Lanczos vectors
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1: 1.6 Vectors and Vector-Valued Functions
§1.6 Vectors and Vector-Valued Functions
►§1.6(i) Vectors
… ►Unit Vectors
… ►Cross Product (or Vector Product)
… ►§1.6(ii) Vectors: Alternative Notations
…2: 3.2 Linear Algebra
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Iterative Refinement
… ►The -norm of a vector is given by … ►§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix
… ►Define the Lanczos vectors and coefficients and by , a normalized vector (perhaps chosen randomly), , , and for by the recursive scheme … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). …3: 1.2 Elementary Algebra
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§1.2(v) Matrices, Vectors, Scalar Products, and Norms
… ►Row and Column Vectors
… ►and the corresponding transposed row vector of length is … ►Two vectors and are orthogonal if … ►Vector Norms
…4: 1.1 Special Notation
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real variables. | |
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inner, or scalar, product for real or complex vectors or functions. | |
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, | column vectors. |
the space of all -dimensional vectors. | |
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5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A complex linear vector space is called an inner product space if an inner product
is defined for all with the properties: (i) is complex linear in ; (ii) ; (iii) ; (iv) if then .
… becomes a normed linear vector space.
If then is normalized.
Two elements and in are orthogonal if .
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►The adjoint of does satisfy where .
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6: Bibliography S
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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation.
Adv. Quantum Chem. 72, pp. 95–127.
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Root-rational-fraction package for exact calculation of vector-coupling coefficients.
Comput. Phys. Comm. 21 (2), pp. 195–205.
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7: 21.1 Special Notation
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►Lowercase boldface letters or numbers are -dimensional real or complex vectors, either row or column depending on the context.
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positive integers. | |
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-dimensional vectors, with all elements in , unless stated otherwise. | |
th element of vector . | |
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scalar product of the vectors and . | |
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set of -dimensional vectors with elements in . | |
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8: 21.6 Products
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►that is, is the number of elements in the set containing all -dimensional vectors obtained by multiplying on the right by a vector with integer elements.
Two such vectors are considered equivalent if their difference is a vector with integer elements.
…where and are arbitrary -dimensional vectors.
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►Then
…Thus is a -dimensional vector whose entries are either or .
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9: 1.3 Determinants, Linear Operators, and Spectral Expansions
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Linear Operators in Finite Dimensional Vector Spaces
►Square matices can be seen as linear operators because for all and , the space of all -dimensional vectors. … ►The adjoint of a matrix is the matrix such that for all . … ►Assuming is an orthonormal basis in , any vector may be expanded as ►
1.3.20
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