//---------------------------------------------------------------------------------------------------
/// <summary>
/// Calculate the power of a complex number
/// </summary>
/// <param name="c"></param>
/// <param name="exponent"></param>
/// <returns></returns>
public static ComplexF Pow( ComplexF c, double exponent )
{
double x = c.Re;
double y = c.Im;
double modulus = Math.Pow( x*x + y*y, exponent * 0.5 );
double argument = Math.Atan2( y, x ) * exponent;
c.Re = (float)( modulus * System.Math.Cos( argument ) );
c.Im = (float)( modulus * System.Math.Sin( argument ) );
return c;
}
//-----------------------------------------------------------------------------------
//-----------------------------------------------------------------------------------
/// <summary>
/// Determine whether two complex numbers are almost (i.e. within the tolerance) equivalent.
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <param name="tolerance"></param>
/// <returns></returns>
public static bool IsEqual( ComplexF a, ComplexF b, float tolerance )
{
return
( Math.Abs( a.Re - b.Re ) < tolerance ) &&
( Math.Abs( a.Im - b.Im ) < tolerance );
}
/// <summary>
/// Swap two complex numbers
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
public static void Swap( ref ComplexF a, ref ComplexF b )
{
ComplexF temp = a;
a = b;
b = temp;
}
private static void UnlockWorkspaceF( ref ComplexF[] workspace )
{
Debug.Assert( _workspaceF == workspace );
Debug.Assert( _workspaceFLocked == true );
_workspaceFLocked = false;
workspace = null;
}
/// <summary>
/// Shift (offset) the elements in the array
/// </summary>
/// <param name="array"></param>
/// <param name="offset"></param>
public static void Shift( ComplexF[] array, int offset )
{
Debug.Assert( array != null );
Debug.Assert( offset >= 0 );
Debug.Assert( offset < array.Length );
if( offset == 0 ) {
return;
}
int length = array.Length;
ComplexF[] workspace = null;
ComplexArray.LockWorkspaceF( length, ref workspace );
for( int i = 0; i < length; i ++ ) {
workspace[ ( i + offset ) % length ] = array[ i ];
}
for( int i = 0; i < length; i ++ ) {
array[ i ] = workspace[ i ];
}
ComplexArray.UnlockWorkspaceF( ref workspace );
}
/// <summary>
/// Multiply each element in the array by a specific value
/// </summary>
/// <param name="array"></param>
/// <param name="scale"></param>
public static void Scale( ComplexF[] array, ComplexF scale )
{
Debug.Assert( array != null );
int length = array.Length;
for( int i = 0; i < length; i ++ ) {
array[i] *= scale;
}
}
/// <summary>
/// Scale and offset the elements in the array so that the
/// overall range is [0, 1]
/// </summary>
/// <param name="array"></param>
public static void Normalize( ComplexF[] array )
{
float min = 0, max = 0;
GetLengthRange( array, ref min, ref max );
Scale( array, ( 1 / ( max - min ) ) );
Offset( array, ( - min / ( max - min ) ) );
}
/// <summary>
/// Multiply each element in target array with corresponding element in rhs array
/// </summary>
/// <param name="target"></param>
/// <param name="rhs"></param>
public static void Multiply( ComplexF[] target, ComplexF[] rhs )
{
ComplexArray.Multiply( target, rhs, target );
}
private static ComplexF SumRecursion( ComplexF[] data, int start, int end )
{
Debug.Assert( 0 <= start, "start = " + start );
Debug.Assert( start < end, "start = " + start + " and end = " + end );
Debug.Assert( end <= data.Length, "end = " + end + " and data.Length = " + data.Length );
if( ( end - start ) <= 1000 ) {
ComplexF sum = ComplexF.Zero;
for( int i = start; i < end; i ++ ) {
sum += data[ i ];
}
return sum;
}
else {
int middle = ( start + end ) >> 1;
return SumRecursion( data, start, middle ) + SumRecursion( data, middle, end );
}
}
private static float SumOfSquaredErrorRecursion( ComplexF[] alpha, ComplexF[] beta, int start, int end )
{
Debug.Assert( 0 <= start, "start = " + start );
Debug.Assert( start < end, "start = " + start + " and end = " + end );
Debug.Assert( end <= alpha.Length, "end = " + end + " and alpha.Length = " + alpha.Length );
Debug.Assert( beta.Length == alpha.Length );
if( ( end - start ) <= 1000 ) {
float sumOfSquaredError = 0;
for( int i = start; i < end; i ++ ) {
ComplexF delta = beta[ i ] - alpha[ i ];
sumOfSquaredError += ( delta.Re * delta.Re ) + ( delta.Im * delta.Im );
}
return sumOfSquaredError;
}
else {
int middle = ( start + end ) >> 1;
return SumOfSquaredErrorRecursion( alpha, beta, start, middle ) + SumOfSquaredErrorRecursion( alpha, beta, middle, end );
}
}
/// <summary>
/// Calculate the variance
/// </summary>
/// <param name="data"></param>
/// <returns></returns>
public static ComplexF Variance( ComplexF[] data )
{
Debug.Assert( data != null );
if( data.Length == 0 ) {
throw new DivideByZeroException( "length of data is zero" );
}
return ComplexStats.SumOfSquares( data ) / data.Length - ComplexStats.Sum( data );
}
//--------------------------------------------------------------------------------------------
//--------------------------------------------------------------------------------------------
/// <summary>
/// Calculate the sum of squares
/// </summary>
/// <param name="data"></param>
/// <returns></returns>
public static ComplexF SumOfSquares( ComplexF[] data )
{
Debug.Assert( data != null );
return SumOfSquaresRecursion( data, 0, data.Length );
}
/// <summary>
/// Calculate the standard deviation
/// </summary>
/// <param name="data"></param>
/// <returns></returns>
public static ComplexF StdDev( ComplexF[] data )
{
Debug.Assert( data != null );
if( data.Length == 0 ) {
throw new DivideByZeroException( "length of data is zero" );
}
return ComplexMath.Sqrt( ComplexStats.Variance( data ) );
}
/// <summary>
/// Create a complex number based on an existing complex number
/// </summary>
/// <param name="c"></param>
public ComplexF(ComplexF c)
{
this.Re = c.Re;
this.Im = c.Im;
}
//-----------------------------------------------------------------------------------
//-----------------------------------------------------------------------------------
/// <summary>
/// Determine whether two complex numbers are almost (i.e. within the tolerance) equivalent.
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <param name="tolerance"></param>
/// <returns></returns>
static public bool IsEqual(ComplexF a, ComplexF b, float tolerance)
{
return
((Math.Abs(a.Re - b.Re) < tolerance) &&
(Math.Abs(a.Im - b.Im) < tolerance));
}
private static void LinearFFT( ComplexF[] data, int start, int inc, int length, FourierDirection direction )
{
Debug.Assert( data != null );
Debug.Assert( start >= 0 );
Debug.Assert( inc >= 1 );
Debug.Assert( length >= 1 );
Debug.Assert( ( start + inc * ( length - 1 ) ) < data.Length );
// copy to buffer
ComplexF[] buffer = null;
LockBufferCF( length, ref buffer );
int j = start;
for( int i = 0; i < length; i ++ ) {
buffer[ i ] = data[ j ];
j += inc;
}
FFT( buffer, length, direction );
// copy from buffer
j = start;
for( int i = 0; i < length; i ++ ) {
data[ j ] = buffer[ i ];
j += inc;
}
UnlockBufferCF( ref buffer );
}
/// <summary>
/// Determine whether the elements in the two arrays are the same
/// </summary>
/// <param name="array1"></param>
/// <param name="array2"></param>
/// <param name="tolerance"></param>
/// <returns></returns>
public static bool IsEqual( ComplexF[] array1, ComplexF[] array2, float tolerance )
{
if ( array1.Length != array2.Length ) {
return false;
}
for( int i = 0; i < array1.Length; i ++ ) {
if( ComplexF.IsEqual( array1[i], array2[i], tolerance ) == false ) {
return false;
}
}
return true;
}
//--------------------------------------------------------------------------------------------
//--------------------------------------------------------------------------------------------
/// <summary>
/// Calculate the mean (average)
/// </summary>
/// <param name="data"></param>
/// <returns></returns>
public static ComplexF Mean( ComplexF[] data )
{
return ComplexStats.Sum( data ) / data.Length;
}
/// <summary>
/// Multiply each element in lhs array with corresponding element in rhs array and
/// put product in result array
/// </summary>
/// <param name="lhs"></param>
/// <param name="rhs"></param>
/// <param name="result"></param>
public static void Multiply( ComplexF[] lhs, ComplexF[] rhs, ComplexF[] result )
{
Debug.Assert( lhs != null );
Debug.Assert( rhs != null );
Debug.Assert( result != null );
Debug.Assert( lhs.Length == rhs.Length );
Debug.Assert( lhs.Length == result.Length );
int length = lhs.Length;
for( int i = 0; i < length; i ++ ) {
result[i] = lhs[i] * rhs[i];
}
}
//--------------------------------------------------------------------------------------------
//--------------------------------------------------------------------------------------------
/// <summary>
/// Calculate the root mean squared (RMS) error between two sets of data.
/// </summary>
/// <param name="alpha"></param>
/// <param name="beta"></param>
/// <returns></returns>
public static float RMSError( ComplexF[] alpha, ComplexF[] beta )
{
Debug.Assert( alpha != null );
Debug.Assert( beta != null );
Debug.Assert( beta.Length == alpha.Length );
return (float) Math.Sqrt( SumOfSquaredErrorRecursion( alpha, beta, 0, alpha.Length ) );
}
/// <summary>
/// Add a specific value to each element in the array
/// </summary>
/// <param name="array"></param>
/// <param name="offset"></param>
public static void Offset( ComplexF[] array, ComplexF offset )
{
int length = array.Length;
for( int i = 0; i < length; i ++ ) {
array[i] += offset;
}
}
public int GetMandelbrotDepth( ComplexF c )
{
int depth = 0;
ComplexF a = ComplexF.Zero;
// iterate until either the limit is reached or it is
// certain that a(k+1) = a(k)^2 + c goes to infinity.
do {
a = a*a + c;
depth += 8;
} while ( ( depth < 512 ) && ( a.GetModulusSquared() < 4.0f ) );
if( a.GetModulusSquared() < 4.0f ) {
return 255;
}
return ( depth % 256 );
}
/// <summary>
/// Multiply each element in the array by a specific value
/// </summary>
/// <param name="array"></param>
/// <param name="scale"></param>
/// <param name="start"></param>
/// <param name="length"></param>
public static void Scale( ComplexF[] array, ComplexF scale, int start, int length )
{
Debug.Assert( array != null );
Debug.Assert( start >= 0 );
Debug.Assert( length >= 0 );
Debug.Assert( ( start + length ) < array.Length );
for( int i = 0; i < length; i ++ ) {
array[i + start] *= scale;
}
}
/// <summary>
/// Copy an array
/// </summary>
/// <param name="dest"></param>
/// <param name="source"></param>
public static void Copy( ComplexF[] dest, ComplexF[] source )
{
Debug.Assert( dest != null );
Debug.Assert( source != null );
Debug.Assert( dest.Length == source.Length );
for( int i = 0; i < dest.Length; i ++ ) {
dest[i] = source[i];
}
}
private static void LockWorkspaceF( int length, ref ComplexF[] workspace )
{
Debug.Assert( _workspaceFLocked == false );
_workspaceFLocked = true;
if( length >= _workspaceF.Length ) {
_workspaceF = new ComplexF[ length ];
}
workspace = _workspaceF;
}
/// <summary>
/// Divide each element in target array with corresponding element in rhs array
/// </summary>
/// <param name="target"></param>
/// <param name="rhs"></param>
public static void Divide( ComplexF[] target, ComplexF[] rhs )
{
ComplexArray.Divide( target, rhs, target );
}
/// <summary>
/// Calculate the square root of a complex number
/// </summary>
/// <param name="c"></param>
/// <returns></returns>
public static ComplexF Sqrt( ComplexF c )
{
double x = c.Re;
double y = c.Im;
double modulus = Math.Sqrt( x*x + y*y );
int sign = ( y < 0 ) ? -1 : 1;
c.Re = (float)( _halfOfRoot2 * Math.Sqrt( modulus + x ) );
c.Im = (float)( _halfOfRoot2 * sign * Math.Sqrt( modulus - x ) );
return c;
}
/// <summary>
/// Divide each element in lhs array with corresponding element in rhs array and
/// put product in result array
/// </summary>
/// <param name="lhs"></param>
/// <param name="rhs"></param>
/// <param name="result"></param>
public static void Divide( ComplexF[] lhs, ComplexF[] rhs, ComplexF[] result )
{
Debug.Assert( lhs != null );
Debug.Assert( rhs != null );
Debug.Assert( result != null );
Debug.Assert( lhs.Length == rhs.Length );
Debug.Assert( lhs.Length == result.Length );
ComplexF zero = ComplexF.Zero;
int length = lhs.Length;
for( int i = 0; i < length; i ++ ) {
if( rhs[i] != zero ) {
result[i] = lhs[i] / rhs[i];
}
else {
result[i] = zero;
}
}
}
/// <summary>
/// Get the range of element lengths
/// </summary>
/// <param name="array"></param>
/// <param name="minimum"></param>
/// <param name="maximum"></param>
public static void GetLengthRange( ComplexF[] array, ref float minimum, ref float maximum )
{
minimum = +float.MaxValue;
maximum = -float.MaxValue;
for( int i = 0; i < array.Length; i ++ ) {
float temp = array[i].GetModulus();
minimum = Math.Min( temp, minimum );
maximum = Math.Max( temp, maximum );
}
}
/// <summary>
/// Invert each element in the array
/// </summary>
/// <param name="array"></param>
public static void Invert( ComplexF[] array )
{
for( int i = 0; i < array.Length; i ++ ) {
array[i] = ((ComplexF) 1 ) / array[i];
}
}
/// <summary>
/// Create a complex number based on an existing complex number
/// </summary>
/// <param name="c"></param>
public ComplexF( ComplexF c )
{
this.Re = c.Re;
this.Im = c.Im;
}
/// <summary>
/// Compute a 1D fast Fourier transform of a dataset of complex numbers.
/// </summary>
/// <param name="data"></param>
/// <param name="length"></param>
/// <param name="direction"></param>
public static void FFT_Quick( ComplexF[] data, int length, FourierDirection direction )
{
/*if( data == null ) {
throw new ArgumentNullException( "data" );
}
if( data.Length < length ) {
throw new ArgumentOutOfRangeException( "length", length, "must be at least as large as 'data.Length' parameter" );
}
if( Fourier.IsPowerOf2( length ) == false ) {
throw new ArgumentOutOfRangeException( "length", length, "must be a power of 2" );
}
Fourier.SyncLookupTableLength( length );*/
int ln = Fourier.Log2( length );
// reorder array
Fourier.ReorderArray( data );
// successive doubling
int N = 1;
int signIndex = ( direction == FourierDirection.Forward ) ? 0 : 1;
for( int level = 1; level <= ln; level ++ ) {
int M = N;
N <<= 1;
float[] uRLookup = _uRLookupF[ level, signIndex ];
float[] uILookup = _uILookupF[ level, signIndex ];
for( int j = 0; j < M; j ++ ) {
float uR = uRLookup[j];
float uI = uILookup[j];
for( int even = j; even < length; even += N ) {
int odd = even + M;
float r = data[ odd ].Re;
float i = data[ odd ].Im;
float odduR = r * uR - i * uI;
float odduI = r * uI + i * uR;
r = data[ even ].Re;
i = data[ even ].Im;
data[ even ].Re = r + odduR;
data[ even ].Im = i + odduI;
data[ odd ].Re = r - odduR;
data[ odd ].Im = i - odduI;
}
}
}
}