CN113255253B - Matrix fast decomposition method based on resistance-capacitance network - Google Patents
Matrix fast decomposition method based on resistance-capacitance network Download PDFInfo
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- G06F30/30—Circuit design
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- G06F30/398—Design verification or optimisation, e.g. using design rule check [DRC], layout versus schematics [LVS] or finite element methods [FEM]
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Abstract
A matrix fast decomposition method based on a resistance-capacitance network comprises the following steps: reordering the formed symmetric invertible matrix; carrying out symbol decomposition on the reordered symmetrical reversible matrix according to columns, establishing the dependency relationship of the matrix, carrying out matrix column pruning according to the dependency relationship, and determining the positions of filling elements in the matrix; and carrying out numerical decomposition on the reordered symmetrical reversible matrix by utilizing a Cholesky formula and a symbol decomposition result. The matrix rapid decomposition method based on the resistance-capacitance network can improve the circuit simulation performance and accelerate the overall simulation speed.
Description
Technical Field
The invention relates to the technical field of Integrated Circuit Computer Aided Design (Integrated Circuit/Computer Aided Design), in particular to LL for quickly solving sparse matrix in Circuit simulationtAnd (3) a decomposition method.
Background
The fundamental problem of transient simulation of an integrated circuit is that a large-scale nonlinear equation set is solved at each time point, and a Newton iteration method is needed for solving the nonlinear equation set. And each step of the Newton iteration method is to solve the linear equation set Ax ═ b. The linear equation system can be solved by a direct method (trigonometric decomposition and back substitution solution) and an iterative method.
Typically, a direct method of linear equations will use a LU decomposition, i.e., a decomposition of A into the product of a lower triangular matrix L (with a diagonal of 1) and an upper triangular matrix U. For solving a direct solution of a sparse linear equation set with high complexity and long time consumption, how to efficiently carry out triangular decomposition on a sparse matrix and carry out back substitution solution becomes a very important research direction.
For a linear circuit, or linear portion of a circuit, a is a symmetric matrix generated by passive devices such as a constant resistance and a constant capacitance, and by this property, a can be decomposed into the product of a lower triangular matrix L and its transpose. Intuitively, due to the symmetry of A, in LLtThe amount of calculation in decomposition is reduced to half of LU. For linear resistor-capacitor networks, this problem translates into how to perform fast cholesky decomposition.
The sparse linear equation system solution occupies the maximum proportion of the total time of the circuit simulation, and becomes the performance bottleneck of the circuit simulation.
Disclosure of Invention
In order to solve the defects in the prior art, the invention aims to provide a matrix fast decomposition method based on a resistance-capacitance network, which can improve the circuit simulation performance and accelerate the overall simulation speed.
In order to achieve the above object, the invention provides a method for fast matrix decomposition based on a resistor-capacitor network, comprising the following steps:
reordering the formed symmetric invertible matrix;
performing symbol decomposition on the reordered symmetrical reversible matrix according to columns, establishing the dependency relationship of the matrix, performing matrix column pruning according to the dependency relationship, and determining the positions of filling elements in the matrix;
and carrying out numerical decomposition on the reordered symmetrical reversible matrix by utilizing a Cholesky formula and a symbol decomposition result.
Further, the step of performing symbol decomposition on the reordered symmetrical reversible matrix by columns to establish a matrix dependency relationship further comprises,
performing symbolic decomposition on the reordered symmetrical reversible matrix from left to right according to columns;
constructing a matrix dependency relationship tree in the symbol decomposition process, and taking the row number of the decomposed column and the next position adjacent to the diagonal element as the father node of the matrix dependency relationship tree;
taking the transposed column of the L matrix as a descendant of the transposed column;
splitting columns according to sub-column computation in the dependency, skipping columns that depend on the conduction.
Further, the step of pruning the matrix columns according to the dependency relationship and determining the positions of the filling elements in the matrix further comprises,
and maintaining a position pointer in each row in the matrix, reading the part from the row position pointer to the tail of the row, and pruning in the row and the column.
Further, the air conditioner is provided with a fan,
the step of performing numerical decomposition on the reordered symmetrical invertible matrix by using a Cholesky formula and a symbolic decomposition result further comprises,
and according to the non-zero position of the matrix determined by the symbolic decomposition, performing numerical decomposition on the lower left corner of the reordered symmetrical reversible matrix from left to right according to columns.
Further, the method also comprises the following steps of,
and calculating the decomposition result of the decomposed column by utilizing a Cholesky formula according to the descendant column in the dependency relationship.
Furthermore, the method also comprises the following steps of,
in the symbol decomposition process and the numerical decomposition process, a start position of an in-column pointer when reading the column next time is recorded for each column of the matrix.
In order to achieve the above object, the present invention further provides an electronic device, which includes a memory and a processor, where the memory stores a computer program running on the processor, and the processor executes the computer program to perform the steps of the above method for fast matrix decomposition based on a resistor-capacitor network.
To achieve the above object, the present invention further provides a computer-readable storage medium, on which a computer program is stored, which when running executes the steps of the resistor-capacitor network-based matrix fast decomposition method as described above.
The matrix rapid decomposition method based on the resistance-capacitance network, the electronic equipment and the computer readable storage medium have the following beneficial effects:
1) for symmetric matrices, only L is calculated, reducing the amount of computation and memory consumption compared to LU.
2) And the pruning of the symbol decomposition is carried out according to the dependency relationship, so that the time complexity is reduced.
3) According to the symmetry of the Cholesky decomposition, the symbol decomposition is correspondingly read and pruned in the column, so that the time complexity is reduced.
4) And the symbol decomposition and the numerical decomposition are separated, so that the memory access efficiency is improved.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a method for fast decomposition of a matrix based on a resistor-capacitor network according to the present invention;
FIG. 2 is a matrix ordering decomposition diagram according to an embodiment of the invention;
FIG. 3 is a matrix reordering decomposition diagram according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a process of sign decomposition looking to the left for a 6-dimensional sparse matrix according to an embodiment of the present invention;
FIG. 5 is a diagram of a dependency tree constructed on a symbolic decomposition edge according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of a left-looking numerical decomposition process for a 6-dimensional matrix according to an embodiment of the invention;
FIG. 7 is a diagram illustrating a sign decomposition and a value decomposition according to an embodiment of the invention.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it will be understood that they are described herein for the purpose of illustration and explanation and not limitation.
Example 1
Fig. 1 is a flowchart of a method for fast decomposing a matrix based on a resistor-capacitor network according to the present invention, and the method for fast decomposing a matrix based on a resistor-capacitor network according to the present invention will be described in detail with reference to fig. 1.
First, in step 101, the formed symmetric invertible matrices are reordered.
In the embodiment of the invention, the formed symmetrical reversible matrix is reordered by adopting a minimum method or a nested splitting method and the like, and the matrix A is multiplied by a permutation matrix on the left and the right respectively to become the permutation matrix
The goal is to reduce the decomposed padding elements.
In step 102, the sorted matrix is subjected to symbolic decomposition from left to right according to columns, i.e. the positions of the padding elements are determined, a dependency relationship is established while decomposing, and pruning is performed by using the obtained dependency relationship, i.e. columns dependent on conduction are skipped.
Preferably, the sign decomposition is performed independently, the order is that the sign decomposition is performed from left to right according to the columns, the dependency tree is constructed by performing the sign decomposition while the parent node is labeled more than the child node, and the line number of the k-th column after the decomposition and the next position adjacent to the diagonal element is taken as the parent of k.
Preferably, the kth row of L, i.e., the kth column of the L transpose, is recorded, which is the descendant of k, i.e., the descendant on which k depends. The purpose is as follows: the calculation is performed only on the basis of the dependency relationship without considering each row 1 to k-1.
In the embodiment of the invention, according to the constructed dependency relationship tree, only the influence of the son columns in the dependency relationship on the decomposed column is considered in the symbol decomposition process, and all descendants do not need to be considered. The reason for this is the dependence transitivity.
In step 103, the reordered symmetric invertible matrix obtained in step 101 is numerically decomposed from left to right directly using the cholesky formula and the symbolic decomposition result, and only the lower left corner is made because of the symmetric matrix, i.e. only L is calculated.
Preferably, numerical decomposition is performed independently, and the sequence is from left to right in a row, so that the final non-zero position of the L is determined during symbol decomposition, memory allocation and the like are not needed at the moment, and the access and storage are friendly.
In the embodiment of the invention, the numerical value decomposition process uses the descendant columns in the dependency relationship and utilizes a Cholesky formula to calculate the decomposition result of the decomposed columns.
In the embodiment of the present invention, in the aspect of pruning, pruning can be performed in step 102 except for using the dependency relationship, that is, skipping the whole column reading; it is also possible to maintain a read pointer for each column in steps 102 and 103, and perform intra-column pruning when reading, i.e. only reading the lower half of a column.
In the embodiment of the invention, in the symbol decomposition process and the numerical decomposition process, an in-column pointer is recorded for each column so as to record the starting position of reading the column next time, thereby facilitating in-column pruning. Thus, the first k-1 column is used to decompose the k-th column, and because the symmetry of the Cholesky decomposition, the first k-1 column is also known as the transpose of the first k-1 column when decomposing the k-th column, so only the k rows and the following portions in the k column need to be updated with the k rows and the following portions in the first k-1 column.
In the embodiment of the invention, after a given reordering method is used, fast cholesky decomposition is performed on a symmetric positive definite nonsingular matrix filled based on a linear circuit network, as shown in fig. 2 and fig. 3, the influence of matrix ordering on the decomposition is shown, wherein (1) is an original matrix, (2) is a matrix after the decomposition, represents a nonzero element, and a null represents a sparse position, namely 0, without consuming a storage unit, represents filling elements added in the decomposition, so that the filling influence of different ordering methods on the matrix decomposition can be seen to be large, the sparse matrix after the decomposition is changed into a full matrix in fig. 2, and the sparse degree and the matrix structure are not changed after the decomposition in fig. 3.
Example 2
The fast matrix decomposition method based on the resistor-capacitor network of the present invention is further described below with reference to a specific embodiment.
FIG. 4 is a schematic diagram illustrating a symbol decomposition process of a 6-dimensional sparse matrix looking to the left according to an embodiment of the present invention, as shown in FIG. 4, where a represents a non-zero element and a null represents a sparse location, i.e., 0, without consuming memory cells. Due to the symmetric matrix, only the lower triangular part is shown, wherein 4(1) is the original sparse matrix, and the cases from 4(2) to 4(7) are respectively the cases of decomposing a to e columns, the columns being decomposed are displayed by bold capital letters, the reading pointer of each column is displayed by a character box, and the reading pointers of some columns stop in the middle of the column when the decomposition is completed and the graph in fig. 4(7) is reached because the symbol decomposition performs the whole-column pruning according to the dependency relationship.
The original sparse matrix of fig. 4(1) is first reordered to reduce the decomposed padding elements, assuming that after reordering, the same as in fig. 4(1) is applied.
And secondly, performing symbol decomposition from left to right according to columns, namely determining the position of the filling element. As shown in fig. 4(2) to 4 (7). One position pointer is maintained for each column and the character box is displayed because due to the symmetry of the cholesky decomposition, only the lower triangular portion needs to be read, i.e., only the portion of the column position pointer to the end of the column, which is intra-column pruning. And meanwhile, constructing a dependency relation tree, and only reading son columns in the offspring, such as skipping the son of the son, so as to carry out whole-column pruning.
Specifically, as shown in fig. 4(2), a is the first column, and the pointer in the column moves down one position without decomposition; FIG. 4(3) shows that b has no dependency on a, so no decomposition is required and the pointer in the column is moved down by one position; as shown in fig. 4(4), c depends on a and b, so that a and b are merged with the original non-zero position of c from the pointer in the column to the non-zero position in the tail of the column, as shown by adding two padding elements, the pointer in the column of a, b, c moves down to the first padding element; FIG. 4(5) shows that d has no dependency on all columns on the left, so no decomposition is required and the pointers in the columns are moved down by one position; as shown in fig. 4(6), e depends on a and c, but c depends on a, because the dependency of c on a is already added to c column when c column is processed in fig. 4(4), only c column is merged from the pointer in column to the non-zero position in the tail of column, with the original non-zero position of e column, a non-zero element is added, and the pointer in column of c and e is moved down by one position; fig. 4(7) shows that f depends on b, c, d, e, and similarly to 3(6), only the influence of d and e on f is considered due to the transitivity of the dependency, and pointers in d and e columns move down.
Then, the numerical decomposition is carried out from left to right according to the column, namely, the Gerrisby formula is utilized
And
the value of the corresponding non-zero position in the k-th column is calculated from the first k-1 columns of L.
Fig. 6 is a schematic diagram illustrating a leftward view of a 6-dimensional matrix according to an embodiment of the present invention, as shown in fig. 6(2) to 6 (7). One position pointer is maintained for each column and the character box is displayed because due to the symmetry of the cholesky decomposition, only the lower triangular portion needs to be read, i.e., only the portion of the column position pointer to the end of the column, which is intra-column pruning. And (4) numerical decomposition, and pruning in an array without dependence. Wherein (1) is the original sparse matrix, and 6(2) to 6(7) are the cases of decomposing a to f columns respectively, the columns being decomposed are displayed by bold capital letters, the reading pointer of each column is displayed by a character box, and as the numerical decomposition has no whole column pruning performed according to the dependency relationship in the symbol decomposition (the numerical decomposition cannot perform the same whole column pruning as in the symbol decomposition, and the pruning can cause a result error), when the decomposition is completed and reaches fig. 5(7), the column reading pointers can read the whole columns completely.
Specifically, as shown in fig. 6(2), a is the first column, the diagonal element is calculated by using the cholesky formula directly, that is, the diagonal element of the a column in L is obtained by squaring the diagonal element, then the other non-diagonal positions of the a column are calculated by using the calculated diagonal element, and the pointer in the column is moved down by one position; FIG. 6(3) shows that b has no dependency relationship with a, so the diagonal is directly calculated and the column is updated, and the pointer in the column is moved down by one position; as shown in fig. 6(4), c depends on a and b, so that a and b are updated from the pointer in the column to the non-zero element in the tail of the column to the column c by using the cholesky formula, then the diagonal element is calculated and the column c is updated by using the diagonal element, and the pointer in the column of a, b and c is moved down by one position; as shown in fig. 6(5), d has no dependency relationship with all columns on the left, so that the diagonal element of the d column in L is obtained by directly squaring the diagonal element, the other non-diagonal positions of the d column are updated by the diagonal element, and the pointer in the column is moved down by one position, as done in the a column and the b column; FIG. 6(6) shows that e depends on a and c, so the non-zero elements of a and c are used to compute e columns, and the pointers in a, c, and e are shifted down by one position; in FIG. 6(7), f depends on b, c, d, e, and is calculated by the Geriski formula, and the pointer in the b, c, d, and e columns moves down. And (6) finishing.
Fig. 7 is a schematic diagram of symbol decomposition and numerical decomposition according to an embodiment of the present invention, in which a trapezoidal shaded area represents a column being decomposed, and a rectangular shaded area represents an area that has been decomposed and needs to be read, and here, a method of pruning within a column is shown, that is, only a rectangular shaded area portion is read, and the upper half portion is not read.
The invention provides an efficient matrix fast decomposition method based on a resistor-capacitor network, which can accelerate the overall simulation speed of an EDA circuit.
Example 3
In an embodiment of the present invention, there is also provided an electronic device, including a memory and a processor, where the memory stores a computer program running on the processor, and the processor executes the computer program to perform the steps of the resistor-capacitor network-based matrix fast decomposition method as described above.
Example 4
In an embodiment of the present invention, a computer-readable storage medium is further provided, on which a computer program is stored, which when running executes the steps of the resistor-capacitor network-based matrix fast decomposition method as described above.
Those of ordinary skill in the art will understand that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that changes may be made in the embodiments and/or equivalents thereof without departing from the spirit and scope of the invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (5)
1. A matrix fast decomposition method based on a resistance-capacitance network comprises the following steps:
reordering the formed symmetric invertible matrix;
carrying out symbol decomposition on the reordered symmetrical reversible matrix according to columns, establishing the dependency relationship of the matrix, carrying out matrix column pruning according to the dependency relationship, and determining the positions of filling elements in the matrix;
the step of performing symbol decomposition on the reordered symmetrical reversible matrix according to columns to establish the dependency relationship of the matrix further comprises,
performing symbolic decomposition on the reordered symmetrical reversible matrix from left to right according to columns;
constructing a matrix dependency relationship tree in the symbol decomposition process, and taking the row number of the decomposed column and the next position adjacent to the diagonal element as the father node of the matrix dependency relationship tree;
taking the transposed column of the L matrix as a descendant of the transposed column;
calculating a split column according to the subcolumns in the dependency relationship, and skipping columns dependent on conduction;
performing numerical decomposition on the reordered symmetrical reversible matrix by using a Cholesky formula and a symbol decomposition result;
the step of performing numerical decomposition on the reordered symmetrical invertible matrix by using a Cholesky formula and a symbolic decomposition result further comprises,
according to the non-zero position of the matrix determined by the symbolic decomposition, performing numerical decomposition on the lower left corner of the reordered symmetrical reversible matrix from left to right according to columns;
also comprises the following steps of (1) preparing,
and calculating the decomposition result of the decomposed column by utilizing a Cholesky formula according to the descendant column in the dependency relationship.
2. The method of claim 1, wherein the step of pruning the matrix columns according to the dependency relationship to determine the positions of the padding elements in the matrix further comprises,
and maintaining a position pointer in each row in the matrix, reading the part from the row position pointer to the tail of the row, and pruning in the row and the column.
3. The method for fast decomposition of a resistor-capacitor network-based matrix according to claim 1, further comprising,
in the symbol decomposition process and the numerical decomposition process, a start position of an in-column pointer when reading the column next time is recorded for each column of the matrix.
4. An electronic device, comprising a memory and a processor, wherein the memory stores a computer program running on the processor, and the processor executes the computer program to perform the steps of the rc network-based matrix fast decomposition method according to any one of claims 1 to 3.
5. A computer-readable storage medium, on which a computer program is stored, wherein the computer program is configured to execute the steps of the method for fast decomposition of a resistor-capacitor network-based matrix according to any one of claims 1 to 3 when running.
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