《计算方法》 李晓红等 第三章 解线性方程组的直接法 例题 代码
#include "stdafx.h"
#include
// 解线性方程组的直接法
void PS(double A[], int n, char *name);
void GaussXiaoQu()
{
printf("高斯直接消元法:\n");
double a[3][3] = {{2,4,-2},{1,-1,5},{4,1,-2}};
double b[3] = {6,0,2};
double x[3];
int n = 3;
int k,i,j;
double l;
// 消元
for (k = 0; k < n-1; k++)
{
for (i = k+1; i < n; i++)
{
l = a[i][k]/a[k][k];
b[i] = b[i] - l*b[k];
for (j = k; j < n; j++)
{
a[i][j] = a[i][j] - l*a[k][j];
}
}
}
// 消元结果
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",a[i][j]);
printf("%f",b[i]);
printf("\n");
}
// 回代
for (i = n-1; i >= 0; i--)
{
x[i] = b[i];
for (j = n-1; j > i; j--)
x[i] = x[i] - a[i][j]*x[j];
x[i] = x[i] / a[i][i];
}
printf("x= ");
for(i = 0; i < n; i++)
printf("%f\t",x[i]);
printf("\n");
}
void GaussLieZhuYuan()
{
printf("高斯列主元法:\n");
double a[3][3] = {{1,2,3},{5,4,10},{3,-0.1,1}};
double b[3] = {1,0,2};
double x[3];
int n = 3; // 系数矩阵维数
int k,i,j;
double l;
double tmp;
int r;
// 消元
for (k = 0; k < n-1; k++)
{
// 找到该列最大元
for (r=k,j=k+1; j < n; j++) {
if (abs(a[j][k]) > abs(a[r][k]))
{
r = j;
}
}
// 行交换
if (r != k)
{
for (j = k; j < n; j++) {
tmp = a[k][j];
a[k][j] = a[r][j];
a[r][j] = tmp;
}
tmp = b[k];
b[k] = b[r];
b[r] = tmp;
}
// 消元
for (i = k+1; i < n; i++)
{
l = a[i][k]/a[k][k];
b[i] = b[i] - l*b[k];
for (j = k; j < n; j++)
{
a[i][j] = a[i][j] - l*a[k][j];
}
}
}
// 消元结果
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",a[i][j]);
printf("%f",b[i]);
printf("\n");
}
// 回代
for (i = n-1; i >= 0; i--)
{
x[i] = b[i];
for (j = n-1; j > i; j--)
x[i] = x[i] - a[i][j]*x[j];
x[i] = x[i] / a[i][i];
}
printf("x= ");
for (i = 0; i < n; i++) {
printf("%f\t",x[i]);
}
printf("\n");
}
void GaussQuanZhuYuan()
{
printf("高斯全主元法:\n");
double a[3][3] = {{1,2,3},{5,4,10},{3,-0.1,1}};
double b[3] = {1,0,2};
double x[3];
double y[3];
int n = 3; // 系数矩阵维数
int k,i,j;
double l;
double tmp;
int r,c;
int p[3] = {0,1,2},q;
// 消元
for (k = 0; k < n-1; k++)
{
// 找到子矩阵内最大元
r = c = k;
for (i = k; i < n; i++) {
for (j = k; j < n; j++) {
if (abs(a[i][j]) > abs(a[r][c]))
{
r = i;
c = j;
}
}
}
// 行交换 k~r
if (r != k)
{
for (j = k; j < n; j++) {
tmp = a[k][j];
a[k][j] = a[r][j];
a[r][j] = tmp;
}
tmp = b[k];
b[k] = b[r];
b[r] = tmp;
}
// 交换列 k~c
if (c != k)
{
for (i = 0; i < n; i++) {
tmp = a[i][k];
a[i][k] = a[i][c];
a[i][c] = tmp;
}
q = p[k];
p[k] = p[c];
p[c] = q;
}
// 消元
for (i = k+1; i < n; i++)
{
l = a[i][k]/a[k][k];
b[i] = b[i] - l*b[k];
for (j = k; j < n; j++)
{
a[i][j] = a[i][j] - l*a[k][j];
}
}
}
// 消元结果
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",a[i][j]);
printf("%f",b[i]);
printf("\n");
}
// 回代
for (i = n-1; i >= 0; i--)
{
x[i] = b[i];
for (j = n-1; j > i; j--)
x[i] = x[i] - a[i][j]*x[j];
x[i] = x[i] / a[i][i];
}
printf("x= ");
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++)
{
if (i == p[j])
{
y[i] = x[j];
}
}
printf("%f\t",y[i]);
}
printf("\n");
}
void GaussYueDangMethod()
{
printf("高斯约当消去法(列主元):\n");
double a[3][3] = {{1,-1,0},{2,2,3},{-1,2,1}};
double b[3] = {1,0,0};
double x[3];
int n = 3; // 系数矩阵维数
int k,i,j;
double l;
double tmp;
int r;
// 消元
for (k = 0; k < n; k++)
{
// 找到该列最大元
for (r=k,j=k+1; j < n; j++) {
if (abs(a[j][k]) > abs(a[r][k]))
{
r = j;
}
}
// 行交换
if (r != k)
{
for (j = k; j < n; j++) {
tmp = a[k][j];
a[k][j] = a[r][j];
a[r][j] = tmp;
}
tmp = b[k];
b[k] = b[r];
b[r] = tmp;
}
// 消元
for (i = 0; i < n; i++) {
if (i == k)
continue;
else
{
l = a[i][k]/a[k][k];
b[i] = b[i] - l*b[k];
for (j = k; j < n; j++)
{
a[i][j] = a[i][j] - l*a[k][j];
}
}
}
// 消去本行
l = a[k][k];
for (j = k; j < n; j++)
a[k][j] = a[k][j] / l;
b[k] = b[k] / l;
// 消元结果
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",a[i][j]);
printf("%f",b[i]);
printf("\n");
}
printf("\n");
}
/* // 消元结果
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",a[i][j]);
printf("%f",b[i]);
printf("\n");
}
*/ // 回代
for (i = n-1; i >= 0; i--)
{
x[i] = b[i];
}
printf("x= ");
for (i = 0; i < n; i++) {
printf("%f\t",x[i]);
}
printf("\n");
}
void ZhuiGanMethod_I()
{
printf("三对角方程追赶法I:\n");
double p[3];
double q[2];
double a[3][3] = {{6,1,0},{1,4,1},{6,1,14}};
double d[3] = {6,24,322};
double x[3];
int k;
int n = 3;
double l;
p[0] = d[0]/a[0][0];
q[0] = a[0][1]/a[0][0];
for (k = 1; k < n; k++) {
l = a[k][k] - a[k][k-1]*q[k-1];
p[k] = (d[k]-a[k][k-1]*p[k-1])/l;
if(k
}
// p,q结果
for (k = 0; k < n; k++)
printf("p[%d]=%f\t",k,p[k]);
printf("\n");
for (k = 0; k < n-1; k++)
printf("q[%d]=%f\t",k,q[k]);
printf("\n");
// 回代
x[n-1] = p[n-1];
for (k = n-2; k >= 0; k--) {
x[k] = p[k]-q[k]*x[k+1];
}
for (k = 0; k < n; k++) {
printf("%f\t",x[k]);
}
printf("\n");
}
void ZhiJieSanJiaoXing()
{
printf("直接三角形分解法:\n");
double a[3][3] = {{2,2,3},{4,7,7},{-2,4,5}};
double b[3] = {3,1,-7};
double l[3][3],u[3][3];
int n = 3;
int i,j;
// LU分解
int k = 0;;
int r;
for (k = 0; k < n; k++) {
// 先计算u
for (j = 0; j < n; j++) {
if(j
else
{
u[k][j] = a[k][j];
for (r = 0; r < k && k > 0; r++) {
u[k][j] -= l[k][r]*u[r][j];
}
}
}
// 后计算l
for (i = 0; i < n; i++) {
if (i
else if(i==k) l[i][k]=1.;
else
{
l[i][k] = a[i][k];
for (r = 0; r < k && k > 0; r++) {
l[i][k] -= l[i][r]*u[r][k];
}
l[i][k] /= u[k][k];
}
}
}
printf("l=\n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",l[i][j]);
printf("\n");
}
printf("u=\n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",u[i][j]);
printf("\n");
}
// 回代 求y
// Ly = b
double y[3];
y[0] = b[0];
for (i = 1; i < n; i++) {
y[i] = b[i];
for (j = 0; j < i; j++) {
y[i] = y[i] - l[i][j]*y[j];
}
}
// 求x
//Ux=y
double x[3];
x[n-1] = y[n-1]/u[n-1][n-1];
for (i = n-2; i >=0; i--) {
x[i] = y[i];
for (j = i+1; j < n; j++) {
x[i] = x[i] - u[i][j]*x[j];
}
x[i] /= u[i][i];
}
printf("x= ");
for (i = 0; i < n; i++) {
printf("%f\t",x[i]);
}
printf("\n");
}
void ZhuiGanMethod_II()
{
printf("三对角方程追赶法II:\n");
double p[3];
double q[2];
double a[3][3] = {{6,1,0},{1,4,1},{0,1,14}};
double d[3] = {6,24,322};
double x[3],y[3];
int i;
int n = 3;
// 追
p[0] = a[0][0];
q[0] = a[0][1]/p[0];
for (i = 1; i < n; i++)
{
p[i] = a[i][i] - a[i][i-1]*q[i-1];
if (i
}
PS(p,n,"p");PS(q,n-1,"q");
// 赶
for (i=0;i
{
y[i] = d[i];
if(i>0) y[i] -= a[i][i-1]*y[i-1];
y[i] /= p[i];
}
for (i=n-1;i>=0;i--) {
x[i] = y[i];
if(i
}
PS(y,n,"y");PS(x,n,"x");
}
void PingFangGenMethod()
{
printf("平方根法: A=LL'\n");
double a[3][3] = {{6,1,0},{1,4,1},{0,1,14}};
double b[3] = {6,24,322};
int n = 3;
int i,j,k;
double l[3][3];
double x[3],y[3];
for (i=0;i
for (j=0;j
l[i][j] = 0.;
}
}
// 分解
for (i=0;i
// 对角元
l[i][i] = a[i][i];
for (k=0;k
l[i][i] -= l[i][k]*l[i][k];
}
l[i][i] = sqrt(l[i][i]);
// 列元
for (j=i+1;j
l[j][i] = a[j][i];
for (k=0;k
l[j][i] -= l[j][k]*l[i][k]; // 转置!
}
l[j][i] /= l[i][i];
}
}
printf("l=\n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",l[i][j]);
printf("\n");
}
// 回代 ,L'y = b
for (i=0;i
{
y[i] = b[i];
for (j=0;j
y[i] -= l[i][j]*y[j];
}
y[i] /= l[i][i];
}
for (i=n-1;i>=0;i--) {
x[i] = y[i];
for(j=n-1;j>i;j--)
x[i]-=l[j][i]*x[j]; // 主意L 转置, L' x = y
x[i] /= l[i][i];
}
PS(y,n,"y");PS(x,n,"x");
}
void GaiJinPingFangMethod() //
{
printf("改进的平方根法: A=LDL'\n");
double a[3][3] = {{6,1,0},{1,4,1},{0,1,14}};
double b[3] = {6,24,322};
int n = 3;
int i,j,k;
double l[3][3],u[3][3];
for (i=0;i
for (j=0;j
u[i][j] = l[i][j] = 0.;
}
}
// LU分解
int r;
for (k = 0; k < n; k++) {
// 先计算u
for (j = k; j < n; j++) {
u[k][j] = a[k][j];
for (r = 0; r < k && k > 0; r++) {
u[k][j] -= l[k][r]*u[r][j];
}
}
// 后计算l, 主意与紧凑格式法差异,运算量减少
l[k][k] = 1.;
for (i = k+1; i < n; i++) {
l[i][k] = u[k][i]/u[k][k];
}
}
printf("l=\n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",l[i][j]);
printf("\n");
}
printf("u=\n");
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
printf("%f\t",u[i][j]);
printf("\n");
}
// 回代 求y
// Ly = b
double y[3];
y[0] = b[0];
for (i = 1; i < n; i++) {
y[i] = b[i];
for (j = 0; j < i; j++) {
y[i] = y[i] - l[i][j]*y[j];
}
}
// 求x
//Ux=y
double x[3];
x[n-1] = y[n-1]/u[n-1][n-1];
for (i = n-2; i >=0; i--) {
x[i] = y[i];
for (j = i+1; j < n; j++) {
x[i] = x[i] - u[i][j]*x[j];
}
x[i] /= u[i][i];
}
PS(x,n,"x");
}