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Math & linear algebra

Linear algebra

a tour of Qu's M2 matrix surface.

catalog/qu_linear_algebra.qu

Runs end-to-end on the reference interpreter (`qu run`). Exercises: §13 matrix `*` (matmul) vs elementwise `.*` §14 NumPy-style broadcasting and orientation contracts (`as vector(r,c)`) §15 2-D indexing: M[i,j], M[i,:], M[:,j], submatrices §24 reductions with an `axis=` keyword built-ins: eye, zeros/ones, reshape, transpose (`'`), matmul, dot, norm

How it works

A = [1, 2; 3, 4]
B = [5, 6; 7, 8]

print("A       = {A}")
print("B       = {B}")
print("A * B   = {A * B}")          # matrix multiplication -> [19,22;43,50]
print("A .* B  = {A .* B}")         # elementwise (Hadamard) -> [5,12;21,32]
print("A'      = {A'}")             # transpose -> [1,3;2,4]
print("I3      = {eye(3)}")

print("A[0, 1] = {A[0, 1]}")        # 2   (zero-based)
print("A[1, :] = {A[1, :]}")        # row 1 -> [3, 4]
print("A[:, 0] = {A[:, 0]}")        # col 0 -> [1, 3]

M = zeros(3, 3)
M[0, :] = [1, 2, 3]                 # write a whole row
M[1, 1] = 9                         # write a single cell
M[2, :] = ones(1, 3)               # broadcast a row of ones
print("M       = {M}")

u = [1, 2, 3]
v = [10, 20]
u as vector(3, 1)                  # column
v as vector(1, 2)                  # row
outer = u * v                      # (3,1) * (1,2) -> (3,2) outer product
print("outer   = {outer}")
print("shape   = {shape(outer)}")

G = [1, 2, 3; 4, 5, 6]
print("colsum  = {sum(G, axis=0)}")   # down columns -> [5, 7, 9]
print("rowmean = {mean(G, axis=1)}")  # across rows  -> [2, 5]

a = [1, 2, 2]
print("dot     = {dot(a, a)}")        # 9
print("norm    = {norm(a):.4f}")      # 3.0000

r = reshape(0 to 5, 2, 3)
print("reshape = {r}")

show plot

Results

A       = [1, 2; 3, 4]
B       = [5, 6; 7, 8]
A * B   = [19, 22; 43, 50]
A .* B  = [5, 12; 21, 32]
A'      = [1, 3; 2, 4]
I3      = [1, 0, 0; 0, 1, 0; 0, 0, 1]
A[0, 1] = 2
A[1, :] = [3, 4]
A[:, 0] = [1, 3]
M       = [1, 2, 3; 0, 9, 0; 1, 1, 1]
outer   = [10, 20; 20, 40; 30, 60]
shape   = [3, 2]
colsum  = [5, 7, 9]
rowmean = [2, 5]
dot     = 9
norm    = 3.0000
reshape = [0, 2, 4; 1, 3, 5]

The whole program

Runnable as it stands: qu run catalog/qu_linear_algebra.qu

# qu_linear_algebra.qu — a tour of Qu's M2 matrix surface.

#

# Runs end-to-end on the reference interpreter (`qu run`). Exercises:

#   §13  matrix `*` (matmul) vs elementwise `.*`

#   §14  NumPy-style broadcasting and orientation contracts (`as vector(r,c)`)

#   §15  2-D indexing: M[i,j], M[i,:], M[:,j], submatrices

#   §24  reductions with an `axis=` keyword

#   built-ins: eye, zeros/ones, reshape, transpose (`'`), matmul, dot, norm


backend auto

#%% Matrix basics

# --- Matrix literals are row-major to read; stored column-major ------------

A = [1, 2; 3, 4]
B = [5, 6; 7, 8]

print("A       = {A}")
print("B       = {B}")
print("A * B   = {A * B}")          # matrix multiplication -> [19,22;43,50]

print("A .* B  = {A .* B}")         # elementwise (Hadamard) -> [5,12;21,32]

print("A'      = {A'}")             # transpose -> [1,3;2,4]

print("I3      = {eye(3)}")

#%% Indexing

# --- Indexing --------------------------------------------------------------

print("A[0, 1] = {A[0, 1]}")        # 2   (zero-based)

print("A[1, :] = {A[1, :]}")        # row 1 -> [3, 4]

print("A[:, 0] = {A[:, 0]}")        # col 0 -> [1, 3]


#%% Assignment

# --- Assignment into a matrix ---------------------------------------------

M = zeros(3, 3)
M[0, :] = [1, 2, 3]                 # write a whole row

M[1, 1] = 9                         # write a single cell

M[2, :] = ones(1, 3)               # broadcast a row of ones

print("M       = {M}")

#%% Broadcasting

# --- Broadcasting: outer product via orientation contracts (§14) ----------

u = [1, 2, 3]
v = [10, 20]
u as vector(3, 1)                  # column

v as vector(1, 2)                  # row

outer = u * v                      # (3,1) * (1,2) -> (3,2) outer product

print("outer   = {outer}")
print("shape   = {shape(outer)}")

#%% Reductions

# --- Reductions along an axis ---------------------------------------------

G = [1, 2, 3; 4, 5, 6]
print("colsum  = {sum(G, axis=0)}")   # down columns -> [5, 7, 9]

print("rowmean = {mean(G, axis=1)}")  # across rows  -> [2, 5]


#%% Vector algebra

# --- Vector algebra --------------------------------------------------------

a = [1, 2, 2]
print("dot     = {dot(a, a)}")        # 9

print("norm    = {norm(a):.4f}")      # 3.0000


#%% Reshape

# --- Reshape preserves column-major order ---------------------------------

r = reshape(0 to 5, 2, 3)
print("reshape = {r}")

show plot