/// <summary>
/// This function solves simultaneous equations in matrix form.
/// The equations are represented by the matrix equation:
///
/// aMatrix xVector = bVector
///
/// The algorithm is from the book:
/// "Mathematical Methods For Digital Computers" Volume 2
/// Edited by Anthony Ralston and Herbert S. Wilf, 1967,
/// John Wiley and Sons, pages 65-93.
/// "The solution of ill-conditioned linear equations"
/// by J. H. Wilkinson.
/// </summary>
/// <param name="numberOfEquations">The number of equations</param>
/// <param name="aMatrix">A square matrix</param>
/// <param name="bVector">A vector that contains the right hand side of the matrix equations shown below.</param>
/// <param name="xVector">A vector to contain the solution of the matrix equations.</param>
/// <returns>
/// This function returns an enumerated value of type Status_T
/// that is the value LinearEquationSolverStatus.Singular if the
/// coefficient matrix is singular to working accuracy. A value of
/// LinearEquationSolverStatus.IllConditioned is returned if the
/// coefficient matrix is singular to working accuracy, i.e. the
/// floating point arithmetic does not have enough precision to
/// guarantee convergence to an accurate solution.
/// The value LinearEquationSolverStatus.Success is returned
/// if the solution vector was calculated successfully.
/// </returns>
public static LinearEquationSolverStatus Solve(int numberOfEquations,
Sparse2DMatrix <int, int, double> aMatrix,
SparseArray <int, double> bVector,
SparseArray <int, double> xVector)
{
//----------------------------------------------------------
// Matrix a_matrix is copied into working matrix aMatrixCopy.
//----------------------------------------------------------
Sparse2DMatrix <int, int, double> aMatrixCopy = new Sparse2DMatrix <int, int, double>(aMatrix);
//----------------------------------------------------------
// The maximum value rowMaximumVector[i], i = 0 to n - 1
// is stored
//----------------------------------------------------------
SparseArray <int, double> rowMaximumVector = new SparseArray <int, double>();
int i;
for (i = 0; i < numberOfEquations; i++)
{
double temp = 0.0;
for (int j = 0; j < numberOfEquations; j++)
{
double test = Math.Abs(aMatrix[i, j]);
if (test > temp)
{
temp = test;
}
}
rowMaximumVector[i] = temp;
//----------------------------------------------------------
// Test for singular matrix.
//----------------------------------------------------------
if (Math.Abs(temp) <= 0.0)
{
return(LinearEquationSolverStatus.Singular);
}
}
//----------------------------------------------------------
// The r'th column of "l", the r'th pivotal position r', and
// the r'th row of "u" are determined.
//----------------------------------------------------------
SparseArray <int, int> pivotRowArray = new SparseArray <int, int>();
for (int r = 0; r < numberOfEquations; r++)
{
double maximumValue = 0.0;
int rowMaximumValueIndex = r;
//----------------------------------------------------------
// The l[i][r], i = r to n - 1 are determined.
// l[i][r] is a lower triangular matrix. It is calculated
// using the variable temp and copied into matrix
// "aMatrixCopy". The variable "maximumValue" contains
// the largest Math.Abs(l[i][r] / pivotRowArray[i]) and
// rowMaximumValueIndex stores the "i" which corresponds
// to the value in variable maximumValue.
//----------------------------------------------------------
double temp;
for (i = r; i < numberOfEquations; i++)
{
temp = aMatrixCopy[i, r];
for (int j = 0; j < r; j++)
{
temp = temp - aMatrixCopy[i, j] * aMatrixCopy[j, r];
}
aMatrixCopy[i, r] = temp;
double test = Math.Abs(temp / rowMaximumVector[i]);
if (test > maximumValue)
{
maximumValue = test;
rowMaximumValueIndex = i;
}
}
//----------------------------------------------------------
// Test for matrix which is singular to working precision.
//----------------------------------------------------------
if (Math.Abs(maximumValue) <= 0.0)
{
return(LinearEquationSolverStatus.IllConditioned);
}
//----------------------------------------------------------
// "rowMaximumValueIndex" = r' and "pivotRowArray[r]"
// is the pivotal row.
//----------------------------------------------------------
rowMaximumVector[rowMaximumValueIndex] = rowMaximumVector[r];
pivotRowArray[r] = rowMaximumValueIndex;
//----------------------------------------------------------
// Rows "r" and "pivotRowArray[r]" are exchanged.
//----------------------------------------------------------
for (i = 0; i < numberOfEquations; i++)
{
temp = aMatrixCopy[r, i];
aMatrixCopy[r, i] = aMatrixCopy[rowMaximumValueIndex, i];
aMatrixCopy[rowMaximumValueIndex, i] = temp;
}
//----------------------------------------------------------
// The u[r][i], i = r + 1 to n - 1 are determined.
// "u[r][i]" is an upper triangular matrix. It is copied
// into aMatrixCopy.
//----------------------------------------------------------
for (i = r + 1; i < numberOfEquations; i++)
{
temp = aMatrixCopy[r, i];
for (int j = 0; j < r; j++)
{
temp = temp - aMatrixCopy[r, j] * aMatrixCopy[j, i];
}
aMatrixCopy[r, i] = temp / aMatrixCopy[r, r];
}
}
//----------------------------------------------------------
// The first solution vector is set equal to the null vector
// and the first residuals are set equal to the equation
// constant vector.
//----------------------------------------------------------
SparseArray <int, double> residualVector = new SparseArray <int, double>();
for (i = 0; i < numberOfEquations; i++)
{
xVector[i] = 0.0;
residualVector[i] = bVector[i];
}
//----------------------------------------------------------
// The iteration counter is initialized.
//----------------------------------------------------------
int iteration = 0;
bool notConvergedFlag = true;
do
{
//----------------------------------------------------------
// The forward substitution is performed and the solution
// of l y = p r is calculated where p r is the current
// residual after interchanges.
//----------------------------------------------------------
for (i = 0; i < numberOfEquations; i++)
{
int pivotRowIndex = pivotRowArray[i];
double temp = residualVector[pivotRowIndex];
residualVector[pivotRowIndex] = residualVector[i];
for (int j = 0; j < i; j++)
{
temp = temp - aMatrixCopy[i, j] * residualVector[j];
}
residualVector[i] = temp / aMatrixCopy[i, i];
}
//----------------------------------------------------------
// The back substitution is performed and the solution of
// u e = y is calculated. The current correction is stored
// in variable residualVector.
//----------------------------------------------------------
for (i = numberOfEquations - 1; i >= 0; i--)
{
double temp = residualVector[i];
for (int j = i + 1; j < numberOfEquations; j++)
{
temp = temp - aMatrixCopy[i, j] * residualVector[j];
}
residualVector[i] = temp;
}
//----------------------------------------------------------
// The norm of the error in the residuals and the norm of
// the present solution vector are calculated.
//----------------------------------------------------------
double normOfX = 0.0;
double normOfError = 0.0;
for (i = 0; i < numberOfEquations; i++)
{
double test = Math.Abs(xVector[i]);
if (test > normOfX)
{
normOfX = test;
}
test = Math.Abs(residualVector[i]);
if (test > normOfError)
{
normOfError = test;
}
}
//----------------------------------------------------------
// If iteration is zero then skip this section because
// no correction exists on the first iteration.
//----------------------------------------------------------
if (iteration != 0)
{
//------------------------------------------------------
// Test for matrix which is singular to working
// precision.
//------------------------------------------------------
if ((iteration == 1) && (normOfError / normOfX > 0.5))
{
return(LinearEquationSolverStatus.IllConditioned);
}
//------------------------------------------------------
// Check to see if "normOfError" / "normOfX" is greater
// than 2 ^ (1 - t), where "t" is the number of bits
// in the mantissa of a double precision number. If
// this is not true then the last correction is almost
// negligible and "notConvergedFlag" is set to false.
//------------------------------------------------------
notConvergedFlag = normOfError / normOfX >= SSmallFloatingPointValue;
#if DEBUGCODE
double normRatioForDebug = normOfError / normOfX;
#endif
}
//----------------------------------------------------------
// The corrections (residuals) are added to the
// solution vector.
//----------------------------------------------------------
for (i = 0; i < numberOfEquations; i++)
{
xVector[i] = xVector[i] + residualVector[i];
}
//----------------------------------------------------------
// The new residuals corresponding to the solution vector
// are calculated.
//----------------------------------------------------------
for (i = 0; i < numberOfEquations; i++)
{
double temp = bVector[i];
for (int j = 0; j < numberOfEquations; j++)
{
temp = temp - aMatrix[i, j] * xVector[j];
}
residualVector[i] = temp;
}
//----------------------------------------------------------
// The iteration counter is incremented and the flag
// "notConvergedFlag" is tested. If notConvergedFlag is false
// then the solution vector has converged to sufficient
// accuracy.
//----------------------------------------------------------
iteration++;
}while (notConvergedFlag);
return(LinearEquationSolverStatus.Success);
}
/// <summary>
/// This method gets a term in the simultaneous equation. The term
/// can be a number, a variable, or a number and a variable. A term
/// cannot be split between lines input to this method.
/// </summary>
/// <param name="inputLine">The input line to be parsed</param>
/// <param name="positionIndex">The current parse position in the input string.</param>
/// <param name="aMatrix">The A matrix for the simultaneous equations.
/// This is updated as each line of input is parsed.
/// </param>
/// <param name="bVector">The B vector for the simultaneous equations.
/// This is updated as each line of input is parsed.</param>
/// <param name="variableNameIndexMap">A map that stores the integer
/// index for a variable using the variable name as a key.</param>
/// <returns>This method returns the value 'true' if and only if a term is found.
/// </returns>
private bool GetTerm(string inputLine,
ref int positionIndex,
Sparse2DMatrix <int, int, double> aMatrix,
SparseArray <int, double> bVector,
SparseArray <string, int> variableNameIndexMap)
{
//------------------------------------------------------------------
// A term is one of the following three patterns.
//
// <Space> <Sign> Number <Space>
// <Space> <Sign> Variable <Space>
// <Space> <Sign> Number Variable <Space>
//
// Check for a plus or a minus sign at the start of a term.
//------------------------------------------------------------------
bool numberIsNegativeFlag = false;
GetSign(inputLine,
ref positionIndex,
ref numberIsNegativeFlag);
//------------------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
//------------------------------------------------------------------
// Check to see if this is a number or a variable.
//------------------------------------------------------------------
string numberString = "";
bool haveNumberFlag = GetNumber(inputLine,
ref positionIndex,
ref numberString);
//------------------------------------------------------------------
// If an error occurred then abort.
//------------------------------------------------------------------
if (LastStatusValue != LinearEquationParserStatus.Success)
{
return(false);
}
//------------------------------------------------------------------
// If there was a number encountered then test to see if the
// number has an exponent.
//------------------------------------------------------------------
if (haveNumberFlag)
{
if (positionIndex < inputLine.Length)
{
//----------------------------------------------------------
// Does the number have an exponent?
//----------------------------------------------------------
if (inputLine[positionIndex] == '^')
{
positionIndex++;
//------------------------------------------------------
// Does the exponent have a sign.
//------------------------------------------------------
bool negativeExponentFlag = false;
GetSign(inputLine,
ref positionIndex,
ref negativeExponentFlag);
//------------------------------------------------------
// Get the exponent digits.
//------------------------------------------------------
string exponentString = "";
if (GetNumber(inputLine,
ref positionIndex,
ref exponentString))
{
//--------------------------------------------------
// Is the exponent a valid exponent.
//--------------------------------------------------
var exponentLength = exponentString.Length;
if (exponentLength <= 2)
{
var exponent_error_flag = false;
for (var i = 0; i < exponentLength; ++i)
{
if (!char.IsDigit(exponentString[i]))
{
exponent_error_flag = true;
}
}
if (!exponent_error_flag)
{
numberString += 'E';
if (negativeExponentFlag)
{
numberString += '-';
}
numberString += exponentString;
}
else
{
SetLastStatusValue(LinearEquationParserStatus.ErrorIllegalExponent,
positionIndex);
return(false);
}
}
else
{
SetLastStatusValue(LinearEquationParserStatus.ErrorIllegalExponent,
positionIndex);
return(false);
}
}
else
{
SetLastStatusValue(LinearEquationParserStatus.ErrorMissingExponent,
positionIndex);
return(false);
}
}
}
}
//------------------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
string variableName = "";
bool haveVariableNameFlag = GetVariableName(inputLine,
ref positionIndex,
ref variableName);
//------------------------------------------------------------------
// Calculate the sign of the value. The sign is negated
// if the equals sign has been encountered.
//------------------------------------------------------------------
bool negativeFlag =
m_equalSignInEquationFlag ^ m_negativeOperatorFlag ^ numberIsNegativeFlag;
double value = 0.0;
if (haveNumberFlag)
{
value = Convert.ToDouble(numberString);
if (negativeFlag)
{
value = -value;
}
}
else
{
value = 1.0;
if (negativeFlag)
{
value = -value;
}
}
//------------------------------------------------------------------
// If a variable was encountered then add to the aMatrix.
//------------------------------------------------------------------
var haveTermFlag = false;
if (haveVariableNameFlag)
{
//--------------------------------------------------------------
// If either a number or a variable is found then a term was
// found.
//--------------------------------------------------------------
haveTermFlag = true;
//--------------------------------------------------------------
// Each equation must have at least one variable.
// Record that a variable was encountered in this equation.
//--------------------------------------------------------------
m_atLeastOneVariableInEquationFlag = true;
//--------------------------------------------------------------
// If this variable has not been encountered before then add
// the variable to the variableNameIndexMap.
//--------------------------------------------------------------
if (!variableNameIndexMap.TryGetValue(variableName, out int variableNameIndex))
{
// This is a new variable. Add it to the map.
variableNameIndex = variableNameIndexMap.Count;
variableNameIndexMap[variableName] = variableNameIndex;
}
aMatrix[m_equationIndex, variableNameIndex] =
aMatrix[m_equationIndex, variableNameIndex] + value;
}
else if (haveNumberFlag)
{
//--------------------------------------------------------------
// If either a number or a variable is found then a term was
// found.
//--------------------------------------------------------------
haveTermFlag = true;
//--------------------------------------------------------------
// Put the value in the B vector.
//--------------------------------------------------------------
bVector[m_equationIndex] = bVector[m_equationIndex] - value;
}
else
{
haveTermFlag = false;
SetLastStatusValue(LinearEquationParserStatus.ErrorNoTermEncountered,
positionIndex);
return(false);
}
//------------------------------------------------------------------
// There must be at least one term on each side of the equal sign.
//------------------------------------------------------------------
if (m_equalSignInEquationFlag)
{
m_termAfterEqualSignExistsFlag = true;
}
else
{
m_termBeforeEqualSignExistsFlag = true;
}
//------------------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
return(true);
}
/// <summary>
/// Copy constructor.
/// </summary>
/// <param name="sparseArray">The sparse array instance to be copied</param>
public SparseArray(SparseArray <TKey, TValue> sparseArray)
{
_mDictionary = new Dictionary <TKey, TValue>();
Initialize(sparseArray);
DefaultValue = sparseArray.DefaultValue;
}
/// <summary>
/// This function parses line that contains all or part of a simple
/// linear equation. The equation contains terms separated by operators.
/// The term can be a number, a variable, or a number and a variable.
/// A term cannot be split between lines input to the parser method.
/// The operators are either the plus character '+', the minus
/// character '-', or the equal sign character '='. Numbers can have
/// up to 15 digits, a decimal point, and an exponent of a power
/// of 10 that follows the '^' character.
/// </summary>
/// <param name="inputLine">The input line of text to be parsed</param>
/// <param name="aMatrix">The A matrix for the simultaneous equations.
/// This is updated as each line of input is parsed.
/// </param>
/// <param name="bVector">The B vector for the simultaneous equations.
/// This is updated as each line of input is parsed.</param>
/// <param name="variableNameIndexMap">A map that stores the integer
/// index for a variable using the variable name as a key.</param>
/// <param name="numberOfEquations">A reference to an integer that
/// will contain the number of equations when this method exits.
/// </param>
/// <returns>An enum of type LinearEquationParserStatus</returns>
public LinearEquationParserStatus Parse(string inputLine,
Sparse2DMatrix <int, int, double> aMatrix,
SparseArray <int, double> bVector,
SparseArray <string, int> variableNameIndexMap,
ref int numberOfEquations)
{
//------------------------------------------------------------------
// Trim any space characters from the end of the line.
//------------------------------------------------------------------
inputLine.TrimEnd(null);
//------------------------------------------------------------------
// Assume success status.
//------------------------------------------------------------------
int positionIndex = 0;
SetLastStatusValue(LinearEquationParserStatus.Success, positionIndex);
//------------------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
//------------------------------------------------------------------
// Save the position of the first non-whitespace character. If
// the first term is not found at this position then set the
// error status to report that it is likely that the last
// equation was not properly terminated.
//------------------------------------------------------------------
m_startPosition = positionIndex;
//------------------------------------------------------------------
// Separate the input string into tokens.
//
// Variables contains the letters A through Z and the underscore
// '_' character.
//
// Operators include plus '+', minus '-', and times '*'.
//
// Delimiters include the equals sign '='.
//
// Numbers include the digits 0 through 9, the decimal point
// (period character) '.', an optional exponent character which
// is the letter '^', and up to two digits for the optional exponent.
//
// Check for sequences of terms and operators.
//
// Term:
// <Space> <Sign> Number <Space>
// <Space> <Sign> Variable <Space>
// <Space> <Sign> Number Variable <Space>
//
// Operator:
// <Space> Plus <Space>
// <Space> Minus <Space>
// <Space> Equals <Space>
//
//--------------------------------------------------------------
bool operatorFoundLast = false;
while (positionIndex < inputLine.Length)
{
if (m_parserState == LinearEquationParserState.ParseTerm)
{
//------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
if (positionIndex < inputLine.Length)
{
if (GetTerm(inputLine,
ref positionIndex,
aMatrix,
bVector,
variableNameIndexMap))
{
m_parserState = LinearEquationParserState.ParseOperator;
operatorFoundLast = false;
}
else
{
if (LastStatusValue == LinearEquationParserStatus.Success)
{
SetLastStatusValue(LinearEquationParserStatus.ErrorIllegalEquation,
positionIndex);
}
break;
}
}
else
{
break;
}
}
else if (m_parserState == LinearEquationParserState.ParseOperator)
{
//------------------------------------------------------
// Skip whitespace characters
//------------------------------------------------------
SkipSpaces(inputLine, ref positionIndex);
if (positionIndex < inputLine.Length)
{
//------------------------------------------------------
// Check for plus sign, minus sign, or an equal sign.
//------------------------------------------------------
if (GetOperator(inputLine, ref positionIndex))
{
m_parserState = LinearEquationParserState.ParseTerm;
operatorFoundLast = true;
}
else
{
if (LastStatusValue != LinearEquationParserStatus.Success)
{
if (positionIndex == m_startPosition)
{
SetLastStatusValue(LinearEquationParserStatus.ErrorIllegalEquation,
positionIndex);
}
}
break;
}
}
}
}
// If an operator was found at
if ((positionIndex >= inputLine.Length) && (positionIndex > 0) && (!operatorFoundLast))
{
ResetForNewEquation();
numberOfEquations = m_equationIndex;
}
return(LastStatusValue);
}
