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340 | #include "Spline.h"
using namespace tk;
// ---------------------------------------------------------------------
// implementation part, which could be separated into a cpp file
// ---------------------------------------------------------------------
// band_matrix implementation
// -------------------------
band_matrix::band_matrix(int dim, int n_u, int n_l)
{
resize(dim, n_u, n_l);
}
void band_matrix::resize(int dim, int n_u, int n_l)
{
assert(dim>0);
assert(n_u>=0);
assert(n_l>=0);
m_upper.resize(n_u+1);
m_lower.resize(n_l+1);
for(size_t i=0; i<m_upper.size(); i++) {
m_upper[i].resize(dim);
}
for(size_t i=0; i<m_lower.size(); i++) {
m_lower[i].resize(dim);
}
}
int band_matrix::dim() const
{
if(m_upper.size()>0) {
return static_cast<int>(m_upper[0].size());
} else {
return 0;
}
}
// defines the new operator (), so that we can access the elements
// by A(i,j), index going from i=0,...,dim()-1
double & band_matrix::operator () (int i, int j)
{
int k=j-i; // what band is the entry
assert( (i>=0) && (i<dim()) && (j>=0) && (j<dim()) );
assert( (-num_lower()<=k) && (k<=num_upper()) );
// k=0 -> diogonal, k<0 lower left part, k>0 upper right part
if(k>=0) return m_upper[k][i];
else return m_lower[-k][i];
}
double band_matrix::operator () (int i, int j) const
{
int k=j-i; // what band is the entry
assert( (i>=0) && (i<dim()) && (j>=0) && (j<dim()) );
assert( (-num_lower()<=k) && (k<=num_upper()) );
// k=0 -> diogonal, k<0 lower left part, k>0 upper right part
if(k>=0) return m_upper[k][i];
else return m_lower[-k][i];
}
// second diag (used in LU decomposition), saved in m_lower
double band_matrix::saved_diag(int i) const
{
assert( (i>=0) && (i<dim()) );
return m_lower[0][i];
}
double & band_matrix::saved_diag(int i)
{
assert( (i>=0) && (i<dim()) );
return m_lower[0][i];
}
// LR-Decomposition of a band matrix
void band_matrix::lu_decompose()
{
int i_max,j_max;
int j_min;
double x;<--- The scope of the variable 'i_max' can be reduced. [+]The scope of the variable 'i_max' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
<--- The scope of the variable 'j_min' can be reduced. [+]The scope of the variable 'j_min' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
// preconditioning
// normalize column i so that a_ii=1
for(int i=0; i<this->dim(); i++) {
assert(this->operator()(i,i)!=0.0);
this->saved_diag(i)=1.0/this->operator()(i,i);
j_min=std::max(0,i-this->num_lower());
j_max=std::min(this->dim()-1,i+this->num_upper());
for(int j=j_min; j<=j_max; j++) {
this->operator()(i,j) *= this->saved_diag(i);
}
this->operator()(i,i)=1.0; // prevents rounding errors
}
// Gauss LR-Decomposition
for(int k=0; k<this->dim(); k++) {
i_max=std::min(this->dim()-1,k+this->num_lower()); // num_lower not a mistake!
for(int i=k+1; i<=i_max; i++) {
assert(this->operator()(k,k)!=0.0);
x=-this->operator()(i,k)/this->operator()(k,k);
this->operator()(i,k)=-x; // assembly part of L
j_max=std::min(this->dim()-1,k+this->num_upper());
for(int j=k+1; j<=j_max; j++) {
// assembly part of R
this->operator()(i,j)=this->operator()(i,j)+x*this->operator()(k,j);
}
}
}
}
// solves Ly=b
std::vector<double> band_matrix::l_solve(const std::vector<double>& b) const
{
assert( this->dim()==(int)b.size() );
std::vector<double> x(this->dim());
int j_start;
double sum;
for(int i=0; i<this->dim(); i++) {<--- The scope of the variable 'j_start' can be reduced. [+]The scope of the variable 'j_start' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
sum=0;<--- The scope of the variable 'sum' can be reduced. [+]The scope of the variable 'sum' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
j_start=std::max(0,i-this->num_lower());
for(int j=j_start; j<i; j++) sum += this->operator()(i,j)*x[j];
x[i]=(b[i]*this->saved_diag(i)) - sum;
}
return x;
}
// solves Rx=y
std::vector<double> band_matrix::r_solve(const std::vector<double>& b) const
{
assert( this->dim()==(int)b.size() );
std::vector<double> x(this->dim());
int j_stop;
double sum;
for(int i=this->dim()-1; i>=0; i--) {<--- The scope of the variable 'j_stop' can be reduced. [+]The scope of the variable 'j_stop' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
sum=0;<--- The scope of the variable 'sum' can be reduced. [+]The scope of the variable 'sum' can be reduced. Warning: Be careful when fixing this message, especially when there are inner loops. Here is an example where cppcheck will write that the scope for 'i' can be reduced:
void f(int x)
{
int i = 0;
if (x) {
// it's safe to move 'int i = 0;' here
for (int n = 0; n < 10; ++n) {
// it is possible but not safe to move 'int i = 0;' here
do_something(&i);
}
}
}
When you see this message it is always safe to reduce the variable scope 1 level.
j_stop=std::min(this->dim()-1,i+this->num_upper());
for(int j=i+1; j<=j_stop; j++) sum += this->operator()(i,j)*x[j];
x[i]=( b[i] - sum ) / this->operator()(i,i);
}
return x;
}
std::vector<double> band_matrix::lu_solve(const std::vector<double>& b,
bool is_lu_decomposed)
{
assert( this->dim()==(int)b.size() );
std::vector<double> x,y;
if(is_lu_decomposed==false) {
this->lu_decompose();
}
y=this->l_solve(b);
x=this->r_solve(y);
return x;
}
// spline implementation
// -----------------------
void spline::set_boundary(spline::bd_type left, double left_value,
spline::bd_type right, double right_value,
bool force_linear_extrapolation)
{
assert(m_x.size()==0); // set_points() must not have happened yet
m_left=left;
m_right=right;
m_left_value=left_value;
m_right_value=right_value;
m_force_linear_extrapolation=force_linear_extrapolation;
}
void spline::set_points(const std::vector<double>& x,
const std::vector<double>& y, bool cubic_spline)
{
assert(x.size()==y.size());
assert(x.size()>2);
m_x=x;
m_y=y;
int n = static_cast<int>(x.size());
// TODO: maybe sort x and y, rather than returning an error
for(int i=0; i<n-1; i++) {
assert(m_x[i]<m_x[i+1]);
}
if(cubic_spline==true) { // cubic spline interpolation
// setting up the matrix and right hand side of the equation system
// for the parameters b[]
band_matrix A(n,1,1);
std::vector<double> rhs(n);
for(int i=1; i<n-1; i++) {
A(i,i-1)=1.0/3.0*(x[i]-x[i-1]);
A(i,i)=2.0/3.0*(x[i+1]-x[i-1]);
A(i,i+1)=1.0/3.0*(x[i+1]-x[i]);
rhs[i]=(y[i+1]-y[i])/(x[i+1]-x[i]) - (y[i]-y[i-1])/(x[i]-x[i-1]);
}
// boundary conditions
if(m_left == spline::second_deriv) {
// 2*b[0] = f''
A(0,0)=2.0;
A(0,1)=0.0;
rhs[0]=m_left_value;
} else if(m_left == spline::first_deriv) {
// c[0] = f', needs to be re-expressed in terms of b:
// (2b[0]+b[1])(x[1]-x[0]) = 3 ((y[1]-y[0])/(x[1]-x[0]) - f')
A(0,0)=2.0*(x[1]-x[0]);
A(0,1)=1.0*(x[1]-x[0]);
rhs[0]=3.0*((y[1]-y[0])/(x[1]-x[0])-m_left_value);
} else {
assert(false);
}
if(m_right == spline::second_deriv) {
// 2*b[n-1] = f''
A(n-1,n-1)=2.0;
A(n-1,n-2)=0.0;
rhs[n-1]=m_right_value;
} else if(m_right == spline::first_deriv) {
// c[n-1] = f', needs to be re-expressed in terms of b:
// (b[n-2]+2b[n-1])(x[n-1]-x[n-2])
// = 3 (f' - (y[n-1]-y[n-2])/(x[n-1]-x[n-2]))
A(n-1,n-1)=2.0*(x[n-1]-x[n-2]);
A(n-1,n-2)=1.0*(x[n-1]-x[n-2]);
rhs[n-1]=3.0*(m_right_value-(y[n-1]-y[n-2])/(x[n-1]-x[n-2]));
} else {
assert(false);
}
// solve the equation system to obtain the parameters b[]
m_b=A.lu_solve(rhs);
// calculate parameters a[] and c[] based on b[]
m_a.resize(n);
m_c.resize(n);
for(int i=0; i<n-1; i++) {
m_a[i]=1.0/3.0*(m_b[i+1]-m_b[i])/(x[i+1]-x[i]);
m_c[i]=(y[i+1]-y[i])/(x[i+1]-x[i])
- 1.0/3.0*(2.0*m_b[i]+m_b[i+1])*(x[i+1]-x[i]);
}
} else { // linear interpolation
m_a.resize(n);
m_b.resize(n);
m_c.resize(n);
for(int i=0; i<n-1; i++) {
m_a[i]=0.0;
m_b[i]=0.0;
m_c[i]=(m_y[i+1]-m_y[i])/(m_x[i+1]-m_x[i]);
}
}
// for left extrapolation coefficients
m_b0 = (m_force_linear_extrapolation==false) ? m_b[0] : 0.0;
m_c0 = m_c[0];
// for the right extrapolation coefficients
// f_{n-1}(x) = b*(x-x_{n-1})^2 + c*(x-x_{n-1}) + y_{n-1}
double h=x[n-1]-x[n-2];
// m_b[n-1] is determined by the boundary condition
m_a[n-1]=0.0;
m_c[n-1]=3.0*m_a[n-2]*h*h+2.0*m_b[n-2]*h+m_c[n-2]; // = f'_{n-2}(x_{n-1})
if(m_force_linear_extrapolation==true)
m_b[n-1]=0.0;
}
double spline::operator() (double x) const
{
size_t n=m_x.size();
// find the closest point m_x[idx] < x, idx=0 even if x<m_x[0]
std::vector<double>::const_iterator it;
it=std::lower_bound(m_x.begin(),m_x.end(),x);
int idx=std::max( int(it-m_x.begin())-1, 0);
double h=x-m_x[idx];
double interpol;
if(x<m_x[0]) {
// extrapolation to the left
interpol=(m_b0*h + m_c0)*h + m_y[0];
} else if(x>m_x[n-1]) {
// extrapolation to the right
interpol=(m_b[n-1]*h + m_c[n-1])*h + m_y[n-1];
} else {
// interpolation
interpol=((m_a[idx]*h + m_b[idx])*h + m_c[idx])*h + m_y[idx];
}
return interpol;
}
double spline::deriv(int order, double x) const
{
assert(order>0);
size_t n=m_x.size();
// find the closest point m_x[idx] < x, idx=0 even if x<m_x[0]
std::vector<double>::const_iterator it;
it=std::lower_bound(m_x.begin(),m_x.end(),x);
int idx=std::max( int(it-m_x.begin())-1, 0);
double h=x-m_x[idx];
double interpol;
if(x<m_x[0]) {
// extrapolation to the left
switch(order) {
case 1:
interpol=2.0*m_b0*h + m_c0;
break;
case 2:
interpol=2.0*m_b0*h;
break;
default:
interpol=0.0;
break;
}
} else if(x>m_x[n-1]) {
// extrapolation to the right
switch(order) {
case 1:
interpol=2.0*m_b[n-1]*h + m_c[n-1];
break;
case 2:
interpol=2.0*m_b[n-1];
break;
default:
interpol=0.0;
break;
}
} else {
// interpolation
switch(order) {
case 1:
interpol=(3.0*m_a[idx]*h + 2.0*m_b[idx])*h + m_c[idx];
break;
case 2:
interpol=6.0*m_a[idx]*h + 2.0*m_b[idx];
break;
case 3:
interpol=6.0*m_a[idx];
break;
default:
interpol=0.0;
break;
}
}
return interpol;
}
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