class SymTriMatrixR extends MatriR with Error with Serializable
The SymTriMatrixR
class stores and operates on symmetric tridiagonal matrices.
The elements are of type of Real
. A matrix is stored as two vectors:
the diagonal vector and the sub-diagonal vector.
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- SymTriMatrixR
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- MatriR
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Instance Constructors
-
new
SymTriMatrixR(b: MatriR)
Construct a symmetric tridiagonal matrix from the given matrix.
Construct a symmetric tridiagonal matrix from the given matrix.
- b
the matrix of values to assign
-
new
SymTriMatrixR(v1: VectoR, v2: VectoR)
Construct a symmetric tridiagonal matrix with the given diagonal and sub-diagonal.
Construct a symmetric tridiagonal matrix with the given diagonal and sub-diagonal.
- v1
the diagonal vector
- v2
the sub-diagonal vector
-
new
SymTriMatrixR(d1: Int)
- d1
the first/row dimension (symmetric => d2 = d1)
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
def
*(x: Real): SymTriMatrixR
Multiply 'this' tridiagonal matrix by scalar 'x'.
Multiply 'this' tridiagonal matrix by scalar 'x'.
- x
the scalar to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
*(u: VectoR): VectorR
Multiply 'this' tridiagonal matrix by vector 'u'.
Multiply 'this' tridiagonal matrix by vector 'u'.
- u
the vector to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
*(b: SymTriMatrixR): MatrixR
Multiply 'this' tridiagonal matrix by matrix 'b'.
Multiply 'this' tridiagonal matrix by matrix 'b'. Requires 'b' to have type
SymTriMatrixR
, but returns a more general type of matrix.- b
the matrix to multiply by
-
def
*(b: MatriR): SymTriMatrixR
Multiply 'this' tridiagonal matrix by matrix 'b'.
Multiply 'this' tridiagonal matrix by matrix 'b'.
- b
the matrix to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
**(u: VectoR): MatrixR
Multiply 'this' tridiagonal matrix by vector 'u' to produce another matrix 'a_ij * u_j'.
Multiply 'this' tridiagonal matrix by vector 'u' to produce another matrix 'a_ij * u_j'. E.g., multiply a diagonal matrix represented as a vector by a matrix.
- u
the vector to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
**:(u: VectoR): MatrixR
Multiply vector 'u' by 'this' tridiagonal matrix to produce another matrix 'u_i * a_ij'.
Multiply vector 'u' by 'this' tridiagonal matrix to produce another matrix 'u_i * a_ij'. E.g., multiply a diagonal matrix represented as a vector by a matrix. This operator is right associative.
- u
the vector to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
**=(u: VectoR): MatrixR
Multiply in-place 'this' tridiagonal matrix by vector 'u' to produce another matrix 'a_ij * u_j'.
Multiply in-place 'this' tridiagonal matrix by vector 'u' to produce another matrix 'a_ij * u_j'. E.g., multiply a diagonal matrix represented as a vector by a matrix.
- u
the vector to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
*:(u: VectoR): VectoR
Multiply (row) vector 'u' by 'this' matrix.
Multiply (row) vector 'u' by 'this' matrix. Note '*:' is right associative. vector = vector *: matrix
- u
the vector to multiply by
- Definition Classes
- MatriR
-
def
*=(x: Real): SymTriMatrixR
Multiply in-place 'this' tridiagonal matrix by scalar 'x'.
Multiply in-place 'this' tridiagonal matrix by scalar 'x'.
- x
the scalar to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
*=(b: MatriR): SymTriMatrixR
Multiply in-place 'this' tridiagonal matrix by matrix 'b'.
Multiply in-place 'this' tridiagonal matrix by matrix 'b'.
- b
the matrix to multiply by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+(x: Real): SymTriMatrixR
Add 'this' tridiagonal matrix and scalar 'x'.
Add 'this' tridiagonal matrix and scalar 'x'.
- x
the scalar to add
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+(u: VectoR): MatrixR
Add 'this' tridiagonal matrix and (row) vector 'u'.
Add 'this' tridiagonal matrix and (row) vector 'u'.
- u
the vector to add
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+(b: MatriR): SymTriMatrixR
Add 'this' tridiagonal matrix and matrix 'b'.
Add 'this' tridiagonal matrix and matrix 'b'.
- b
the matrix to add (requires 'leDimensions')
- Definition Classes
- SymTriMatrixR → MatriR
-
def
++(b: MatriR): SymTriMatrixR
Concatenate (row-wise) 'this' matrix and matrix 'b'.
Concatenate (row-wise) 'this' matrix and matrix 'b'.
- b
the matrix to be concatenated as the new last rows in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
def
++^(b: MatriR): SymTriMatrixR
Concatenate (column-wise) 'this' matrix and matrix 'b'.
Concatenate (column-wise) 'this' matrix and matrix 'b'.
- b
the matrix to be concatenated as the new last columns in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+:(u: VectoR): SymTriMatrixR
Concatenate (row) vector 'u' and 'this' matrix, i.e., prepend 'u' to 'this'.
Concatenate (row) vector 'u' and 'this' matrix, i.e., prepend 'u' to 'this'.
- u
the vector to be prepended as the new first row in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+=(x: Real): SymTriMatrixR
Add in-place 'this' tridiagonal matrix and scalar 'x'.
Add in-place 'this' tridiagonal matrix and scalar 'x'.
- x
the scalar to add
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+=(u: VectoR): MatrixR
Add in-place 'this' tridiagonal matrix and (row) vector 'u'.
Add in-place 'this' tridiagonal matrix and (row) vector 'u'.
- u
the vector to add
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+=(b: MatriR): SymTriMatrixR
Add in-place 'this' tridiagonal matrix and matrix 'b'.
Add in-place 'this' tridiagonal matrix and matrix 'b'.
- b
the matrix to add (requires 'leDimensions')
- Definition Classes
- SymTriMatrixR → MatriR
-
def
+^:(u: VectoR): SymTriMatrixR
Concatenate (column) vector 'u' and 'this' matrix, i.e., prepend 'u' to 'this'.
Concatenate (column) vector 'u' and 'this' matrix, i.e., prepend 'u' to 'this'.
- u
the vector to be prepended as the new first column in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-(x: Real): SymTriMatrixR
From 'this' tridiagonal matrix subtract scalar 'x'.
From 'this' tridiagonal matrix subtract scalar 'x'.
- x
the scalar to subtract
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-(u: VectoR): MatrixR
From 'this' tridiagonal matrix subtract (row) vector 'u'.
From 'this' tridiagonal matrix subtract (row) vector 'u'.
- u
the vector to subtract
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-(b: MatriR): SymTriMatrixR
From 'this' tridiagonal matrix subtract matrix 'b'.
From 'this' tridiagonal matrix subtract matrix 'b'.
- b
the matrix to subtract (requires 'leDimensions')
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-=(x: Real): SymTriMatrixR
From 'this' tridiagonal matrix subtract in-place scalar 'x'.
From 'this' tridiagonal matrix subtract in-place scalar 'x'.
- x
the scalar to subtract
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-=(u: VectoR): MatrixR
From 'this' tridiagonal matrix subtract in-place (row) vector 'u'.
From 'this' tridiagonal matrix subtract in-place (row) vector 'u'.
- u
the vector to subtract
- Definition Classes
- SymTriMatrixR → MatriR
-
def
-=(b: MatriR): SymTriMatrixR
From 'this' tridiagonal matrix subtract in-place matrix 'b'.
From 'this' tridiagonal matrix subtract in-place matrix 'b'.
- b
the matrix to subtract (requires 'leDimensions')
- Definition Classes
- SymTriMatrixR → MatriR
-
def
/(x: Real): SymTriMatrixR
Divide 'this' tridiagonal matrix by scalar 'x'.
Divide 'this' tridiagonal matrix by scalar 'x'.
- x
the scalar to divide by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
/=(x: Real): SymTriMatrixR
Divide in-place 'this' tridiagonal matrix by scalar 'x'.
Divide in-place 'this' tridiagonal matrix by scalar 'x'.
- x
the scalar to divide by
- Definition Classes
- SymTriMatrixR → MatriR
-
def
:+(u: VectoR): SymTriMatrixR
Concatenate 'this' matrix and (row) vector 'u', i.e., append 'u' to 'this'.
Concatenate 'this' matrix and (row) vector 'u', i.e., append 'u' to 'this'.
- u
the vector to be appended as the new last row in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
def
:^+(u: VectoR): SymTriMatrixR
Concatenate 'this' matrix and (column) vector 'u', i.e., append 'u' to 'this'.
Concatenate 'this' matrix and (column) vector 'u', i.e., append 'u' to 'this'.
- u
the vector to be appended as the new last column in new matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
apply(ir: Range, jr: Range): SymTriMatrixR
Get a slice 'this' tridiagonal matrix row-wise on range 'ir' and column-wise on range 'jr'.
Get a slice 'this' tridiagonal matrix row-wise on range 'ir' and column-wise on range 'jr'. Ex: b = a(2..4, 3..5)
- ir
the row range
- jr
the column range
- Definition Classes
- SymTriMatrixR → MatriR
-
def
apply(i: Int): VectorR
Get 'this' tridiagonal matrix's vector at the 'i'-th index position ('i'-th row).
Get 'this' tridiagonal matrix's vector at the 'i'-th index position ('i'-th row).
- i
the row index
- Definition Classes
- SymTriMatrixR → MatriR
-
def
apply(i: Int, j: Int): Real
Get 'this' tridiagonal matrix's element at the 'i,j'-th index position.
Get 'this' tridiagonal matrix's element at the 'i,j'-th index position.
- i
the row index
- j
the column index
- Definition Classes
- SymTriMatrixR → MatriR
-
def
apply(i: Int, jr: Range): VectoR
Get a slice 'this' matrix row-wise at index 'i' and column-wise on range 'jr'.
Get a slice 'this' matrix row-wise at index 'i' and column-wise on range 'jr'. Ex: u = a(2, 3..5)
- i
the row index
- jr
the column range
- Definition Classes
- MatriR
-
def
apply(ir: Range, j: Int): VectoR
Get a slice 'this' matrix row-wise on range 'ir' and column-wise at index j.
Get a slice 'this' matrix row-wise on range 'ir' and column-wise at index j. Ex: u = a(2..4, 3)
- ir
the row range
- j
the column index
- Definition Classes
- MatriR
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
-
def
at(i: Int, j: Int): Real
Get 'this' tridiagonal matrix's element at the 'i,j'-th index position, returning 0, if off tridiagonal.
Get 'this' tridiagonal matrix's element at the 'i,j'-th index position, returning 0, if off tridiagonal.
- i
the row index
- j
the column index
-
def
bsolve(y: VectoR): VectorR
Solve for 'x' using back substitution in the equation 'u*x = y' where 'this' matrix ('u') is upper triangular (see 'lud' above).
Solve for 'x' using back substitution in the equation 'u*x = y' where 'this' matrix ('u') is upper triangular (see 'lud' above).
- y
the constant vector
- Definition Classes
- SymTriMatrixR → MatriR
-
def
clean(thres: Double, relative: Boolean = true): SymTriMatrixR
Clean values in 'this' tridiagonal matrix at or below the threshold by setting them to zero.
Clean values in 'this' tridiagonal matrix at or below the threshold by setting them to zero. Iterative algorithms give approximate values and if very close to zero, may throw off other calculations, e.g., in computing eigenvectors.
- thres
the cutoff threshold (a small value)
- relative
whether to use relative or absolute cutoff
- Definition Classes
- SymTriMatrixR → MatriR
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
def
col(col: Int, from: Int = 0): VectorR
Get column 'col' from 'this' tridiagonal matrix, returning it as a vector.
Get column 'col' from 'this' tridiagonal matrix, returning it as a vector.
- col
the column to extract from the matrix
- from
the position to start extracting from
- Definition Classes
- SymTriMatrixR → MatriR
-
def
copy(): SymTriMatrixR
Create an exact copy of 'this' m-by-n symmetric tridiagonal matrix.
Create an exact copy of 'this' m-by-n symmetric tridiagonal matrix.
- Definition Classes
- SymTriMatrixR → MatriR
- val d1: Int
-
def
det: Real
Compute the determinant of 'this' tridiagonal matrix.
Compute the determinant of 'this' tridiagonal matrix.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
dg: VectorR
Get the diagonal of 'this' tridiagonal matrix.
-
def
dg_(v: VectorR): Unit
Set the diagonal of 'this' tridiagonal matrix.
Set the diagonal of 'this' tridiagonal matrix.
- v
the vector to assign to the diagonal
-
def
diag(p: Int, q: Int): SymTriMatrixR
Form a matrix '[Ip, this, Iq]' where Ir is a r-by-r identity matrix, by positioning the three matrices 'Ip', 'this' and 'Iq' along the diagonal.
Form a matrix '[Ip, this, Iq]' where Ir is a r-by-r identity matrix, by positioning the three matrices 'Ip', 'this' and 'Iq' along the diagonal. Fill the rest of matrix with zeros.
- p
the size of identity matrix Ip
- q
the size of identity matrix Iq
- Definition Classes
- SymTriMatrixR → MatriR
-
def
diag(b: MatriR): MatriR
Combine 'this' tridiagonal matrix with matrix 'b', placing them along the diagonal and filling in the bottom left and top right regions with zeros: '[this, b]'.
Combine 'this' tridiagonal matrix with matrix 'b', placing them along the diagonal and filling in the bottom left and top right regions with zeros: '[this, b]'.
- b
the matrix to combine with 'this' tridiagonal matrix
- Definition Classes
- SymTriMatrixR → MatriR
-
lazy val
dim1: Int
Dimension 1
Dimension 1
- Definition Classes
- SymTriMatrixR → MatriR
-
lazy val
dim2: Int
Dimension 2
Dimension 2
- Definition Classes
- SymTriMatrixR → MatriR
-
def
dot(b: MatriR): VectorR
Compute the dot product of 'this' matrix with matrix 'b' to produce a vector.
Compute the dot product of 'this' matrix with matrix 'b' to produce a vector.
- b
the second matrix of the dot product
- Definition Classes
- SymTriMatrixR → MatriR
-
def
dot(b: SymTriMatrixR): VectorR
Compute the dot product of 'this' matrix with matrix 'b' to produce a vector.
Compute the dot product of 'this' matrix with matrix 'b' to produce a vector.
- b
the second matrix of the dot product
-
def
dot(u: VectoR): VectorR
Compute the dot product of 'this' tridiagonal matrix and vector 'u', by first transposing 'this' tridiagonal matrix and then multiplying by 'u' (i.e., 'a dot u = a.t * u').
Compute the dot product of 'this' tridiagonal matrix and vector 'u', by first transposing 'this' tridiagonal matrix and then multiplying by 'u' (i.e., 'a dot u = a.t * u'). Since 'this' is symmetric, the result is the same as 'a * u'.
- u
the vector to multiply by (requires same first dimensions)
- Definition Classes
- SymTriMatrixR → MatriR
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
equals(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
val
fString: String
Format string used for printing vector values (change using 'setFormat')
Format string used for printing vector values (change using 'setFormat')
- Attributes
- protected
- Definition Classes
- MatriR
-
def
finalize(): Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
final
def
flaw(method: String, message: String): Unit
Show the flaw by printing the error message.
Show the flaw by printing the error message.
- method
the method where the error occurred
- message
the error message
- Definition Classes
- Error
-
def
foreach[U](f: (Array[Real]) ⇒ U): Unit
Iterate over 'this' matrix row by row applying method 'f'.
Iterate over 'this' matrix row by row applying method 'f'.
- f
the function to apply
- Definition Classes
- MatriR
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
-
def
getDiag(k: Int = 0): VectorR
Get the 'k'th diagonal of 'this' tridiagonal matrix.
Get the 'k'th diagonal of 'this' tridiagonal matrix. Assumes 'dim2 >= dim1'.
- k
how far above the main diagonal, e.g., (-1, 0, 1) for (sub, main, super)
- Definition Classes
- SymTriMatrixR → MatriR
-
def
hashCode(): Int
- Definition Classes
- AnyRef → Any
-
def
inverse: MatriR
Invert 'this' tridiagonal matrix.
Invert 'this' tridiagonal matrix.
- Definition Classes
- SymTriMatrixR → MatriR
- See also
www.amm.shu.edu.cn/EN/article/downloadArticleFile.do?attachType=PDF&id=4339
-
def
inverse_ip(): SymTriMatrixR
Invert in-place 'this' matrix (requires a 'squareMatrix') and use partial pivoting.
Invert in-place 'this' matrix (requires a 'squareMatrix') and use partial pivoting.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
isBidiagonal: Boolean
Check whether 'this' matrix is bidiagonal (has non-zero elements only in main diagonal and super-diagonal).
Check whether 'this' matrix is bidiagonal (has non-zero elements only in main diagonal and super-diagonal). The method may be overriding for efficiency.
- Definition Classes
- MatriR
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
def
isNonnegative: Boolean
Check whether 'this' tridiagonal matrix is nonnegative (has no negative elements).
Check whether 'this' tridiagonal matrix is nonnegative (has no negative elements).
- Definition Classes
- SymTriMatrixR → MatriR
-
def
isRectangular: Boolean
Check whether 'this' tridiagonal matrix is rectangular (all rows have the same number of columns).
Check whether 'this' tridiagonal matrix is rectangular (all rows have the same number of columns).
- Definition Classes
- SymTriMatrixR → MatriR
-
def
isSquare: Boolean
Check whether 'this' matrix is square (same row and column dimensions).
Check whether 'this' matrix is square (same row and column dimensions).
- Definition Classes
- MatriR
-
def
isSymmetric: Boolean
Check whether 'this' tridiagonal matrix is symmetric.
Check whether 'this' tridiagonal matrix is symmetric.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
isTridiagonal: Boolean
Check whether 'this' tridiagonal matrix is bidiagonal (has non-zero elements only in main diagonal and super-diagonal).
Check whether 'this' tridiagonal matrix is bidiagonal (has non-zero elements only in main diagonal and super-diagonal). The method may be overriding for efficiency.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
leDimensions(b: MatriR): Boolean
Check whether 'this' matrix dimensions are less than or equal to 'le' those of the other matrix 'b'.
Check whether 'this' matrix dimensions are less than or equal to 'le' those of the other matrix 'b'.
- b
the other matrix
- Definition Classes
- MatriR
-
def
lowerT: SymTriMatrixR
Return the lower triangular of 'this' matrix (rest are zero).
Return the lower triangular of 'this' matrix (rest are zero).
- Definition Classes
- SymTriMatrixR → MatriR
-
def
lud_ip(): (MatriR, MatriR)
Factor in-place 'this' matrix into the product of lower and upper triangular matrices '(l, u)' using the 'LU' Decomposition algorithm.
Factor in-place 'this' matrix into the product of lower and upper triangular matrices '(l, u)' using the 'LU' Decomposition algorithm.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
lud_npp: (MatriR, MatriR)
Factor 'this' tridiagonal matrix into the product of lower and upper triangular matrices '(l, u)' using the 'LU' Factorization algorithm.
Factor 'this' tridiagonal matrix into the product of lower and upper triangular matrices '(l, u)' using the 'LU' Factorization algorithm. 'l' is lower bidiagonal and 'u' is upper bidiagonal. FIX: would be more efficient to use tridiagonal matrices than dense matrices.
- Definition Classes
- SymTriMatrixR → MatriR
- See also
www.webpages.uidaho.edu/~barannyk/Teaching/LU_factorization_tridiagonal.pdf
-
def
mag: Real
Find the magnitude of 'this' matrix, the element value farthest from zero.
Find the magnitude of 'this' matrix, the element value farthest from zero.
- Definition Classes
- MatriR
-
def
max(e: Int = dim1): Real
Find the maximum element in 'this' tridiagonal matrix.
Find the maximum element in 'this' tridiagonal matrix.
- e
the ending row index (exclusive) for the search
- Definition Classes
- SymTriMatrixR → MatriR
-
def
mdot(b: MatriR): MatrixR
Compute the matrix dot product of 'this' matrix with matrix 'b' to produce a matrix.
Compute the matrix dot product of 'this' matrix with matrix 'b' to produce a matrix.
- b
the second matrix of the dot product
- Definition Classes
- SymTriMatrixR → MatriR
-
def
mdot(b: SymTriMatrixR): MatrixR
Compute the matrix dot product of 'this' matrix with matrix 'b' to produce a matrix.
Compute the matrix dot product of 'this' matrix with matrix 'b' to produce a matrix.
- b
the second matrix of the dot product
-
def
mean: VectoR
Compute the column means of this matrix.
Compute the column means of this matrix.
- Definition Classes
- MatriR
-
def
min(e: Int = dim1): Real
Find the minimum element in 'this' tridiagonal matrix.
Find the minimum element in 'this' tridiagonal matrix.
- e
the ending row index (exclusive) for the search
- Definition Classes
- SymTriMatrixR → MatriR
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
norm1: Real
Compute the 1-norm of 'this' matrix, i.e., the maximum 1-norm of the column vectors.
Compute the 1-norm of 'this' matrix, i.e., the maximum 1-norm of the column vectors. This is useful for comparing matrices '(a - b).norm1'.
- Definition Classes
- MatriR
-
final
def
notify(): Unit
- Definition Classes
- AnyRef
-
final
def
notifyAll(): Unit
- Definition Classes
- AnyRef
-
def
nullspace: VectorR
Compute the (right) nullspace of 'this' 'm-by-n' matrix (requires 'n = m+1') by performing Gauss-Jordan reduction and extracting the negation of the last column augmented by 1.
Compute the (right) nullspace of 'this' 'm-by-n' matrix (requires 'n = m+1') by performing Gauss-Jordan reduction and extracting the negation of the last column augmented by 1.
nullspace (a) = set of orthogonal vectors v s.t. a * v = 0
The left nullspace of matrix 'a' is the same as the right nullspace of 'a.t'. FIX: need a more robust algorithm for computing nullspace (@see Fac_QR.scala). FIX: remove the 'n = m+1' restriction.
- Definition Classes
- SymTriMatrixR → MatriR
- See also
/solving-ax-0-pivot-variables-special-solutions/MIT18_06SCF11_Ses1.7sum.pdf
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces
-
def
nullspace_ip(): VectorR
Compute in-place the (right) nullspace of 'this' 'm-by-n' matrix (requires 'n = m+1') by performing Gauss-Jordan reduction and extracting the negation of the last column augmented by 1.
Compute in-place the (right) nullspace of 'this' 'm-by-n' matrix (requires 'n = m+1') by performing Gauss-Jordan reduction and extracting the negation of the last column augmented by 1.
nullspace (a) = set of orthogonal vectors v s.t. a * v = 0
The left nullspace of matrix 'a' is the same as the right nullspace of 'a.t'. FIX: need a more robust algorithm for computing nullspace (@see Fac_QR.scala). FIX: remove the 'n = m+1' restriction.
- Definition Classes
- SymTriMatrixR → MatriR
- See also
/solving-ax-0-pivot-variables-special-solutions/MIT18_06SCF11_Ses1.7sum.pdf
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces
-
val
range1: Range
Range for the storage array on dimension 1 (rows)
Range for the storage array on dimension 1 (rows)
- Definition Classes
- MatriR
-
val
range2: Range
Range for the storage array on dimension 2 (columns)
Range for the storage array on dimension 2 (columns)
- Definition Classes
- MatriR
-
def
reduce: SymTriMatrixR
Use Gauss-Jordan reduction on 'this' matrix to make the left part embed an identity matrix.
Use Gauss-Jordan reduction on 'this' matrix to make the left part embed an identity matrix. A constraint on 'this' m by n matrix is that n >= m.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
reduce_ip(): Unit
Use Gauss-Jordan reduction in-place on 'this' matrix to make the left part embed an identity matrix.
Use Gauss-Jordan reduction in-place on 'this' matrix to make the left part embed an identity matrix. A constraint on 'this' m by n matrix is that n >= m.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
sameCrossDimensions(b: MatriR): Boolean
Check whether 'this' matrix and the other matrix 'b' have the same cross dimensions.
Check whether 'this' matrix and the other matrix 'b' have the same cross dimensions.
- b
the other matrix
- Definition Classes
- MatriR
-
def
sameDimensions(b: MatriR): Boolean
Check whether 'this' matrix and the other matrix 'b' have the same dimensions.
Check whether 'this' matrix and the other matrix 'b' have the same dimensions.
- b
the other matrix
- Definition Classes
- MatriR
-
def
sd: VectorR
Get the sub-diagonal of 'this' tridiagonal matrix.
-
def
sd_(v: VectorR): Unit
Set the sub-diagonal of 'this' tridiagonal matrix.
Set the sub-diagonal of 'this' tridiagonal matrix.
- v
the vector to assign to the sub-diagonal
-
def
selectCols(colIndex: Array[Int]): SymTriMatrixR
Select columns from 'this' tridiagonal matrix according to the given index/basis.
Select columns from 'this' tridiagonal matrix according to the given index/basis. Ex: Can be used to divide a matrix into a basis and a non-basis.
- colIndex
the column index positions (e.g., (0, 2, 5))
- Definition Classes
- SymTriMatrixR → MatriR
-
def
selectRows(rowIndex: Array[Int]): SymTriMatrixR
Select rows from 'this' tridiagonal matrix according to the given index/basis.
Select rows from 'this' tridiagonal matrix according to the given index/basis.
- rowIndex
the row index positions (e.g., (0, 2, 5))
- Definition Classes
- SymTriMatrixR → MatriR
-
def
set(i: Int, u: VectoR, j: Int = 0): Unit
Set 'this' tridiagonal matrix's 'i'th row starting at column 'j' to the vector 'u'.
Set 'this' tridiagonal matrix's 'i'th row starting at column 'j' to the vector 'u'.
- i
the row index
- u
the vector value to assign
- j
the starting column index
- Definition Classes
- SymTriMatrixR → MatriR
-
def
set(u: Array[Array[Real]]): Unit
Set all the values in 'this' tridiagonal matrix as copies of the values in 2D array 'u'.
Set all the values in 'this' tridiagonal matrix as copies of the values in 2D array 'u'. Ignore parts of array not corresponding to tridiagonal.
- u
the 2D array of values to assign
- Definition Classes
- SymTriMatrixR → MatriR
-
def
set(x: Real): Unit
Set all the elements in 'this' tridiagonal matrix to the scalar 'x'.
Set all the elements in 'this' tridiagonal matrix to the scalar 'x'.
- x
the scalar value to assign
- Definition Classes
- SymTriMatrixR → MatriR
-
def
setCol(col: Int, u: VectoR): Unit
Set column 'col' of 'this' tridiagonal matrix to a vector.
Set column 'col' of 'this' tridiagonal matrix to a vector.
- col
the column to set
- u
the vector to assign to the column
- Definition Classes
- SymTriMatrixR → MatriR
-
def
setDiag(x: Real): Unit
Set the main diagonal of 'this' tridiagonal matrix to the scalar 'x'.
Set the main diagonal of 'this' tridiagonal matrix to the scalar 'x'. Assumes 'dim2 >= dim1'.
- x
the scalar to set the diagonal to
- Definition Classes
- SymTriMatrixR → MatriR
-
def
setDiag(u: VectoR, k: Int = 0): Unit
Set the 'k'th diagonal of 'this' tridiagonal matrix to the vector 'u'.
Set the 'k'th diagonal of 'this' tridiagonal matrix to the vector 'u'. Assumes 'dim2 >= dim1'.
- u
the vector to set the diagonal to
- k
how far above the main diagonal, e.g., (-1, 0, 1) for (sub, main, super)
- Definition Classes
- SymTriMatrixR → MatriR
-
def
setFormat(newFormat: String): Unit
Set the format to the 'newFormat'.
-
def
slice(r_from: Int, r_end: Int, c_from: Int, c_end: Int): SymTriMatrixR
Slice 'this' tridiagonal matrix row-wise 'r_from' to 'r_end' and column-wise 'c_from' to 'c_end'.
Slice 'this' tridiagonal matrix row-wise 'r_from' to 'r_end' and column-wise 'c_from' to 'c_end'.
- r_from
the start of the row slice
- r_end
the end of the row slice
- c_from
the start of the column slice
- c_end
the end of the column slice
- Definition Classes
- SymTriMatrixR → MatriR
-
def
slice(from: Int, end: Int): SymTriMatrixR
Slice 'this' tridiagonal matrix row-wise 'from' to 'end'.
Slice 'this' tridiagonal matrix row-wise 'from' to 'end'.
- from
the start row of the slice (inclusive)
- end
the end row of the slice (exclusive)
- Definition Classes
- SymTriMatrixR → MatriR
-
def
sliceCol(from: Int, end: Int): SymTriMatrixR
Slice 'this' tridiagonal matrix column-wise 'from' to 'end'.
Slice 'this' tridiagonal matrix column-wise 'from' to 'end'.
- from
the start column of the slice (inclusive)
- end
the end column of the slice (exclusive)
- Definition Classes
- SymTriMatrixR → MatriR
-
def
sliceExclude(row: Int, col: Int): SymTriMatrixR
Slice 'this' tridiagonal matrix excluding the given 'row' and 'col'umn.
Slice 'this' tridiagonal matrix excluding the given 'row' and 'col'umn.
- row
the row to exclude
- col
the column to exclude
- Definition Classes
- SymTriMatrixR → MatriR
-
def
solve(b: VectoR): VectoR
Solve for 'x' in the equation 'a*x = b' where 'a' is 'this' tridiagonal matrix, using the Thomas Algorithm.
Solve for 'x' in the equation 'a*x = b' where 'a' is 'this' tridiagonal matrix, using the Thomas Algorithm. Caveat: Stability vs. diagonal dominance. This method is more efficient, since a 'lud_npp' creates dense matrices.
- b
the constant vector
- Definition Classes
- SymTriMatrixR → MatriR
- See also
en.wikibooks.org/wiki/Algorithm_Implementation/Linear_Algebra/Tridiagonal_matrix_algorithm
-
def
solve(l: MatriR, u: MatriR, b: VectoR): VectoR
Solve for 'x' in the equation 'a*x = l*u*x = b' (see 'lud_npp' above).
Solve for 'x' in the equation 'a*x = l*u*x = b' (see 'lud_npp' above).
- l
the lower triangular matrix
- u
the upper triangular matrix
- b
the constant vector
- Definition Classes
- SymTriMatrixR → MatriR
-
def
solve(lu: (MatriR, MatriR), b: VectoR): VectoR
Solve for 'x' in the equation 'l*u*x = b' (see 'lud' above).
Solve for 'x' in the equation 'l*u*x = b' (see 'lud' above).
- lu
the lower and upper triangular matrices
- b
the constant vector
- Definition Classes
- MatriR
-
def
sum: Real
Compute the sum of 'this' tridiagonal matrix, i.e., the sum of its elements.
Compute the sum of 'this' tridiagonal matrix, i.e., the sum of its elements.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
sumAbs: Real
Compute the 'abs' sum of 'this' tridiagonal matrix, i.e., the sum of the absolute value of its elements.
Compute the 'abs' sum of 'this' tridiagonal matrix, i.e., the sum of the absolute value of its elements. This is useful for comparing matrices '(a - b).sumAbs'.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
sumLower: Real
Compute the sum of the lower triangular region of 'this' tridiagonal matrix.
Compute the sum of the lower triangular region of 'this' tridiagonal matrix.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
swap(i: Int, k: Int, col: Int = 0): Unit
Swap the elements in rows 'i' and 'k' starting from column 'col'.
Swap the elements in rows 'i' and 'k' starting from column 'col'.
- i
the first row in the swap
- k
the second row in the swap
- col
the starting column for the swap (default 0 => whole row)
- Definition Classes
- MatriR
-
def
swapCol(j: Int, l: Int, row: Int = 0): Unit
Swap the elements in columns 'j' and 'l' starting from row 'row'.
Swap the elements in columns 'j' and 'l' starting from row 'row'.
- j
the first column in the swap
- l
the second column in the swap
- row
the starting row for the swap (default 0 => whole column)
- Definition Classes
- MatriR
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
t: SymTriMatrixR
Transpose 'this' tridiagonal matrix (rows => columns).
Transpose 'this' tridiagonal matrix (rows => columns). Note, since the matrix is symmetric, it returns itself.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
toDense: MatrixR
Convert 'this' tridiagonal matrix to a dense matrix.
Convert 'this' tridiagonal matrix to a dense matrix.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
toInt: SymTriMatrixI
Convert 'this'
SymTriMatrixR
into a SymTriMatrixI.
Convert 'this'
SymTriMatrixR
into a SymTriMatrixI.
- Definition Classes
- SymTriMatrixR → MatriR
-
def
toString(): String
Convert 'this' tridiagonal matrix to a string showing the diagonal vector followed by the sub-diagonal vector.
Convert 'this' tridiagonal matrix to a string showing the diagonal vector followed by the sub-diagonal vector.
- Definition Classes
- SymTriMatrixR → AnyRef → Any
-
def
trace: Real
Compute the trace of 'this' tridiagonal matrix, i.e., the sum of the elements on the main diagonal.
Compute the trace of 'this' tridiagonal matrix, i.e., the sum of the elements on the main diagonal. Should also equal the sum of the eigenvalues.
- Definition Classes
- SymTriMatrixR → MatriR
- See also
Eigen.scala
-
def
update(ir: Range, jr: Range, b: MatriR): Unit
Set a slice 'this' tridiagonal matrix row-wise on range 'ir' and column-wise on range 'jr'.
Set a slice 'this' tridiagonal matrix row-wise on range 'ir' and column-wise on range 'jr'. Ex: a(2..4, 3..5) = b
- ir
the row range
- jr
the column range
- b
the matrix to assign
- Definition Classes
- SymTriMatrixR → MatriR
-
def
update(i: Int, u: VectoR): Unit
Set 'this' tridiagonal matrix's row at the 'i'-th index position to the vector 'u'.
Set 'this' tridiagonal matrix's row at the 'i'-th index position to the vector 'u'.
- i
the row index
- u
the vector value to assign
- Definition Classes
- SymTriMatrixR → MatriR
-
def
update(i: Int, j: Int, x: Real): Unit
Set 'this' tridiagonal matrix's element at the 'i,j'-th index position to the scalar 'x'.
Set 'this' tridiagonal matrix's element at the 'i,j'-th index position to the scalar 'x'.
- i
the row index
- j
the column index
- x
the scalar value to assign
- Definition Classes
- SymTriMatrixR → MatriR
-
def
update(i: Int, jr: Range, u: VectoR): Unit
Set a slice of 'this' matrix row-wise at index 'i' and column-wise on range 'jr' to vector 'u'.
Set a slice of 'this' matrix row-wise at index 'i' and column-wise on range 'jr' to vector 'u'. Ex: a(2, 3..5) = u
- i
the row index
- jr
the column range
- u
the vector to assign
- Definition Classes
- MatriR
-
def
update(ir: Range, j: Int, u: VectoR): Unit
Set a slice of 'this' matrix row-wise on range 'ir' and column-wise at index 'j' to vector 'u'.
Set a slice of 'this' matrix row-wise on range 'ir' and column-wise at index 'j' to vector 'u'. Ex: a(2..4, 3) = u
- ir
the row range
- j
the column index
- u
the vector to assign
- Definition Classes
- MatriR
-
def
upperT: SymTriMatrixR
Return the upper triangular of 'this' matrix (rest are zero).
Return the upper triangular of 'this' matrix (rest are zero).
- Definition Classes
- SymTriMatrixR → MatriR
-
final
def
wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
def
write(fileName: String): Unit
Write 'this' tridiagonal matrix to a CSV-formatted text file with name 'fileName'.
Write 'this' tridiagonal matrix to a CSV-formatted text file with name 'fileName'.
- fileName
the name of file to hold the data
- Definition Classes
- SymTriMatrixR → MatriR
-
def
zero(m: Int = dim1, n: Int = dim2): SymTriMatrixR
Create an m-by-n symmetric tridiagonal matrix with all elements initialized to zero.
Create an m-by-n symmetric tridiagonal matrix with all elements initialized to zero.
- m
the number of rows
- n
the number of columns
- Definition Classes
- SymTriMatrixR → MatriR
-
def
~^(p: Int): SymTriMatrixR
Raise 'this' tridiagonal matrix to the 'p'th power (for some integer 'p' >= 2).
Raise 'this' tridiagonal matrix to the 'p'th power (for some integer 'p' >= 2).
- p
the power to raise this tridiagonal matrix to
- Definition Classes
- SymTriMatrixR → MatriR