Matrices: High Performance¶

CalcpadCE offers a parallel high-performance matrix mode for the larger linear-algebra workloads.

Wrap any matrix with hp([...]) (or use the hp creational helpers) and the engine switches to a packed numerical layout with SIMD-friendly element-wise operations, dropping per-element interpretation overhead — the gain scales with matrix size and is most visible on dense products, decompositions and reductions.

The six worksheets in this group are direct counterparts of the regular Matrices group, rewritten in HP form: literal construction and indexing, creational helpers, row- and column-wise data toolkit, linear-algebra primitives, structural plumbing, and the matrix and matrix–vector operators. The semantics match the plain mode one-to-one, so you can drop hp(...) around an existing pipeline and trade only the storage layout.

Matrix Initialization and Indexing¶

HP foundations: literal matrices wrapped with $hp(...)$, $isHp$ checks, indexing, slicing and submatrix assignment on packed numerical storage.

Code:

'Direct:
A = hp([1|2; 3|4; 5; 6|7; 8])','isHp(A)
'Direct with vectors:
a = [1; 2; 4]
A = hp([a|2*a + 1|3*a + 2])','isHp(A)
'From a custom function (with vector arguments):
A(x) = transp(hp([x^0|x|x^2|x^3|x^4]))
x = hp([1; 2; 3; 4])
A = A(x)','isHp(A)

"Indexing
'<hr/>
A.(3; 2)'- simple dot operator (constant index)
i = 2', 'j = 3
A.(2*i - 1; j + 1)'- expressions with variables
A.(i; j) = 0' - assign a new value to an element:'A
'Initalize values in a block loop
x = hp([1; 2; 3; 4])
A = matrix_hp(len(x); 7)
#hide
#for i = 1 : n_rows(A)
    #for j = 1 : n_cols(A)
        A.(i; j) = x.i^(j - 1)
    #loop
#loop
#show
A
'Inline loop
B = matrix_hp(len(x); 7)
$Repeat{$Repeat{B.(i; j) = x.i^(j - 1) @ j = 1 : n_cols(B)} @ i = 1 : n_rows(B)}
B

Rendered Output:

Direct:

A = hp ( [1 | 2; 3 | 4; 5; 6 | 7; 8] ) = 100 230 456 780 , isHp ( A ) = 1

Direct with vectors:

⃗a = [1; 2; 4] = [1 2 4]

A = hp ( [⃗a | 2 · ⃗a + 1 | 3 · ⃗a + 2] ) = 124 359 5814 , isHp ( A ) = 1

From a custom function (with vector arguments):

A ( x ) = transp ( hp ( [x⁰ | x | x² | x³ | x⁴] ) )

⃗x = hp ( [1; 2; 3; 4] ) = [1 2 3 4]

A = A ( ⃗x ) = 11111 124816 1392781 141664256 , isHp ( A ) = 1

Indexing

A_3,2 = 3 - simple dot operator (constant index)

i = 2 , j = 3

A_3,4 = 27 - expressions with variables

A_2,3 = 0 - assign a new value to an element: A = 11111 120816 1392781 141664256

Initalize values in a block loop

⃗x = hp ( [1; 2; 3; 4] ) = [1 2 3 4]

A = matrix_hp ( len ( ⃗x ) ; 7 ) = 0000000 0000000 0000000 0000000

A = 1111111 1248163264 1392781243729 14166425610244096

Inline loop

B = matrix_hp ( len ( ⃗x ) ; 7 ) = 0000000 0000000 0000000 0000000

$Repeat{$Repeat{B_i,j = ⃗x_i^{j − 1} for j = 1...n_cols ( B ) } for i = 1...n_rows ( B ) } = 4096

B = 1111111 1248163264 1392781243729 14166425610244096

Matrix Creational Functions¶

HP variant of the creational helpers: $matrix$, $identity$, $diagonal$, banded, random and Vandermonde constructors emitting packed storage.

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Matrix</b>(<em>m</em>; <em>n</em>)</p><hr/>
matrix_hp(3; 4)'- creates a 3x4 full (rectangular) matrix

'<p><b>Identity</b>(<em>n</em>)</p><hr/>
identity_hp(3)'- creates  a 3x3 identity matrix (diagonal filled with ones)

'<p><b>Diagonal</b>(<em>n</em>, <em>value</em>)</p><hr/>
diagonal_hp(3; 2)'- creates a 3x3 diagonal matrix filled with 2

'<p><b>Column</b>(<em>m</em>, <em>value</em>)</p><hr/>
column_hp(3; 2)'- creates a 3x1 column matrix filled with 2

'<p><b>Utriang</b>(<em>n</em>, <em>value</em>)</p><hr/>
U = utriang_hp(3)'- creates a 3x3 upper triangular matrix
mfill(U; 1)'- fills the matrix with value of 1

'<p><b>Ltriang</b>(<em>n</em>, <em>value</em>)</p><hr/>
L = ltriang_hp(3)'- creates a 3x3 lower triangular matrix
mfill(L; 1)'- fills the matrix with value of 1

'<p><b>Symmetric</b>(<em>n</em>)</p><hr/>
A = symmetric_hp(4)'- creates a 4x4 empty symmetrix matrix
A.(1; 1) = 5', 'A.(1; 2) = 4'- assign some values
A.(2; 2) = 3', 'A.(2; 3) = 2
A.(4; 2) = 1', 'A.(4; 4) = 1
A'- symmetric values has also changed to preserve symmetry

'<p><b>Vec2diag</b>(<em>vector</em>)</p><hr/>
a = vec2diag(hp([1; 2; 3]))'- creates a 3x3 diagonal matrix from a vector
isHp(a)

'<p><b>Vec2col</b>(<em>vector</em>)</p><hr/>
a = vec2col(hp([1; 2; 3]))'- creates a 3x3 column matrix from a vector
isHp(a)

'<p><b>Join_Cols</b>(<em>list of vectors</em>)</p><hr/>
a = join_cols(hp([1]); hp([4; 5; 6]); hp([7; 8]); hp([10; 11; 12]))'- creates a general rectangular matrix by joining the input vectors into columns
isHp(a)
a = hp(join_cols([1]; [4; 5; 6]; [7; 8]; [10; 11; 12]))'- creates a general rectangular matrix by joining the input vectors into columns
isHp(a)

'<p><b>Join_Rows</b>(<em>list of vectors</em>)</p><hr/>
a = join_rows(hp([1; 2; 3; 4]); hp([6; 7; 8; 9; 10]))'- creates a general rectangular matrix by joining the input vectors into rows
isHp(a)

'<p><b>Join_Rows</b>(<em>list of vectors</em>)</p><hr/>
a = join_rows(hp([1; 2; 3; 4]); hp([6; 7; 8; 9; 10]))'- creates a general rectangular matrix by joining the input vectors into rows
isHp(a)

'<p><b>Augment</b>(<em>list of matrices</em>)</p><hr/>
A = hp([1; 2|3; 4])
B = hp([1; 2; 3|4; 5; 6|7; 8; 9])
c = hp([10; 11; 12; 13])
D = augment(A; B; c)'- creates a general rectangular matrix by joining the input matrices side to side
isHp(D)

'<p><b>Stack</b>(<em>list of matrices</em>)</p><hr/>
A = hp([1; 2|3; 4])
B = hp([1; 2; 3|4; 5; 6|7; 8; 9])
c = hp([10; 11])
D = stack(A; B; c)'- creates a general rectangular matrix by joining the input matrices one below the other
isHp(D)

Rendered Output:

Matrix(m; n)

matrix_hp ( 3; 4 ) = 0000 0000 0000 - creates a 3x4 full (rectangular) matrix

Identity(n)

identity_hp ( 3 ) = 100 010 001 - creates a 3x3 identity matrix (diagonal filled with ones)

Diagonal(n, value)

diagonal_hp ( 3; 2 ) = 200 020 002 - creates a 3x3 diagonal matrix filled with 2

Column(m, value)

column_hp ( 3; 2 ) = 2 2 2 - creates a 3x1 column matrix filled with 2

Utriang(n, value)

U = utriang_hp ( 3 ) = 000 000 000 - creates a 3x3 upper triangular matrix

mfill ( U; 1 ) = 111 011 001 - fills the matrix with value of 1

Ltriang(n, value)

L = ltriang_hp ( 3 ) = 000 000 000 - creates a 3x3 lower triangular matrix

mfill ( L; 1 ) = 100 110 111 - fills the matrix with value of 1

Symmetric(n)

A = symmetric_hp ( 4 ) = 0000 0000 0000 0000 - creates a 4x4 empty symmetrix matrix

A_1,1 = 5 , A_1,2 = 4 - assign some values

A_2,2 = 3 , A_2,3 = 2

A_4,2 = 1 , A_4,4 = 1

A = 5400 4321 0200 0101 - symmetric values has also changed to preserve symmetry

Vec2diag(vector)

a = vec2diag ( hp ( [1; 2; 3] ) ) = 100 020 003 - creates a 3x3 diagonal matrix from a vector

isHp ( a ) = 1

Vec2col(vector)

a = vec2col ( hp ( [1; 2; 3] ) ) = 1 2 3 - creates a 3x3 column matrix from a vector

isHp ( a ) = 1

Join_Cols(list of vectors)

a = join_cols ( hp ( [1] ) ; hp ( [4; 5; 6] ) ; hp ( [7; 8] ) ; hp ( [10; 11; 12] ) ) = 14710 05811 06012 - creates a general rectangular matrix by joining the input vectors into columns

isHp ( a ) = 1

a = hp ( join_cols ( [1]; [4; 5; 6]; [7; 8]; [10; 11; 12] ) ) = 14710 05811 06012 - creates a general rectangular matrix by joining the input vectors into columns

isHp ( a ) = 1

Join_Rows(list of vectors)

a = join_rows ( hp ( [1; 2; 3; 4] ) ; hp ( [6; 7; 8; 9; 10] ) ) = 12340 678910 - creates a general rectangular matrix by joining the input vectors into rows

isHp ( a ) = 1

Join_Rows(list of vectors)

a = join_rows ( hp ( [1; 2; 3; 4] ) ; hp ( [6; 7; 8; 9; 10] ) ) = 12340 678910 - creates a general rectangular matrix by joining the input vectors into rows

isHp ( a ) = 1

Augment(list of matrices)

A = hp ( [1; 2 | 3; 4] ) = 12 34

B = hp ( [1; 2; 3 | 4; 5; 6 | 7; 8; 9] ) = 123 456 789

⃗c = hp ( [10; 11; 12; 13] ) = [10 11 12 13]

D = augment ( A; B; ⃗c ) = 1212310 3445611 0078912 0000013 - creates a general rectangular matrix by joining the input matrices side to side

isHp ( D ) = 1

Stack(list of matrices)

A = hp ( [1; 2 | 3; 4] ) = 12 34

B = hp ( [1; 2; 3 | 4; 5; 6 | 7; 8; 9] ) = 123 456 789

⃗c = hp ( [10; 11] ) = [10 11]

D = stack ( A; B; ⃗c ) = 120 340 123 456 789 1000 1100 - creates a general rectangular matrix by joining the input matrices one below the other

isHp ( D ) = 1

Matrix Data Functions¶

HP variant of the data toolkit: row- and column-wise sorting, ordering, search and reductions, all on packed numerical storage.

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Sort_Cols</b>(<em>matrix</em>; <em>row index</em>) and <b>Rsort_Cols</b>(<em>matrix</em>; <em>row index</em>)</p><hr/>
A = hp([5; 2; 3|4; 9; 1|6; 8; 7])
B = sort_cols(A; 2)'- the matrix sorted by row 2 in ascending order
isHp(B)
C = rsort_cols(A; 2)'- the matrix sorted by row 2 in descending order
isHp(C)
A' - the original matrix remains unchanged

'<p><b>Sort_Rows</b>(<em>matrix</em>; <em>column index</em>) and <b>Rsort_Rows</b>(<em>matrix</em>; <em>column index</em>)</p><hr/>
A = hp([5; 2; 3|4; 9; 1|6; 8; 7])
B = sort_rows(A; 2)'- the matrix sorted by column 2 in ascending order
isHp(B)
C = rsort_rows(A; 2)'- the matrix sorted by column 2 in descending order
isHp(C)
A' - the original matrix remains unchanged

'<p><b>Order_Cols</b>(<em>matrix</em>; <em>row index</em>) and <b>Revorder_Cols</b>(<em>matrix</em>; <em>row index</em>)</p><hr/>
A = hp([5; 2; 3|4; 9; 1|6; 8; 7])
b = order_cols(A; 2)'- the indexes of matrix columns ordered by the values in row 2 in ascending order
isHp(b)
B = extract_cols(A; b)'- the matrix sorted by row 2 in ascending order
isHp(B)
c = revorder_cols(A; 2)'- the indexes of matrix columns ordered by the values in row 2 in descending order
isHp(c)
C = extract_cols(A; c)'- the matrix sorted by row 2 in descending order
isHp(C)

'<p><b>Order_Rows</b>(<em>matrix</em>; <em>column index</em>) and <b>Revorder_Rows</b>(<em>matrix</em>; <em>column index</em>)</p><hr/>
A = hp([5; 2; 3|4; 9; 1|6; 8; 7])
b = order_rows(A; 2)'- the indexes of matrix rows ordered by the values in column 2 in ascending order
isHp(b)
B = extract_rows(A; b)
isHp(B)
c = revorder_rows(A; 2)'- the matrix sorted by row 2 in descending order
isHp(c)
C = extract_rows(A; c)'- the indexes of matrix rows ordered by the values in column 2 in descending order
isHp(C)

'<p><b>Mcount</b>(<em>matrix</em>; <em>value</em>)</p><hr/>
A = hp([1; 0; 1|2; 1; 2|1; 3; 1])
n = mcount(A; 1)'- the number of occurances of value 1 in matrix A

'<p><b>Msearch</b>(<em>matrix</em>; <em>value</em>; <em>starting row index</em>; <em>starting column index</em>)</p><hr/>
A = hp([1; 2; 3|1; 5; 6|1; 8; 9])
b = msearch(A; 1; 1; 1)' - the row and column indexes of the first occurance of value 1 in A, starting from (1,1)
isHp(b)
c = msearch(A; 1; 2; 2)' - the same are previous, but starting  from (2,2)
isHp(c)
d = msearch(A; 4; 1; 1)' - if the value is not found [0 0] is returned
isHp(d)

'<p><b>Mfind</b>(<em>matrix</em>; <em>value</em>)</p><hr/>
A = hp([1; 2; 3|4; 1; 6|1; 8; 9])
B = mfind(A; 1)' - the row and column indexes of all elements in A, that are equal to 1
isHp(B)
C = mfind_ne(A; 1)' - the row and column indexes of all elements in A, that are not equal to 1
isHp(C)
D = mfind_lt(A; 5)' - the row and column indexes of all elements in A, that are less than 5
isHp(D)
E = mfind(A; 5)'- if the value is not found
isHp(E)

'<p><b>Hlookup</b>(<em>matrix</em>; <em>value</em>; <em>reference row</em>; <em>return row</em>)</p><hr/>
A = hp([0; 1; 0; 1|1; 2; 3; 4; 5|6; 7; 8; 9; 10])
b = hlookup(A; 0; 1; 3)'- the values in row 3, where row 1 contains zeros
isHp(b)
c = hlookup_ne(A; 0; 1; 3)'- the values in row 3, where row 1  does not contain zeros
isHp(c)
d = hlookup_lt(A; 9; 3; 2)'- the values in row 2, where the respective values in row 3 are &lt; 9
isHp(d)
e = hlookup_ge(A; 8; 3; 2)'- the values in row 2, where the respective values in row 3 are &ge; 8
isHp(e)
f = hlookup(A; 0; 1; 3)'- the values in row 3, where row 1 contains zeros
isHp(f)
g = hlookup(A; 2; 1; 3)'- since 2 is not found in row 1, an empty vector is returned
isHp(g)

'<p><b>Vlookup</b>(<em>matrix</em>; <em>value</em>; <em>reference column</em>; <em>return column</em>)</p><hr/>
A = hp([1; 2|3; 4; 1|5; 6|7; 8; 1|9; 10])
b = vlookup(A; 0; 3; 1)'- the values in column 1, where column 3 contains zeros
isHp(b)
c = vlookup_ne(A; 0; 3; 2)'- the values in column 2, where column 3 does not contain zeros
isHp(c)
d = vlookup_le(A; 5; 1; 2)'- the values in column 2, where the respective values in column 1 are &le; 5
isHp(d)
e = vlookup_gt(A; 3; 1; 2)'- the values in column 2, where the respective values in column 1 are &gt; 3
isHp(e)
f = vlookup(A; 2; 3; 1)'- since 2 is not found in column 3, an empty vector is returned
isHp(f)

Rendered Output:

Sort_Cols(matrix; row index) and Rsort_Cols(matrix; row index)

A = hp ( [5; 2; 3 | 4; 9; 1 | 6; 8; 7] ) = 523 491 687

B = sort_cols ( A; 2 ) = 352 149 768 - the matrix sorted by row 2 in ascending order

isHp ( B ) = 1

C = rsort_cols ( A; 2 ) = 253 941 867 - the matrix sorted by row 2 in descending order

isHp ( C ) = 1

A = 523 491 687 - the original matrix remains unchanged

Sort_Rows(matrix; column index) and Rsort_Rows(matrix; column index)

A = hp ( [5; 2; 3 | 4; 9; 1 | 6; 8; 7] ) = 523 491 687

B = sort_rows ( A; 2 ) = 523 687 491 - the matrix sorted by column 2 in ascending order

isHp ( B ) = 1

C = rsort_rows ( A; 2 ) = 491 687 523 - the matrix sorted by column 2 in descending order

isHp ( C ) = 1

A = 523 491 687 - the original matrix remains unchanged

Order_Cols(matrix; row index) and Revorder_Cols(matrix; row index)

A = hp ( [5; 2; 3 | 4; 9; 1 | 6; 8; 7] ) = 523 491 687

⃗b = order_cols ( A; 2 ) = [3 1 2] - the indexes of matrix columns ordered by the values in row 2 in ascending order

isHp ( ⃗b ) = 1

B = extract_cols ( A; ⃗b ) = 352 149 768 - the matrix sorted by row 2 in ascending order

isHp ( B ) = 1

⃗c = revorder_cols ( A; 2 ) = [2 1 3] - the indexes of matrix columns ordered by the values in row 2 in descending order

isHp ( ⃗c ) = 1

C = extract_cols ( A; ⃗c ) = 253 941 867 - the matrix sorted by row 2 in descending order

isHp ( C ) = 1

Order_Rows(matrix; column index) and Revorder_Rows(matrix; column index)

A = hp ( [5; 2; 3 | 4; 9; 1 | 6; 8; 7] ) = 523 491 687

⃗b = order_rows ( A; 2 ) = [1 3 2] - the indexes of matrix rows ordered by the values in column 2 in ascending order

isHp ( ⃗b ) = 1

B = extract_rows ( A; ⃗b ) = 523 687 491

isHp ( B ) = 1

⃗c = revorder_rows ( A; 2 ) = [2 3 1] - the matrix sorted by row 2 in descending order

isHp ( ⃗c ) = 1

C = extract_rows ( A; ⃗c ) = 491 687 523 - the indexes of matrix rows ordered by the values in column 2 in descending order

isHp ( C ) = 1

Mcount(matrix; value)

A = hp ( [1; 0; 1 | 2; 1; 2 | 1; 3; 1] ) = 101 212 131

n = mcount ( A; 1 ) = 5 - the number of occurances of value 1 in matrix A

Msearch(matrix; value; starting row index; starting column index)

A = hp ( [1; 2; 3 | 1; 5; 6 | 1; 8; 9] ) = 123 156 189

⃗b = msearch ( A; 1; 1; 1 ) = [1 1] - the row and column indexes of the first occurance of value 1 in A, starting from (1,1)

isHp ( ⃗b ) = 1

⃗c = msearch ( A; 1; 2; 2 ) = [3 1] - the same are previous, but starting from (2,2)

isHp ( ⃗c ) = 1

⃗d = msearch ( A; 4; 1; 1 ) = [0 0] - if the value is not found [0 0] is returned

isHp ( ⃗d ) = 1

Mfind(matrix; value)

A = hp ( [1; 2; 3 | 4; 1; 6 | 1; 8; 9] ) = 123 416 189

B = mfind ( A; 1 ) = 123 121 - the row and column indexes of all elements in A, that are equal to 1

isHp ( B ) = 1

C = mfind_ne ( A; 1 ) = 112233 231323 - the row and column indexes of all elements in A, that are not equal to 1

isHp ( C ) = 1

D = mfind_lt ( A; 5 ) = 111223 123121 - the row and column indexes of all elements in A, that are less than 5

isHp ( D ) = 1

E = mfind ( A; 5 ) = 0 0 - if the value is not found

isHp ( E ) = 1

Hlookup(matrix; value; reference row; return row)

A = hp ( [0; 1; 0; 1 | 1; 2; 3; 4; 5 | 6; 7; 8; 9; 10] ) = 01010 12345 678910

⃗b = hlookup ( A; 0; 1; 3 ) = [6 8 10] - the values in row 3, where row 1 contains zeros

isHp ( ⃗b ) = 1

⃗c = hlookup_ne ( A; 0; 1; 3 ) = [7 9] - the values in row 3, where row 1 does not contain zeros

isHp ( ⃗c ) = 1

⃗d = hlookup_lt ( A; 9; 3; 2 ) = [1 2 3] - the values in row 2, where the respective values in row 3 are < 9

isHp ( ⃗d ) = 1

⃗e = hlookup_ge ( A; 8; 3; 2 ) = [3 4 5] - the values in row 2, where the respective values in row 3 are ≥ 8

isHp ( ⃗e ) = 1

⃗f = hlookup ( A; 0; 1; 3 ) = [6 8 10] - the values in row 3, where row 1 contains zeros

isHp ( ⃗f ) = 1

⃗g = hlookup ( A; 2; 1; 3 ) = [] - since 2 is not found in row 1, an empty vector is returned

isHp ( ⃗g ) = 1

Vlookup(matrix; value; reference column; return column)

A = hp ( [1; 2 | 3; 4; 1 | 5; 6 | 7; 8; 1 | 9; 10] ) = 120 341 560 781 9100

⃗b = vlookup ( A; 0; 3; 1 ) = [1 5 9] - the values in column 1, where column 3 contains zeros

isHp ( ⃗b ) = 1

⃗c = vlookup_ne ( A; 0; 3; 2 ) = [4 8] - the values in column 2, where column 3 does not contain zeros

isHp ( ⃗c ) = 1

⃗d = vlookup_le ( A; 5; 1; 2 ) = [2 4 6] - the values in column 2, where the respective values in column 1 are ≤ 5

isHp ( ⃗d ) = 1

⃗e = vlookup_gt ( A; 3; 1; 2 ) = [6 8 10] - the values in column 2, where the respective values in column 1 are > 3

isHp ( ⃗e ) = 1

⃗f = vlookup ( A; 2; 3; 1 ) = [] - since 2 is not found in column 3, an empty vector is returned

isHp ( ⃗f ) = 1

Matrix Math Functions¶

HP variant of the linear-algebra toolkit: Hadamard product, determinant, inverse, transpose, norms and decomposition helpers on packed numerical storage.

Code:

'<style>em {font-family:"Times New Roman";}</style>
'<p><b>Hprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = hp([1; 2|3; 4|5; 6])'- first matrix
B = hp([9; 8|7; 6|5; 4])'- second matrix
C = hprod(A; B)'- The Hadamard product of matrices A and B

'<p><b>Fprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = hp([1; 2|3; 4|5; 6])'- first matrix
B = hp([9; 8|7; 6|5; 4])'- second matrix
C = fprod(A; B)'- the Frobenius product of matrices A and B

'<p><b>Kprod</b>(<em>first matrix</em>; <em>second matrix</em>)</p><hr/>
A = hp([1; 2|3; 4])'- first matrix
B = hp([5; 6; 7|8; 9; 10])'- second matrix
C = kprod(A; B)'- the Kronecker product of matrices A and B

'<p><b>Mnorm_1</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
mnorm_1(A)'- the L1 (Manhattan or taxicab) norm of matrix A

'<p><b>Mnorm</b>(<em>matrix</em>) or <b>Mnorm_2</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
mnorm(A)' or
mnorm_2(A)'- the L2 (spectral) norm of matrix A

'<p><b>Mnorm_е</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
mnorm_e(A)'- the Frobenius (a.k.a. Euclidean) norm of matrix A

'<p><b>Mnorm_i</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
mnorm_i(A)'- the L∞ (infinity) norm of matrix A

'<p><b>Cond_1</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
cond_1(A)'- condition number based on the L1 norm of matrix A

'<p><b>Cond</b>(<em>matrix</em>) or <b>Cond_2</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
cond(A)' or
cond_2(A)'- condition number based on the L2 norm of matrix A

'<p><b>Cond_е</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
cond_e(A)'- condition number based on  the Frobenius norm of matrix A

'<p><b>Cond_i</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
cond_i(A)'- condition number based on the L∞ norm of matrix A

'<p><b>Det</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
det(A)'- the determinant of matrix A

'<p><b>Rank</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
rank(A)'- the rank of matrix A

'<p><b>Trace</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
trace(A)'- the trace of matrix A

'<p><b>Transp</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2; 3|4; 5; 6|7; 8; 9])
transp(A)'- the transpose of matrix A

'<p><b>Adj</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2|3; 4])
adj(A)'- the adjugate of matrix A
D = det(A)'- the determinant of matrix A
A_I = inverse(A)'- the inverse of matrix A
A_I*D'- the inverse multiplied by the determinant shoud give the adjugate

'<p><b>Cofactor</b>(<em>matrix</em>)</p><hr/>
A = hp([1; 2|3; 4])
C = cofactor(A)'- the cofactor of matrix A
transp(C)'- the transpose of the cofactor shoud give the adjugate'
adj(A)

'<p><b>Eigenvals</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric matrix by copying a general initializer
eigenvals(A; 0)' - the eigenvalues of matrix A

'<p><b>Eigenvecs</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric matrix by copying a general initializer
eigenvecs(A; 2)' - the eigenvectors of matrix A

'<p><b>Eigen</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric_hp(3); 1; 1)' creates symmetric matrix by copying a general initializer
eigen(A; 2)' - the eigenvalues (first column) and eigenvectors (other columns) of matrix A

'<p><b>Cholesky</b>(<em>matrix</em>)</p><hr/>
 A = copy( _
    [4; 12; -16| _
    12; 37; -43| _
    - 16; -43; 98]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric matrix by copying a general initi_hpalizer
LT = cholesky(A)'- the upper triangular matrix LT by Cholesky decomposition
L = transp(LT)'- the lower triangular matrix L
L*LT'- check: multiplying both matrices should give the original matrix

'<p><b>LU</b>(<em>matrix</em>)</p><hr/>
 A = hp([4; 12; -16|12; 37; -43| - 16; -43; 98])'- creates general matrix
LU = lu(A)'- the combined matrix by LU decomposition
ind'- the index permutation vector
D = not(identity_hp(3))'- a helper matrix to remove the diagonal elements from L

L = hprod(mfill(ltriang_hp(3); 1); LU)^D'- extracts the lower triangular matrix
U = hprod(mfill(utriang_hp(3); 1); LU)'- extracts the upper triangular matrix
extract_rows(L*U; order(ind))'- check: multiplying both matrices and reordering by the permutation vector should give the original matrix

'<p><b>QR</b>(<em>matrix</em>)</p><hr/>
A = hp([4; 12; -16|12; 37; -43| - 16; -43; 98])'- creates a general matrix

QR = qr(A)'- the combined QR matrix
Q = submatrix(QR; 1; 3; 1; 3)'- extracts the Q matrix
R = submatrix(QR; 1; 3; 4; 6)'- extracts the R matrix
Q*R'- check: multiplying both matrices should give the original matrix

'<p><b>SVD</b>(<em>matrix</em>)</p><hr/>
A = hp([4; 12; -16|12; 37; -43| - 16; -43; 98])'- creates a general matrix
SVD = svd(A)'- the combined UΣV matrix, obtained by singular value decomposition
U = submatrix(SVD; 1; 3; 1; 3)'- extracts the U matrix
V = submatrix(SVD; 1; 3; 5; 7)'- extracts the V matrix
σ = col(SVD; 4)'- extracts the singular values
Σ = hp(vec2diag(σ))'- composes the singular value matrix
'Multiplying U,Σ and V may not give the original matrix A, due to the sign ambiguity problem

'<p><b>Inverse</b>(<em>matrix</em>)</p><hr/>
A = hp([4; 12; -16|12; 37; -43| - 16; -43; 98])'- creates a general matrix
B = inverse(A)'- the inverse of A
A*B'- check: multiplying both matrices should give the identity matrix
'Symmetric matrix inverse
A = copy([4; 12; -16|12; 37; -43| - 16; -43; 98]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric matrix
B = inverse(A)'- the inverse of A
A*B'- check: multiplying both matrices should give the identity matrix

'<p><b>Lsolve</b>(<em>matrix</em>, <em>vector</em>)</p><hr/>
A = hp([8; 6; -4|6; 12; -3| - 4; -3; 9])'- creates a general matrix
b = hp([10; 20; 30])'- the right hand side vector
x = lsolve(A; b)'- the solution vector obtained by LU (LDLT) decomposition
A*x'- check: multiplying the matrix by the solution vector should give the right hand side vector

'<p><b>Clsolve</b>(<em>matrix</em>, <em>vector</em>)</p><hr/>
A = copy([8; 6; -4|6; 12; -3| - 4; -3; 9]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric matrix
b = hp([10; 20; 30])'- the right hand side vector
x = clsolve(A; b)'- the solution vector obtained by Cholesky decomposition
x = slsolve(A; b)'- the solution vector obtained by PCG method
A*x'- check: multiplying the matrix by the solution vector should give the right hand side vector

'<p><b>Msolve</b>(<em>matrix</em>, <em>matrix</em>)</p><hr/>
A = hp([8; 6; -4|6; 12; -3| - 4; -3; 9])'- creates a general coefficients matrix
B = join_cols(hp([10; 20; 30]); hp([40; 50; 60]))'- the right hand side matrix
X = msolve(A; B)'- the solution matrix obtained by LU (LDLT) decomposition
A*X'- check: multiplying the coefficients matrix by the solution matrix should give the right hand side matrix

'<p><b>Cmsolve</b>(<em>matrix</em>, <em>matrix</em>)</p><hr/>
A = copy([8; 6; -4|6; 12; -3| - 4; -3; 9]; _
    symmetric_hp(3); 1; 1)'- creates a symmetric coefficients  matrix
B = join_cols(hp([10; 20; 30]); hp([40; 50; 60]))'- the right hand side matrix
X = cmsolve(A; B)'- the solution matrix obtained by Cholesky decomposition
X = smsolve(A; B)'- the solution matrix obtained by PCG method
A*X'- check: multiplying the coefficients matrix by the solution matrix should give the right hand side matrix

#rad
'Trigonometric:<hr/>
x = hp([0; 30; 45|60; 90; 180])*(π/180)
isHp(x)
sin(x)'- sine
cos(x)'- cosine
tan(x)'- tangent
csc(x)'- cosecant
sec(x)'- secant
cot(x)'- cotangent

'Hyperbolic:<hr/>
x = hp([-0.5; 0|0.5; 1])
sinh(x)'- hyperbolic sine
cosh(x)'- hyperbolic cosine
tanh(x)'- hyperbolic tangent
csch(x)'- hyperbolic cosecant
sech(x)'- hyperbolic secant
coth(x)'- hyperbolic cotangent

'Inverse trigonometric:<hr/>
x = hp([-0.5; 0; 0.5|sqrt(2)/2; sqrt(3)/2; 1])
y = hp([1; sqrt(3)/2; sqrt(2)/2|0.5; 0; -0.5])
asin(x)'- inverse sine
acos(x)'- inverse cosine
atan(x)'- inverse tangent
atan2(x; y)'- the angle whose tangent is the quotient of y and x
acsc(1/x)'- inverse cosecant
asec(1/x)'- inverse secant
acot(x)'- inverse cotangent

'Inverse hyperbolic:<hr/>
x = hp([1; 2|4; 8])
asinh(x)'- inverse hyperbolic sine
acosh(x)'- inverse hyperbolic cosine
atanh(x/5)'- inverse hyperbolic tangent
acsch(1/x)'- inverse hyperbolic cosecant
asech(1/x)'- inverse hyperbolic secant
acoth(5/x)'- inverse hyperbolic cotangent

'Logarithmic, exponential and roots:<hr/>
log(x)'- decimal logarithm
ln(x)'- natural logarithm
log_2(x)'- binary logarithm
exp(x)'- natural exponent = eˣ
sqr(x)'- square root
cbrt(x)'- cubic root
root(x; 3)'- n-th root

'Rounding:<hr/>
round(x)'- round to the nearest integer
floor(x)'- round to the smaller integer(towards - ∞)
ceiling(x)'- round to the greater integer(towards + ∞)
trunc(x)'- round to the smaller integer(towards zero)

'Integer:<hr/>
x
y = x + 2
mod(x; y)'- the remainder of an integer division
gcd(y)'- the greatest common divisor of several integers
lcm(y)'- the least common multiple of several integers

'Aggregate and interpolation:<hr />
A = hp([0; 2; 4|6; 8; 12])
b = hp([5; 3; 1])
min(A)'- minimum of multiple values
max(A)'- maximum of multiple values
sum(A)'- sum of multiple values
sumsq(A)'- sum of squares
srss(A)'- square root of sum of squares
average(A)'- average of multiple values
product(A)'- product of multiple values
mean(A)'- geometric mean
take(5; A)'- the 5-th element from the matrix, linearized to list by rows
line(5.5; A)'- linear interpolation along the linearized matrix
spline(5.5; A)'- Hermite spline interpolation along the linearized matrix
take(2; 2; A)'- the element on row 2 and column 2
line(1.5; 2.5; A)'- double linear interpolation
spline(1.5; 2.5; A)'- double Hermite spline interpolation

Rendered Output:

Hprod(first matrix; second matrix)

A = hp ( [1; 2 | 3; 4 | 5; 6] ) = 12 34 56 - first matrix

B = hp ( [9; 8 | 7; 6 | 5; 4] ) = 98 76 54 - second matrix

C = hprod ( A; B ) = 916 2124 2524 - The Hadamard product of matrices A and B

Fprod(first matrix; second matrix)

A = hp ( [1; 2 | 3; 4 | 5; 6] ) = 12 34 56 - first matrix

B = hp ( [9; 8 | 7; 6 | 5; 4] ) = 98 76 54 - second matrix

C = fprod ( A; B ) = 119 - the Frobenius product of matrices A and B

Kprod(first matrix; second matrix)

A = hp ( [1; 2 | 3; 4] ) = 12 34 - first matrix

B = hp ( [5; 6; 7 | 8; 9; 10] ) = 567 8910 - second matrix

C = kprod ( A; B ) = 567101214 8910161820 151821202428 242730323640 - the Kronecker product of matrices A and B

Mnorm_1(matrix)