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jvx.numeric

## Class PnMatrix

• java.lang.Object
• jvx.numeric.PnMatrix

• ```public class PnMatrix
extends java.lang.Object```
Numerical routines for bigger matrix linear algebra problems. Methods are taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.
See Also:
`PdMatrix`
• ### Constructor Summary

Constructors
Constructor and Description
`PnMatrix()`
• ### Method Summary

Methods
Modifier and Type Method and Description
`static boolean` ```cyclic(double[] a, double[] b, double[] c, double[] r, double[] x, double[] bb, double[] u, double[] z, double[] tmp, int n, double eps)```
Solves the equation `Au=r`, where `A` is a cyclic tridiagonal matrix with diagonal `b`, lower offdiagonal `a` and upper offdiagonal `c`.
`static double` ```determinant(double[][] aIn, int n)```
Compute determinant of a given NxN matrix A.
`static boolean` ```invert(double[][] y, double[][] aIn, int n)```
Invert a given NxN matrix A and return its invers.
`static void` ```lubksb(double[][] a, int n, int[] indx, double[] b)```
Solves the set of N linear equations AX=B.
`static double` ```ludcmp(double[][] a, int n, int[] indx)```
Given an NxN matrix A this routine replaces it by the LU decomposition of a rowwise permutation of itself.
`static void` `main(java.lang.String[] args)`
Test routine for matrix inversion.
`static boolean` ```penpes(int n, double[] a, double[] b, double[] c, double[] f, double[] d, double[] e, double[] h, double eps)```
Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix with additional perodicity coefficients (1,n-1),(1,n) and (2,n).
`static boolean` ```pentas(int n, double[] a, double[] b, double[] c, double[] f, double eps)```
Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix.
`static boolean` ```tridag(double[] a, double[] b, double[] c, double[] r, double[] u, double[] tmp, int n, double eps)```
Solves the equation `Au=r`, where `A` is a tridiagonal matrix with diagonal `b`, lower offdiagonal `a` and upper offdiagonal `c`.
• ### Methods inherited from class java.lang.Object

`equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### PnMatrix

`public PnMatrix()`
• ### Method Detail

• #### main

`public static void main(java.lang.String[] args)`
Test routine for matrix inversion.
• #### invert

```public static boolean invert(double[][] y,
double[][] aIn,
int n)```
Invert a given NxN matrix A and return its invers. Apply LU decomposition. A and N are input. Y is output containing the invers of matrix A. Input matrix is not modified since copied to a temporary matrix. All array indices start with 0 rather than the standard fortran 1.
Returns:
false if A was singular and therefore LU failed.
See Also:
`ludcmp(double[][], int, int[])`, `lubksb(double[][], int, int[], double[])`
• #### determinant

```public static double determinant(double[][] aIn,
int n)```
Compute determinant of a given NxN matrix A. Apply LU decomposition. A and N are input. Y is output containing the invers of matrix A. Input matrix is not modified since copied to a temporary matrix.

All array indices start with 0 rather than the standard fortran 1.

Returns:
determinant, or 0. if matrix is singular.
See Also:
`ludcmp(double[][], int, int[])`, `lubksb(double[][], int, int[], double[])`
• #### ludcmp

```public static double ludcmp(double[][] a,
int n,
int[] indx)
throws java.lang.ArithmeticException```
Given an NxN matrix A this routine replaces it by the LU decomposition of a rowwise permutation of itself. A and N are input. A is output, arranged as in equation (2.3.14) described in Numerical Recipes; INDX is an output vector which record the row permutation effected by the partial pivoting; the return value is +/-1 depending on whether the number of row interchanges was even (1.) or odd (-1.). This routine is used in combination with LUBKSB to solve linear equations or invert a matrix.

This version is taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.

Throws:
`java.lang.ArithmeticException` - if A was singular.
See Also:
`lubksb(double[][], int, int[], double[])`
• #### lubksb

```public static void lubksb(double[][] a,
int n,
int[] indx,
double[] b)```
Solves the set of N linear equations AX=B. Here A is input, not as the matrix A but rather as its LU decomposition, determined by the routine LUDCMP. INDX is input as the permutation vector returned by LUDCMP. B is input as the right-hand side vector B, and returns with the solution vector X. A, N, NP and INDX are not modified by the routine and can be left in place for successive calls with different right-hand sides B. This routine takes into account the possibility that B will begin with many zero elements, so it is efficient for the use in matrix inversion.

This version is taken from Numerical Recipes and adapted such that all array indices start with 0 rather than the standard fortran 1.

See Also:
`ludcmp(double[][], int, int[])`
• #### pentas

```public static final boolean pentas(int n,
double[] a,
double[] b,
double[] c,
double[] f,
double eps)```
Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix. The arrays a, b, c and f are modified by the routine.
Parameters:
`n` - dimension of matrix; must be greater or equal to 3
`a` - diagonal of length n
`b` - 1st side diagonal of length n-1 (but array must be of length n)
`c` - 2nd side diagonal of length n-2 (but array must be of length n)
`f` - input: right hand side; output: solution
`eps` - precision; recommended is 10^(-t+2) where t is machine precision
Returns:
true at success; false if matrix is (numerically) singular
• #### penpes

```public static final boolean penpes(int n,
double[] a,
double[] b,
double[] c,
double[] f,
double[] d,
double[] e,
double[] h,
double eps)```
Solves a linear system of equations with penta-diagonal, symmetric coefficient matrix with additional perodicity coefficients (1,n-1),(1,n) and (2,n). The arrays a, b, c and f are modified by the routine.
Parameters:
`n` - dimension of matrix; must be greater or equal to 5
`a` - diagonal of length n
`b` - 1st side diagonal of length n-1; and b[n-1] must contain matrix element (1,n)
`c` - 2nd side diagonal of length n-2; and b[n-2],b[n-1] must contain matrix elements (1,n-1),(2,n)
`f` - input: right hand side; output: solution
`d` - helper array of length n
`e` - helper array of length n
`h` - helper array of length n
`eps` - precision; recommended is 10^(-t+2) where t is machine precision
Returns:
true at success; false if matrix is (numerically) singular
• #### tridag

```public static boolean tridag(double[] a,
double[] b,
double[] c,
double[] r,
double[] u,
double[] tmp,
int n,
double eps)```
Solves the equation `Au=r`, where `A` is a tridiagonal matrix with diagonal `b`, lower offdiagonal `a` and upper offdiagonal `c`. Input vectors `a`, `b`, `c` and `r` are not modified.

The method only succeeds if no pivoting is required!

Parameters:
`a` - lower offdiagonal; only the entries 1,...,n-1 are used
`b` - diagonal
`c` - upper offdiagonal; only the entries 0,...,n-2 are used
`r` - right hand side
`u` - output: solution
`tmp` - temporary vector of size `n`; may be null
`n` - dimension of matrix/vectors
`eps` - precision
Returns:
true at success
• #### cyclic

```public static boolean cyclic(double[] a,
double[] b,
double[] c,
double[] r,
double[] x,
double[] bb,
double[] u,
double[] z,
double[] tmp,
int n,
double eps)```
Solves the equation `Au=r`, where `A` is a cyclic tridiagonal matrix with diagonal `b`, lower offdiagonal `a` and upper offdiagonal `c`. Input vectors `a`, `b`, `c` and `r` are not modified.

The method only succeeds if no pivoting is required!

Parameters:
`a` - lower offdiagonal; contains the element in upper right corner as `a[0]`
`b` - diagonal
`c` - upper offdiagonal; contains the element in lower left corner as `c[n-1]`
`r` - right hand side
`x` - output: solution
`bb` - temporary vector of size `n`; may be null
`u` - temporary vector of size `n`; may be null
`z` - temporary vector of size `n`; may be null
`tmp` - temporary vector of size `n`; may be null
`n` - dimension of matrix/vectors
`eps` - precision
Returns:
true at success
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