEAR ALGEBRA
CONTENTS SOLUTION OF AX = D MATRIX MODELS COLUMN VECTORS AND RN MATRICES APPLICATION TO VISUALIZATION OF MINIMUM COST APPLICATION TO TWO-BAR TRUSS APPLICATION TO TWO-LOOP CIRCUIT EXERCISES MATRIX PRODUCTS MATRIX-VECTOR PRODUCTS MATRIX-MATRIX PRODUCTS APPLICATION TO HEAT CONDUCTION MATRIX COMPUTATIONS USING MATLAB R ? SPECIAL CASES OF AX = D FIVE POSSIBLE CLASSES OF “SOLUTIONS” TRIANGULAR MATRICES APPLICATION TO HEAT IN WIRE WITH CURRENT ROW OPERATIONS AND GAUSS ELIMINATION INTRODUCTORY ILLUSTRATION THREE TYPES OF ROW OPERATIONS GAUSS ELIMINATION FOR SOLVING AX = D APPLICATION TO SIX-BAR TRUSS GAUSS ELIMINATION USING MATLAB R ? INVERSE MATRICES EXAMPLES OF INVERSE MATRICES GAUSS–JORDAN METHOD TO ?ND INVERSE MATRICES PROPERTIES OF INVERSE MATRICES INVERSE MATRICES AND MATLAB R ? DETERMINANTS AND CRAMER’S RULE DETERMINANTS FOR 2 × 2 AND 3 × 3 MATRICES DETERMINANT OF AN N × N MATRIX CRAMER’S RULE AND INVERSES DETERMINANTS USING MATLAB R ? MATRIX FACTORIZATIONS THE SCHUR COMPLEMENT HEAT DI?USION IN ?N WITH TWO DIRECTIONS PA = LU AND A NONSINGULAR A = LU, A?1 ?0 AND M-MATRIX A = GGT AND A SPD SPD AND MINIMIZATION LEAST SQUARES AND NORMAL EQUATIONS NORMAL EQUATIONS MATLAB R ?CODE PRICE EXPDATAM BASIS OF SUBSPACE PROJECTION TO SUBSPACE AX = D WITH M < N ROW ECHELON FORM SOLUTIONS IN R4 GENERAL SOLUTION OF AX = D RELATIONSHIP OF R(A), N(AT ) AND R(AT ), N(A) CONSTRUCTION OF BASES NULL SPACE METHOD FOR EQUILIBRIUM EQUATIONS BLOCK GAUSS ELIMINATION METHOD APPLICATION TO THREE-LOOP CIRCUIT APPLICATION TO ?UID ?OW ORTHOGONAL SUBSPACES AND BASES ORTHOGONAL SUBSPACE FUNDAMENTAL THEOREM: RN = N(A) ?R(AT ) A = QR FACTORIZATION MATLAB R ?CODE QR COLM ORTHONORMAL BASIS FOUR METHODS FOR QR FACTORS CLASSICAL GRAM–SCHMIDT GIVENS TRANSFORM HOUSEHOLDER TRANSFORM EIGENVECTORS AND ORTHONORMAL BASIS EIGENVECTORS OF SYMMETRIC MATRIX APPROXIMATION OF EIGENVALUES GERSCHGORIN CIRCLES POWER ITERATIONS QR ITERATION SPECTRAL THEOREM FACTORS AQ = QD APPLICATIONS NONSINGULAR AX = D SINGULAR VALUE DECOMPOSITION “SMALL” SVD “FULL” SVD MATLAB R ?CODE SVD EXM “TRUNCATED” SVD THREE APPLICATIONS OF SVD IMAGE COMPRESSION MATLAB R ?CODE SVDIMAGEM SEARCH ENGINES MATLAB R ?CODES SENGINEM, SENGINESPARSEM NOISE FILTER MATLAB R ?CODE IMAGE1DSVDM PSEUDOINVERSE OF A ?† AND A† = V ?†U T A† AND LEAST SQUARES ILL-CONDITIONED LEAST SQUARES APPLICATION TO HAZARD IDENTI?CATION MATLAB R ?CODE HAZIDSVD1M 10 GENERAL INNER PRODUCT VECTOR SPACES 101 VECTOR SPACES 1011 EXERCISES 102 INNER PRODUCTS AND ORTHOGONAL VECTORS 1021 GENERAL INNER PRODUCTS 1022 ORTHONORMAL VECTORS 1023 NORMS ON VECTOR SPACES 1024 EXERCISES 103 SCHUR DECOMPOSITION 1031 NORMS AND SPECTRAL RADIUS 1032 NORMAL MATRICES 1033 CAYLEY–HAMILTON THEOREM 1034 EXERCISES 104 SELF-ADJOINT DI?ERENTIAL OPERATORS 1041 LINEAR OPERATORS 1042 STURM-LIOUVILLE PROBLEM 1043 EXERCISES 105 SELF-ADJOINT POSITIVE DE?NITE BVP 1051 EXERCISES 11 ITERATIVE METHODS 111 INVERSE MATRIX APPROXIMATIONS 1111 EXERCISES 112 REGULAR SPLITTINGS FOR M-MATRICES 1121 EXERCISES 113 P-REGULAR SPLITTINGS FOR SPD MATRICES 1131 SOR FOR DI?USION IN 3D CONTENTS 1132 MATLAB R ?IMPLEMENTATION OF SOR 1133 EXERCISES 114 CONJUGATE GRADIENT FOR SPD MATRICES 1141 MATLAB R ?IMPLEMENTATIONS OF CG 1142 EXERCISES 115 GENERALIZED MINIMUM RESIDUAL 1151 MATLAB R ?IMPLEMENTATIONS OF GMRES 1152 EXERCISES 12 NONLINEAR PROBLEMS AND LEAST SQUARES 121 PICARD APPROXIMATION 1211 MATLAB R ?CODE PICCOOLM 122 NEWTON METHOD 1221 MATLAB R ?CODE NEWTCOOLM 123 LEVENBERG-MARQUARDT METHOD 1231 MATLAB R ?CODE LEVMARQPRICEM 124 SIRD EPIDEMIC MODELS 1241 MATLAB R ?CODE SIRD PARIDM 125 THE CUMULATED INFECTION VERSION OF SIRD 1251 US COVID-19: AN AGGREGATED MODEL