| 翻訳と辞書 |
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.==Band matrix==Bandwidth (sparse matrix), matrix bandwidth, bandwidth (matrix), bandwidth (matrix theory)redirect here -->Formally, consider an ''n''×''n'' matrix ''A''=(''a'i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the '''lower''' and '''upper''' '''bandwidth''', respectively. The '''bandwidth''' of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a '''band matrix''' or '''banded matrix''' if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.
==Band matrix==
Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2:
:
then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that if .
A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.
A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.
==Applications==
In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.
Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「In mathematics, particularly matrix theory, a '''band matrix''' is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.==Band matrix==Bandwidth (sparse matrix), matrix bandwidth, bandwidth (matrix), bandwidth (matrix theory)redirect here -->Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the '''lower''' and '''upper''' '''bandwidth''', respectively. The '''bandwidth''' of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a '''band matrix''' or '''banded matrix''' if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.」の詳細全文を読む
i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.
==Band matrix==
Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2:
:
then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that if .
A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.
A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.
==Applications==
In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.
Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「In mathematics, particularly matrix theory, a '''band matrix''' is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.==Band matrix==Bandwidth (sparse matrix), matrix bandwidth, bandwidth (matrix), bandwidth (matrix theory)redirect here -->Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the '''lower''' and '''upper''' '''bandwidth''', respectively. The '''bandwidth''' of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a '''band matrix''' or '''banded matrix''' if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.」の詳細全文を読む
i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.">ウィキペディア（Wikipedia）』
■ウィキペディアで「In mathematics, particularly matrix theory, a '''band matrix''' is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.==Band matrix==Bandwidth (sparse matrix), matrix bandwidth, bandwidth (matrix), bandwidth (matrix theory)redirect here -->Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the '''lower''' and '''upper''' '''bandwidth''', respectively. The '''bandwidth''' of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a '''band matrix''' or '''banded matrix''' if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.」の詳細全文を読む
i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.">ウィキペディアで「In mathematics, particularly matrix theory, a band matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side.==Band matrix==Bandwidth (sparse matrix), matrix bandwidth, bandwidth (matrix), bandwidth (matrix theory)redirect here -->Formally, consider an ''n''×''n'' matrix ''A''=(''a'i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the '''lower''' and '''upper''' '''bandwidth''', respectively. The '''bandwidth''' of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a '''band matrix''' or '''banded matrix''' if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.」の詳細全文を読む
i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\,then the quantities ''k''1 and ''k''2 are called the lower and upper bandwidth, respectively. The bandwidth of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if |i-j| > k .A matrix is called a band matrix or banded matrix if its bandwidth is reasonably small.A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix; a band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix; when ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. If one puts ''k''1 = 0, ''k''2 = ''n''−1, one obtains the definition of an upper triangular matrix; similarly, for ''k''1 = ''n''−1, ''k''2 = 0 one obtains a lower triangular matrix.==Applications==In numerical analysis, matrices from finite element or finite difference problems are often banded. Such matrices can be viewed as descriptions of the coupling between the problem variables; the bandedness corresponds to the fact that variables are not coupled over arbitrarily large distances. Such matrices can be further divided - for instance, banded matrices exist where every element in the band is nonzero. These often arise when discretising one-dimensional problems.Problems in higher dimensions also lead to banded matrices, in which case the band itself also tends to be sparse. For instance, a partial differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension, but inside the band only 5 diagonals are nonzero. Unfortunately, applying Gaussian elimination (or equivalently an LU decomposition) to such a matrix results in the band being filled in by many non-zero elements.」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|